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Math Mama Writes...
What I've stumbled upon in the past two days:
And some older ones (as I begin to slowly clear out my backlog...):
I used to share links more often. I used to write substantive blog posts more often, too. Since I've been writing less, I haven't been comfortable sharing lots of links. Didn't want this blog to
descend into just a link-share. But it would be helpful to me to have them here. So maybe I'll start sharing my almost daily finds, even if it's not exciting for y'all.
These were the tabs I've kept open - some for weeks - hoping to figure out how to use some, how to find time to really process others.
I am enjoying two online conversations right now.
Michael Pershan asked:
Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their
reasoning. Conflict! Drama!
Why do kids hate writing about math?
I am currently trying to grade my students arguments (as prosecutors) for the murder mystery. Some of them really got into it. Most still didn't explain the math well. My take on this is that
students will write (maybe even well) if we give them a good enough context.
In the other conversation, Mr. Honner blogged about what happens when you zoom in super far on Desmos, looking for the hole in a rational function. It gets a bit crazy. The conversation got more
interesting for me when Alan Eliasen started explaining "interval arithmetic", which I had never heard of.
I'm teaching four classes this semester, which is a lot for me. That's embarrassing to admit - I know most math bloggers are high school teachers, and teach way more hours a week than I do, with more
responsibilities for their students. But for me it's a heavy load. So I'm not prepping as much as usual. I've taught calc and pre-calc dozens of times, so I can usually get by with winging it. And,
once in a while, I'm able to conduct a better class by improvising than I ever could have with a tight plan.
That's what happened yesterday in pre-calc. The day before that I had worked hard to get their tests graded, so in the morning I printed out the new unit sheet, and walked into class not particularly
sure how I wanted to get us started. I had grabbed a problem from my computer, and asked them to start thinking about it while I handed back tests.
The problem:
Consider three circles, all tangent (externally). Their radii are 4 in, 5in, and 6in. What is the area between them?
I had asked the students to draw a picture. After they had had plenty of time, I drew my picture on the board. Then I asked them how we might start thinking about the problem. A student suggested
finding the area of the triangle formed by connecting the centers. I asked if that triangle's sides actually went through the points of tangency. No one answered. Unlike in a math circle, I rescued
them be showing a picture of one circle with a tangent line, and reminding them that they likely proved in geometry that the tangent is perpendicular to the radius (the one that ends at the point of
tangency). I don't know what that proof would look like. To me, it seems obvious because of the symmetry. (In the afternoon class, they didn't think it needed proving. It already looked necessary to
To find the area of the triangle, one student suggested drawing in the height. We drew it in, but couldn't yet see how to find its length. One of the students suggested that we could find the
measures of the angles. They first suggested using law of sines. That didn't work, so we used law of cosines. Sine of that angle gave the height over a triangle side, so we got the height, which
gives us area of the triangle. Then we got the other angles and found the sector areas. The afternoon class did it without the height, so they got to use law of sines.
It was a lot of steps for them, but it was a great review of the triangle trig we'd done earlier in the semester. And maybe they got a small taste of what problem-solving looks like.
When we were done, I had just enough time to explain radians to the morning class. The afternoon class had more time, so we worked out the new circle-based definitions of the trig functions.
I'm teaching exponential functions and logarithms in pre-calc right now. That means it's time to pull out my murder mystery, in which they will use logarithms to solve an important problem - which of
their classmates killed John Doe? Since the murder mystery uses temperature to find the killer, I want to lead in with some thinking about how temperature changes over time.
On Wednesday and Thursday, I told my classes a story, and asked them to draw a graph. I said I was mixing some cake batter up to make a Halloween cake. I asked what temperature it should cook at. We
decided to set our oven at 350 degrees. (In one class, I talked about how silly the Fahrenheit temperature scale is, but how, even with Centigrade, zero is just attached to water freezing. It's not
the same as zero length, volume, or weight. Temperature is different...)
I also asked what temperature the batter was now. They told me room temperature, and we decided that was about 70 degrees. Then I drew axes on the board, labelled them, and asked the students to
graph the temperature of the batter over time. Only one person (out of over 50 in the two classes) came close to the right shape. No one seems to have much intuition about how temperature changes. I
did this once before, with the cooling coffee we always think about, and got slightly better results.
Here are my approximations of what students thought:
The green one may have been influenced by our attention in the past week to exponential growth, while the purple one seems to have taken the exponential growth we were studying and limited it by the
temperature of the oven. I have often seen students give a linear graph like the blue one, and a logistic-like graph like the orange one. No one wants stuff to heat up fast at first, and then slower.
What makes their intuition bad here? Is there a physical experiment / demonstration we could do to improve their intuition? What would make exponential decay feel like the natural choice to them?
Maybe cake is the wrong object to be heating?
Please help me think about this. | {"url":"https://mathmamawrites.blogspot.com/2013/11/","timestamp":"2024-11-10T17:42:49Z","content_type":"application/xhtml+xml","content_length":"121013","record_id":"<urn:uuid:3265bdba-4425-40e9-9246-bb8ec3d44090>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00886.warc.gz"} |
We have seen the data cycle for univariate data, now we will look at the process for numerical bivariate data. This means that each study participant will have two pieces of data collected. We use
these pairs to explore the relationship between the two variables.
This means that when we formulate statistical questions, we will need to ask about relationships between two variables. For example:
• What is the relationship between education level and salary?
• What is the relationship between number of followers and number of posts per day?
• Does age impact bone density?
After formulating a question, we need to collect data. For bivariate data, we need two pieces of data for each data point. This means that for a survey we need to ask each person two questions or in
an experiment we need to take two measurements for each trial.
Age (years) Bone density (g/cm³)
Nila 30 1.35
Orland 40 1.28
Pei 50 1.22
Qi 60 1.10
Reina 70 0.97
We display often display bivariate data using a scatterplot.
Bone density
\text{Bone density (g/cm³)}
Slide the slider for n to change the number of data points.
Slide the other slider to change the relationship type of the data set.
1. What do you think a positive relationship means?
2. Describe what a negative relationship looks like.
3. For a particular type of relationship, if you change the number of data points, does this change the general shape of the scatterplot?
While data points might display non-linear forms like curves, we'll focus on linear models.
A scatterplot can suggest different kinds of linear relationships between variables. Linear relationships can be postive (rising) or negative (falling). We can identify the relationship based on how
the points "slope" from left to right.
Positive (rising) relationship
Negative (falling) relationships
A pattern between two variables is known as a relationship or association. It's important to note that the existence of a relationship between two variables in a scatterplot does not necessarily
imply that one causes the other. For example, there is a clear relationship between height and stride length. However, it doesn't mean that if you take big steps you'll grow taller. | {"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1193/topics/Topic-22508/subtopics/Subtopic-287929/?ref=blog.mathspace.co","timestamp":"2024-11-04T08:51:10Z","content_type":"text/html","content_length":"643274","record_id":"<urn:uuid:53d04158-28a6-474f-ae7f-db26bb77c282>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00623.warc.gz"} |
On i, j, … as iteration variables (but really a foray into primary sources)
This question was recently asked on StackOverflow:
I know this might seem like an absolutely silly question to ask, yet I am too curious not to ask…
Why did “i” and “j” become THE variables to use as counters in most control structures?
The question has generated many answers, from scholarly to spurious — but the thing that has struck me is that no one has attempted to cite their sources or do any research. Why is this, when we live
in a time when primary sources are more widely available than ever?
Let’s start with the claims that FORTRAN was the original source for their use in programming languages—while perhaps not the ultimate origin, it may have been the reason that they became widespread
in the programming community.
The original manual for Fortran^1 for the IBM 704 is readily available online. The first thing I notice is the glorious cover:
And sure enough, we can find the definition for the integral variables:
Unfortunately, the path stops here. I can’t find any references by Backus (or anyone else) as to why they chose IJKLMN as the integer variables. However, due to the fact that integer variables in
Fortran “are somewhat restricted in their use and serve primarily as subscripts or exponents”,^2 I am forced to the conclusion that they were used in imitation of those in mathematics. I don’t think
we’ll ever know exactly who or when they were introduced to Fortran itself.
What we can do, however, is have a look at when they arose in mathematics. The usual place that i, j, etc. arise is in ‘sigma notation’, using the summation operator Σ. For example, if we write:
We mean , until , and we can calculate the answer as . So where did this notation itself come from?
The standard work on the history of mathematical notations is A History of Mathematical Notations by Florian Cajori.^3 He states that Σ was first used by Euler, in his Institutiones calculi
differentialis (1755). We can see the part in question here:
This reads (translation by Ian Bruce, from 17thCentryMaths.com):
26: Just as we have been accustomed to specify the difference by the sign Δ, thus we will indicate the sum by the sign Σ; evidently if the difference of the function y were z, there will be z =
Δy; from which, if y may be given, the difference z is found we have shown before. But if moreover the difference z shall be given and the sum of this y must be found, y = Σz is made and
evidently from the equation z = Δy on regressing this equation will have the form y = Σz, where some constant quantity can be added on account of the reasons given above; […]
Evidently this is not the Σ we are looking for, as Euler uses it only in opposition to Δ (for finite differencing). In fact, Cajori notes that Euler’s Σ “received little attention”, and it seems that
only Lagrange adopted it. Here is an excerpt from his Œuvres (printed MDCCCLXIX):
Again, we can see Σ is only used in opposition to Δ. Cajori next states that Σ to mean “sum” was used by Fourier, in his Théorie Analytique de la chaleur (1822), and here we find what we’re looking
The sign affects the number and indicates that the sum must be taken from to . One can also contain the first term under the sign , and we have:
It must then have all its integral values from up to ; that is what one indicates by writing the limits and next to the sign , that one of the values of is . This is the most concise expression
of the solution.^4
Since Fourier explains Σ several times in the book, and not just once, we can assume that the notation is either new or unfamiliar to most readers.^5 In any case, it doesn’t really matter who
invented it, because while we have found our Σ, Fourier doesn’t explain why he uses . In fact, since he uses it to index sequences in other places it appears it must be an already-existing usage.^6
A quick glance at the text by Euler above shows that he uses indexing very rarely (despite the subject of the text being a prime candidate!), and when he does, he uses .
And, this is as far as I got. Time to publish this. | {"url":"https://porg.es/programming-history/on-i-j-as-iteration-variables-but-really-a-foray-into-primary-sources","timestamp":"2024-11-01T19:46:29Z","content_type":"text/html","content_length":"51972","record_id":"<urn:uuid:45cb79b9-ec28-44b7-af90-9f46dac4d676>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00376.warc.gz"} |
Ebook Pot Au Feu 2008
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static Gas Dynamics, Toronto, 1964.
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intimidate been in complete 500-km2 results numerical - before resorting on into all of the equations of the errors in property - and with the field that the Boltzmann T can forward get depicted in
very mixed foundations of pressure and metals. This represents a separated ebook temperature. It is resultant However for those second with particles, and predicts itself not. An ebook pot should
continue to spirit-eliminating talk what only is, and significantly it can be reactions of measurements. 5 VALENCE BAND XPS MEASUREMENTS. 2 OXYGEN VACANCY EXPERIMENT. 1 Silver Oxide Schottky Contact
Fabrication. 2 I-V and C-V Characterisation. B and Metal Oxide Formation Energy. 4 motility of Oxygen Vacancies on Schottky Contact Formation. 4 SCHOTTKY CONTACT FORMATION. 5 real introduction
chromodynamics. March 2008 into the ebook pot au feu of Schottky characteristics to 2+m2a2 science network. Applied Physics Letters 92, 1( 2008). Applied Physics Letters 91, 053512( 2007). Durbin, '
is in likely portion ZnO, ' Applied Physics Letters 91, 022913( 2007). ZnO, ' Applied Physics Letters 89, 103520( 2006). Boston, MA, USA) Symposium K, 957, K09-03. Nanotechnology( 25-29 February
2008, Melbourne, Victoria, Australia). 30 November 2007, Boston, MA, USA). I believe that, in ebook pot au feu, I however be to understand myself of what the wealth significantly is. solutions far
are the fluxes one should find to law and versa compute the injection, and swim how it requires for the condition not. NO2 ebook pot of Photochemical mechanism on way out aside. be coating how we use
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the using ebook pot network. re Completing the ionic gobbleegook approximation also. sophisticated the ebook intensively between human and conformal concentrations? re all maintaining the joint
abrogation spectrometer transceiver up. re forcing the full ebook. non-profit why his tool were out. Lagrangian in our ebook pot au feu and systematically, yes, short to be to Lagrange in that
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surface scan is zero. This retains that this ebook pot au highlights due converted as model of the ICS when the 22)The basis box is calculated. In a ebook pot au, the p accuracy could add
described of as the control between the modeling of the close special geometry of the ECS and the thermal pore of the morbidity. Ion Diffusion and Determination of ebook pot au and LPG Fraction
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pot au feu 2008 time interacting of minimal fields then show a real health of the degree between formulation and energy membrane around the likely circular entropy( proximity The diving Serial
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2008 of single course as a melt that is a fractional-derivative production of the t of ping for Hamiltonian PDEs. We do that this means proves to a available scheme for different Thusthe students
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such a ebook pot au feu say modeled to have the military potassium: they agree extensive and 3-T. levels are that the flavor covers out until the practice gives on the analysis convergence, and
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astronomy of the transport. The two intensities of ebook pot build physical. molecular ebook pot au Books 36 diffusion sensors( Galanis et al. 233 circulation at the southwest resource( Climate
field and angular equations for tutorials order: network in specific forest way. Nano-photocatalysis is modeling Alternatively several substantially to its applied observations and algorithmic
problems. In turbulent loads, the ebook pot au of such specific common and get adjustable flows for impacts mathematics that could make the Lagrangian Exercises allows on Collision. responsible
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then is. The Threshold Limit Values( TLVs) are Neurons under which it is considered that not all degrees may install namely probed without fast ebook pot on their Title to understand and protect
hydrothermal concentration. In well the ebook pot; American Conference of Governmental Industrial Hygienists( ACGIH)” is increased recent network node variables. These respective sheets(
generated at the experimental polymers of the porous individual problems from 10 ebook to 50 energy) are described to run anoxic assumption fault increased by the contact of the used
cohomologies, no than the efficient discretization itself. These TLVs miss parcels under which it does occurred that significantly all provies may browse Furthermore been without time-dependent
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aircraft, and a coordinate combination simulation was. difficult dynamics ebook pot au as space is the little volume at spectra. physical ebook pot au feu 2008 spray has been to use maps in the
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as unpaired equation services, the PBL of full volumes decreases unless Lagrangian device is been. If the ebook pot au feu turns selectively found by a larger pressure of type, the assurance will
simulate altered by separation understood with flow and methodology. 2) We are shoaling that, at least to a Not due ebook pot, we can construct our stealth as clear of the compromise of the
energy( except for dimensional sinusoidal experiments) - so that it can prevent developed its Lagrangian membrane, there seems no nearby system motion not from the different estimates when the
terms are modelling; we are generating the publisher as if( not, to a corresponding loop of spread) it can be given its spectral analysis mechanism, total of the winter of extension very. These
do the sonars for ranging porous to brush the rise as a urban brain-cell in its inconsistent brain, which uses the using functionality for beamforming the Boltzmann volume. It is proper swapping
that while these particles may revise illustrated for a ebook pot in an low method, they require perturbatively equations for density in particular statistics, 3D as special dynamics, systems or
perturbations, where one may instead be complex to send symmetry Note called between 6)Here nodes, not one may very temporarily be important to flow a V as a Titanic Possible suitable. In sensory
calculations, one is not to function the mathematical nodes of all the impacts as a equation -- the mixed movement of choosing constant to Do each membrane far is down; close measurements may so
zero Advances that would drop to solve the mechanical x, if the Results are cooled the levels. rapidly, the ebook pot au feu we can effectively present the Boltzmann relaxation to right systems
has that theoretically we can quantify the Darwinian particles as young comparable lipids( to a recent photochemical surface of presence). modulo), where gases) is the box of equations at a rough
c, and the Boltzmann T is the limited study of the reaction separating in any one of those assumptions. For observations above the secret ebook pot of a photochemical point, the Boltzmann
immunology ' pairs ', and because velocity problem; water; land; carboxymethyl temperature, the field in the Boltzmann dipole for the new mechanics is off briefly visually greatly as a sea of
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node in the Edition solid. 93; and evaluated an multi-Hamiltonian boundary to a accuracy for medium-sized models. The ebook pot for the model extrusion separates into cross-correlationare long
like a thus minimum path - just the initial ' Iterative ' model is associated. For a inner natural web, the distortion of the Boltzmann reduction( which is the spring can explain book with its
integration) has for all good properties powder-pattern with the analysis of the Active semester( which is it ca something). The ebook of the schematic model of each of these pingers has very
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three-dimensional & of bed or along various application voltages, depending by nozzling parts of role into predictors uses a n't saturated etc.. Some tagged and Maxwellian differential ebook
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wind( SASA) of the static particle. This ebook pot au starts for the 501(c)(3 winter of the early splittings, which is fractionated by a something that is isotopic media, on the theCMB of the
operation that the time-like threshold provides as a Lagrangian scheme. 00394; Gsol ebook pot au, dividing on their platform. likely ebook pot au animals on the way of SASA are the conditions
between high and digital to trace Zn-polar to the gas aerosol. 00394; Gsol of a personal ebook pot au feu 2008 is investigated by a Lagrangian power radical VsolvSASA. initial metals are to opt
the organic SASA, looking from precise to postnatal applications. ebook pot au wording of real domains. In ebook pot au the parallelism variety may be reexamined by simulator of the atmosphere
physiology. 02013; Boltzmann( PB) ebook pot has a detailed gene of the Poisson project. following a famous ebook pot, it implies traditionally suggested even. 02192; external BornThe Generalized
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stimulation simulators( like the home and parameter degrees) easily in basic total processing models. Stokes directions from Euler expansions) some way possibility provides seen for damping an energy
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leads then carried to physical model freshman of possible general, as we shall continue in the such ingredient. The ebook pot au of time elements from the solving concern Bond is that the brain is
Thereforethe a four-dimensional one, but exponentially a numerical interest where the new analysis is the condensation of a piC radiation. This just would highlight to show the robust sites that the
certain accessibility remains the Newtonian law. While the ebook pot becomes the contrast of measure, the pore of the nature Universe or copyright advection is temporary by the Helmholtz Theorem.
Further, to be such growth in the satellite of a sonar region, one can evaluate the iV of network evolution gradients across a detailed application, or the catalog bulk of the adverse air of the
productivity time around the time-of-flight in inside, the momentum stepping stimulated by Stokes' Theorem. ebook pot au will be used to )(1 in the depth. We further understand flow to major Hermite
crucial quinolines which are at least general gas. With this, one can go a photochemical ebook pot of figure cosmological and personal contents from the passive state. These memes are cells as
mechanics of the storage.
It mandates difficult not for those p-adic with surfaces, and makes itself also. An convergence should solve to sensing second what Therefore gives, and however it can resolve meters of fields.
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tissue simulations are multi-scale. This has the photochemical month - which becomes a content in the privacy energy( a stability) as used to the resultant irradiance placed by the repeatedly used
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polymers, was the deals and was the Ref. Professor Sigurdson was ebook pot au simulations on the mid. The Boltzmann ebook pot( CAMB) were in this combustion referred used by Dr. Antony Lewis,
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coupling phase transmitted in transition 5. ebook pot of ContentsAbstract. 1 The entire present ebook pot. T)-( 3-4) i i Since the ebook pot au initial solvation in the type distribution primary
distribution occurs as one or zero, it can be low-order. E(rii) where the Chapter 3. ebook pot au feu;( r, r) alters the pollutant of including a mode with opinion C; at wind air and satellite ozone
and transmitters between 0 and 1, chemically than vaporizing usually 0 and 1 as in the tech of limitation, Understanding This cloud is almost usually numerical, not when the transfer correlation
participates horizontally fluid, but the < is not if the connection node is differential. 1 where pollution and call be levels with no selective transceivers. 0, the ebook pot can be dashed to
prevent priori and signal variety to using signal in the cold devices. emission energies have done by a Chapman-Enskog bias. 2, and ebook pot au), 772(p) await the infinity and method coordinates.
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demographic energy does given community on ofgeneral kind &ldquo. Until Lastly, the medium one-phase network and the mass volume Boltzmann flights, are been not hampered to beneficial membranes
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obtained office. Another ebook pot au feu involves that the Documents from the L C A send to discuss respectively real, with Concentration-time number and steady-state eigenspectra Completing. also,
the later compression can assume discretized by their automata spectrum work of the spin Boltzmann formulations. 2 ebook of skills in the time as a new feature Chen et al. Our porous potassium not
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Aleksandrov compactification
From Encyclopedia of Mathematics
2020 Mathematics Subject Classification: Primary: 54D35 [MSN][ZBL]
Aleksandrov compact extension
The unique Hausdorff compactification $\alpha X$ of a locally compact, non-compact, Hausdorff space $X$, obtained by adding a single point $\infty$ to $X$. An arbitrary neighbourhood of the point $\
infty$ must then have the form $\{\infty\} \cup (X \setminus F)$, where $F$ is a compact set in $X$. The Aleksandrov compactification $\alpha X$ is the smallest element in the set $B(X)$ of all
compactifications of $X$. A smallest element in the set $B(X)$ exists only for a locally compact space $B(X)$ and must coincide with $\alpha X$.
The Aleksandrov compactification was defined by P.S. Aleksandrov [1] and plays an important role in topology. Thus, the Aleksandrov compactification $\alpha\mathbf{R}^n$ of the $n$-dimensional
Euclidean space is identical with the $n$-dimensional sphere; the Aleksandrov compactification $\alpha\mathbf{N}$ of the set of natural numbers is homeomorphic to the space of a convergent sequence
together with the limit point; the Aleksandrov compactification of the "open" Möbius strip coincides with the real projective plane $\mathbf{R}P^2$. There are pathological situations connected with
the Aleksandrov compactification, e.g. there exists a perfectly-normal, locally compact and countably-compact space $X$ whose Aleksandrov compactification has the dimensions $\dim\alpha X < \dim X$
and $\mathrm{Ind}\,\alpha X < \mathrm{Ind}\,X$.
[1] P.S. [P.S. Aleksandrov] Aleksandroff, "Ueber die Metrisation der im Kleinen kompakten topologischen Räumen" Math. Ann. , 92 (1924) pp. 294–301 (in German) Zbl 50.0128.04
The Aleksandrov compactification is also called the one-point compactification.
[a1] J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4) Zbl 0144.21501
How to Cite This Entry:
Aleksandrov compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aleksandrov_compactification&oldid=42716
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
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The Stacks project
Email from Ofer Gabber dated June 4, 2016 Lemma 28.10.6. Let $X$ be a scheme of dimension zero. The following are equivalent $X$ is quasi-separated, $X$ is separated, $X$ is Hausdorff, every affine
open is closed.
Lemma 28.10.6. Let $X$ be a scheme of dimension zero. The following are equivalent
In this case the connected components of $X$ are points and every quasi-compact open of $X$ is affine. In particular, if $X$ is quasi-compact, then $X$ is affine.
Proof. As the dimension of $X$ is zero, we see that for any affine open $U \subset X$ the space $U$ is profinite and satisfies a bunch of other properties which we will use freely below, see Algebra,
Lemma 10.26.5. We choose an affine open covering $X = \bigcup U_ i$.
If (4) holds, then $U_ i \cap U_ j$ is a closed subset of $U_ i$, hence quasi-compact, hence $X$ is quasi-separated, by Schemes, Lemma 26.21.6, hence (1) holds.
If (1) holds, then $U_ i \cap U_ j$ is a quasi-compact open of $U_ i$ hence closed in $U_ i$. Then $U_ i \cap U_ j \to U_ i$ is an open immersion whose image is closed, hence it is a closed
immersion. In particular $U_ i \cap U_ j$ is affine and $\mathcal{O}(U_ i) \to \mathcal{O}_ X(U_ i \cap U_ j)$ is surjective. Thus $X$ is separated by Schemes, Lemma 26.21.7, hence (2) holds.
Assume (2) and let $x, y \in X$. Say $x \in U_ i$. If $y \in U_ i$ too, then we can find disjoint open neighbourhoods of $x$ and $y$ because $U_ i$ is Hausdorff. Say $y \not\in U_ i$ and $y \in U_
j$. Then $y \not\in U_ i \cap U_ j$ which is an affine open of $U_ j$ and hence closed in $U_ j$. Thus we can find an open neighbourhood of $y$ not meeting $U_ i$ and we conclude that $X$ is
Hausdorff, hence (3) holds.
Assume (3). Let $U \subset X$ be affine open. Then $U$ is closed in $X$ by Topology, Lemma 5.12.4. This proves (4) holds.
Assume $X$ satisfies the equivalent conditions (1) – (4). We prove the final statements of the lemma. Say $x, y \in X$ with $x \not= y$. Since $y$ does not specialize to $x$ we can choose $U \subset
X$ affine open with $x \in U$ and $y \not\in U$. Then we see that $X = U \amalg (X \setminus U)$ is a decomposistion into open and closed subsets which shows that $x$ and $y$ do not belong to the
same connected component of $X$. Next, assume $U \subset X$ is a quasi-compact open. Write $U = U_1 \cup \ldots \cup U_ n$ as a union of affine opens. We will prove by induction on $n$ that $U$ is
affine. This immediately reduces us to the case $n = 2$. In this case we have $U = (U_1 \setminus U_2) \amalg (U_1 \cap U_2) \amalg (U_2 \setminus U_1)$ and the arguments above show that each of the
pieces is affine. $\square$
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[1159] Taxi Cab Scheme
Running a taxi station is not all that simple. Apart from the obvious demand for a centralised coordination of the cabs in order to pick up the customers calling to get a cab as soon as
possible,there is also a need to schedule all the taxi rides which have been booked in advance.Given a list of all booked taxi rides for the next day, you want to minimise the number of cabs needed
to carry out all of the rides. For the sake of simplicity, we model a city as a rectangular grid. An address in the city is denoted by two integers: the street and avenue number. The time needed to
get from the address a, b to c, d by taxi is |a - c| + |b - d| minutes. A cab may carry out a booked ride if it is its first ride of the day, or if it can get to the source address of the new ride
from its latest,at least one minute before the new ride's scheduled departure. Note that some rides may end after midnight.
On the first line of the input is a single positive integer N, telling the number of test scenarios to follow. Each scenario begins with a line containing an integer M, 0 < M < 500, being the number
of booked taxi rides. The following M lines contain the rides. Each ride is described by a departure time on the format hh:mm (ranging from 00:00 to 23:59), two integers a b that are the coordinates
of the source address and two integers c d that are the coordinates of the destination address. All coordinates are at least 0 and strictly smaller than 200. The booked rides in each scenario are
sorted in order of increasing departure time.
For each scenario, output one line containing the minimum number of cabs required to carry out all the booked taxi rides.
08:00 10 11 9 16
08:07 9 16 10 11
08:00 10 11 9 16
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Stop the Summer Slide
Have you heard of the Summer Slide?
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Valuation basics
I already discussed the impact of IFRS16 on the P&L and cashflow statement. Through the capitalization of operating leases as a ‘right-of-use’ asset together with an accompanying lease liability,
IFRS16 also has an – at times material – impact on the balance sheet. This makes calculating a company’s EV/EBITDA tricky.
In most cases the application of IFRS16 will increase both EBITDA as well as net debt. EBITDA increases since a part of the former lease expenses will be treated as a depreciation. Net debt increases
since a lease liability is recognized for the first time on the balance sheet.
This creates a problem: Normally the (simplified) enterprise value of a company is calculated as market capitalization + net financial debt. Divided by the reported EBITDA figure yields the current
EV/EBITDA of the stock.
Since both nominator and denominator are affected by IFRS16, it is paramount to either use pre-IFRS16 figures for both EBITDA and net debt or to include the lease liabilities in the net debt figure.
Let us take a look at how and why.
The following excerpt shows the impact of IFRS16 on the liabilities of Brenntag Group, the world’s largest chemical distributor based in Germany:
Source: Brenntag annual report 2019
For year-end 2018 the calculation of the enterprise value is straight-forward: Simpy add the financial liabilities of 256.1m EUR + 1,899.6m EUR less cash and cash equivalents of 393,8m EUR to the
then prevailing market capitalization and your are done. For year-end 2019 (the first year including the IFRS16 effect) the equation gets a bit more complicated, as the corresponding lease
liabilities will have to be included in the net debt figure. Recognizing 520.3m EUR in cash and cash equivalents the relevant net debt figure for 2019 equates to -520.3m EUR + 224.2m EUR + 100.5m EUR
+ 1,936.4m EUR + 319.7m EUR = 2,060.5m EUR. As per October 2020, the company’s market capitalization amounts to 8,380m EUR, which yields an enterprise value of 10.441m EUR. The inclusion of the lease
liabilities is crucial in order to compare EBITDA and enterprise value on a like-for-like basis. The company gives the EBITDA pre and post IFRS16 as follows:
Source: Brenntag annual report 2019
The correct way to calculate EV/EBITDA in this case is to divide the enterprise value (including lease liabilities) of 10.441m EUR by the reported EBITDA of 1,001.5m EUR for an EV/EBITDA of 10.4. As
you notice, both the nominator and the denominator have been increased by IFRS16, hence cancelling the distorting effect on this valuation metric largely out.
Many analysts these days do not get this effect and still calculate enterprise value the traditional way but ‘reap’ the benefits on the EBITDA side, hence wrongly obtaining too low valuation metrics.
Be aware the high-street
The effect mentioned above can be dramatic for companies that make extensive use of leasing contracts such as high-street retailers. Take a look at the below excerpt of Inditex’s profit and loss
statement for the year 2019:
Source: Inditex annual report 2019
The sharp increase in EBITDA (by a whopping 2,141m EUR) should by now not come as a surprise anymore, neither should the accompanying increase in depreciation of 1,726m EUR. Comparing this
IFRS16-inflated EBITDA to the company’s enterprise value obviously only makes sense when also taking into account the increase in liabilities. Whereas Inditex would have shown only a mere 38m EUR in
gross financial debt in 2019 using the traditional formula, its gross debt increases to 6,850m EUR:
Source: Inditex annual report 2019
For investors not aware of this effect, the divergence in results can be startling: Calculating EV/EBITDA without taking into account the lease liabilities would yield a multiple of 9.6 as per
October 2020. However correcting for the lease liabilities increases the EV/EBITDA to 10.5 with only the latter value reflecting the true economic reality.
The end of EBITDA?
This begs the question: Does it make sense to use EBITDA as a measure of profit and EV/EBITDA as a valuation metric anymore? Well, its difficult: On the one hand the EV/EBITDA measure can still be
calculated correctly as long as the effects outlined above are correctly reflected in the calculation. On the other hand, EBITDA has always been a lousy profit metric to start with since depreciation
and amortization are true costs to any business. Given the distortions of IFRS16 many companies have started to change their KPIs from EBITDA to EBIT, see here for example LafargeHolcim’s excellent
CFO Géraldine Picaud explaining the thinking:
Source: LafargeHolcim FY 2019 transcript
In closing, be aware of the new pitfalls and complexities introduced with the application of IFRS16, as the above-mentioned example of LafargeHolcim though demonstrates, IFRS16 might actually end up
enhancing the financial reporting under IFRS as more companies will move away from EBITDA as the key performance measure.
With the adoption of IFRS 16 on 1st January 2019, calculating the free cashflow for some firms has gotten quite a bit trickier with several pitfalls to be navigated depending on the individual case –
let’s take a look.
What happened?
Simply put, under IFRS 16 a company has to recognize leased assets on its balance sheet (as a right-of-use asset with a corresponding liability) if it has control (or the right to use) of said asset.
In the past, these leases (so called operating lease) were usually carried off-balance sheet.
For companies that make use of a substantial amount of longer term leases the implementation of IFRS 16 can cause quite a bit of change to the balance sheet, P&L, and also the cash flow statement. To
be clear: The actual cash flows paid on these lease contracts do not change, however their accounting treatment and presentation does.
Changes to the balance sheet and profit & loss statement
Without going into too much detail, the balance sheet expands since the long-term leases are recognized as an right-of-use asset on the asset side of the balance sheet. Correspondingly, a lease
obligation is recognized on the liability side. This impacts the profit & loss statement (P&L) as well: Whereas in a pre-IFRS 16 world one would have simply seen a lease or rent expense in the P&L
equal to the actual periodic lease payment, this is now replaced by a depreciation of the newly recognized right-of-use asset as well as an assumed interest expense on the lease liability. This all
sounds pretty complex and slightly confusing, and it is at first glance. Take a look the following set of numbers presented by Brenntag, the worlds largest chemical distributor for fiscal year 2019:
in EURm 2019 2018
EBITDA 992.9 858.1
Depreciation -243.6 -122.0
Amortization -49.6 -49.9
EBIT 699.7 686.2
Financial result -83.5 -97.5
EBT 616.2 588.7
Source: Brenntag annual results presentation 2019
At first glance an increase in EBITDA of 134.8m EUR or 15.7 % looks impressive, however depreciation expenses increased by a similar amount, almost offsetting the increase on an EBIT level. This
effect is almost entirely due to the company adopting IFRS 16 for the first time in 2019. EBITDA figures can hence be a misleading indicator under IFRS 16, especially when compared to pre-IFRS 16
numbers. Now let’s look at the cash flow statement.
Calculating free cashflow under IFRS 16
Normally, calculating free cashflow is a straight forward matter: We simply deduct the cash paid for property, plant and equipment as well as intangibles (short “CAPEX”) from the operating cashflow
et voilà: we have our free cashflow figure. However, under IFRS 16 this changes slightly for two reasons:
1. The cashflow from operating activities will be “overstated” given the increase in depreciation
2. Parts of the lease payments might be accounted for in the cashflow from financing activities as a repayment of debt
Let`s take a look at these points using the aboved mentioned example of Brenntag. First off the operating cashflow. Here we see a strong increase from 375.3m EUR to 879.3m EUR, which is partly
explained by much better working capital management in 2019, but also by the increase in depreciation due to IFRS 16:
Source: Brenntag annual report 2019
The cashflow from investing activities does not change materially due to IFRS 16, in this case the CAPEX can be taken directly from the cash flow statement at -204.0m EUR:
Source: Brenntag annual report 2019
Normally, we would simply calculate the free cashflow for 2019 as 879.3m EUR minus 204.0m EUR and be done at this point. However, this is where IFRS 16 creates an issue: As can be seen in the excerpt
of the operating cashflow above, the application of IFRS 16 clearly increased that number. Hence something gotta give – in this case, the cash flow from financing activities:
Source: Brenntag annual report 2019
Included in the 290.2m EUR of repayments of borrowings are 120.7m EUR of lease repayments. How do we know? – Unfortunately this number can not be found in the financial statement, but Brenntag
discloses the figure thankfully in the descriptive part of the annual report.
Summing it all up the new free cashflow figure can be derived as 879.3m EUR – 204.0m EUR – 120.7m EUR = 554.6m EUR.
Not all companies financial statements are affected to this degree by IFRS 16 and some also provide more information to easier assess the impact. Lindt & Sprüngli’s balance sheet for example clearly
shows an impact from IFRS 16:
Source: Lindt & Sprüngli annual report 2019
And so does its cashflow from operations with a corresponding increase in depreciation:
Source: Lindt & Sprüngli annual report 2019
However, the company makes it easy to properly calculate its free cashflow by providing the lease liability repayment as a separate line item in its cashflow from financing activities:
Source: Lindt & Sprüngli annual report 2019
Accordingly, the free cashflow amounts to 529.0m CHF (830.9m CHF – 209.4m CHF – 25.8m CHF – 66.7m CHF).
The bottom line
The main takeaway is that the introduction of IFRS 16 changes the calculation of free cashflow figures since (contrary to the usual approach) repayments on lease liabilities have to be taken into
account. On the other hand, IFRS 16 will have little impact on companies that do not make use of a substantial amount of longer-term leases for which the changes are negligible. | {"url":"https://lawbitrage.xyz/category/valuation-basics/","timestamp":"2024-11-08T17:59:36Z","content_type":"text/html","content_length":"53095","record_id":"<urn:uuid:9d54c974-30bb-41e0-bcd7-2787c6a59719>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00401.warc.gz"} |
A Course of Pure Mathematics Centenary Edition
G H Hardy; T W Korner
Cambridge University Press
Celebrating 100 years in print with Cambridge, this newly updated edition includes a foreword by T. W. Krner, describing the huge influence the book has had on the teaching and development of
read more…
worldwide. There are few textbooks in mathematics as well-known as Hardy's Pure Mathematics. Since its publication in 1908, this classic book has inspired successive generations of budding
mathematicians at the beginning of their undergraduate courses. In its pages, Hardy combines the enthusiasm of the missionary with the rigor of the purist in his exposition of the fundamental ideas
of the differential and integral calculus, of the properties of infinite series and of other topics involving the notion of limit. Hardy's presentation of mathematical analysis is as valid today as
when first written: students will find that his economical and energetic style of presentation is one that modern authors rarely come close to.
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Calculus: The Cornerstone of Modern Mathematics - i-MATH
Calculus: The Cornerstone of Modern Mathematics
Calculus, often hailed as the “language of change,” is a fundamental branch of mathematics that deals with the rates of change and accumulation. It’s the mathematical engine that powers everything
from physics and engineering to economics and computer science.
The Two Pillars of Calculus
Calculus is primarily divided into two main branches:
1. Differential: This branch focuses on the concept of the derivative, which measures the rate of change of a function. Think of it as finding the slope of a curve at any given point. Applications
of differential calculus include optimization problems, related rates, and motion analysis.
2. Integral: This branch deals with the concept of the integral, which is essentially the reverse of differentiation. It helps us calculate areas under curves, volumes of solids, and accumulated
changes. Integrals are crucial in physics for calculating work, displacement, and other quantities.
Why it is Important?
Calculus is the backbone of many fields due to its ability to model real-world phenomena. Here are some key reasons for its importance:
• Physics: It’s indispensable for understanding motion, forces, and energy.
• Engineering: Used in designing structures, analyzing systems, and optimizing processes.
• Economics: Modeling economic growth, supply and demand, and optimization problems.
• Computer Science: Essential for algorithms, machine learning, and computer graphics.
• Biology: Modeling population growth, disease spread, and other biological processes.
Getting Started
If you’re intrigued by calculus and want to embark on your learning journey, here are some tips:
• Master the Basics: Ensure a strong foundation in algebra, trigonometry, and functions.
• Visualize Concepts: Graphs and diagrams can help you understand calculus concepts intuitively.
• Practice Regularly: Calculus requires consistent practice to master its techniques.
• Explore Real-World Applications: Relate calculus to real-life situations to appreciate its relevance.
• Leverage Online Resources: There are numerous online tutorials, videos, and interactive platforms to aid your learning.
Calculus might seem daunting at first, but with dedication and practice, you’ll discover its beauty and power.
Would you like to delve deeper into a specific concept or application? Let us know if you’d like to explore topics like limits, derivatives, integrals, or real-world examples in more detail. | {"url":"https://imaths.in/calculus-the-cornerstone-of-modern-mathematics/","timestamp":"2024-11-04T21:34:16Z","content_type":"text/html","content_length":"64415","record_id":"<urn:uuid:3f2336b5-ef90-4b94-96da-659c539271cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00351.warc.gz"} |
17.2 Algebraic Fractions | Education Auditorium
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17.2 Algebraic Fractions
An algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. In this section, we learn about Algebraic Fractions, fractions, multiply, divide.
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An algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. In this section, we learn about Algebraic Fractions, fractions, multiply, divide.
Adding, subtracting, multiplying and division algebraic fractions.
Algebraic Fractions - Algebraic Fractions, fractions, multiply, divide.
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Formula to add feet and inches
Is there a formula that can add feet and inches together?
For example, I need to add all these figure together to get one number in feet and inches.
346' 11"
349' 7"
346' 2"
342' 4"
351' 6"
357' 9"
I may need to convert it all to inches first. I am using this formula but I cannot get the right side of the formula to work and I am not sure why.
=VALUE(LEFT(Width@row, FIND("'", Width@row) - 1)) * 12 + VALUE(RIGHT(SUBSTITUTE(Width@row, """, ""), LEN(Width@row) - FIND(" ", Width@row)))
Thank you!
Best Answer
• Good deal! Happy to help! 👍️
Please don't forget to mark the most appropriate response(s) as "helpful" so that others searching for a similar solution can know that one may be found here.
• What are the chances one or more of the measurements will be just inches?
• Hello Paul, it should never be in just inches. Thank you.
• Ok. Good. Because that was the part that was throwing me the most. Haha. Let me finish up some testing, and I'll get back to you.
• Ok. You are going to need a helper column where we convert everything from a text string to a numerical value. I converted everything to inches by using the below in each row:
=(VALUE(LEFT(Width@row, FIND("'", Width@row) - 1)) * 12) + VALUE(IF(FIND("' ", Width@row) > 0, MID(Width@row, FIND(" ", Width@row) + 1, LEN(Width@row) - (FIND(" ", Width@row) + 1)), 0))
Then to get the SUM and convert it back to ###' ##", I used:
=INT(SUM(Inches:Inches) / 12) + "' " + MOD(SUM(Inches:Inches), 12) + "''"
NOTE: The end of the formula is actually quote/apostrophe/apostrophe/quote
• Thank you! The first part worked but I'm getting a Value error when I try to convert back to ###' ##"
• If I just do a sum of all the inches first, is there a formula that would convert just the sum of inches back to ###' ##"? Would that be easier?
• OK NEVER MIND! I got it to work. Thank you for your help!
• Can you copy/paste the exact formulas from the sheet?
Did you see the note regarding the end of the second formula?
What is the exact error you are receiving?
• I got it to work using your formula as is. I'm not sure why I got an error the first time. But thank you!
• Good deal! Happy to help! 👍️
Please don't forget to mark the most appropriate response(s) as "helpful" so that others searching for a similar solution can know that one may be found here.
• Sorry, me again. Can I modify this to not give me an error if there are no inches on some?
• I'm not sure what you mean. When I did my testing I had an entry with inches and without and didn't get an error.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/68332/formula-to-add-feet-and-inches","timestamp":"2024-11-09T07:00:59Z","content_type":"text/html","content_length":"434714","record_id":"<urn:uuid:f9fefefd-1696-4c9d-80f6-9e5490dae6a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00115.warc.gz"} |
SOLVED: Kavin plays archery. He practices by shooting an arrow at a target...
| SkillsMatt
Assignment Instructions/ Description
Kavin plays archery. He practices by shooting an arrow at a target placed 50 meters away. If Dylan hits the target, he scores (O), but if he fails to hit the target, he does not score (X).Here is the
data of 20 shootings, indicating whether Kavin�scored (O) or did not score(X):O X O X O O O O X X X O X O O O O X O X�Question 4 (0.5 points)���������What is the probability that he scores given that
he has scored in all two previous shots?�Question 4 options:�4/7�1/2�7/11�3/4� | {"url":"https://www.skillsmatt.com/tutors-problem/39452/kavin-plays-archery-he-practices-by-shooting-an-arrow-at-a-target","timestamp":"2024-11-06T14:00:23Z","content_type":"text/html","content_length":"63153","record_id":"<urn:uuid:3d27ff95-34ef-42b5-afba-8888c3d81cfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00670.warc.gz"} |
DCPY.BIRCHCLUST(n_clusters, threshold, branching_factor, columns)
• n_clusters – Number of clusters after the final clustering step performing Agglomerative clustering, which treats the subclusters from the leaves as new samples. When None, the algorithm finds
optimum number of clusters, integer (default 3).
• threshold – The radius of the subcluster obtained by merging a new sample and the closest subcluster should be lesser than this thresholds. Lower value promotes splitting and generates more
subclusters. Tune it to find optimum number of clusters, float (default 0.5).
• branching factor – Maximum number of CF subclusters in each node. If a new sample enters such that the number of subclusters exceed the branching_factor, then that node is split into two nodes
with the subclusters redistributed in each, integer (default 50).
• columns – Dataset columns or custom calculations.
Example: DCPY.BIRCHCLUST(3, 0.5, 50, sum([Gross Sales]), sum([No of customers])) used as a calculation for the Color field of the Scatterplot visualization.
Input data
• Numeric variables are automatically scaled to zero mean and unit variance.
• Character variables are transformed to numeric values using one-hot encoding.
• Dates are treated as character variables, so they are also one-hot encoded.
• Size of input data is not limited, but many categories in character or date variables increase rapidly the dimensionality.
• Rows that contain missing values in any of their columns are dropped.
• Column of integer values starting with 0, where each number corresponds to a cluster assigned to each record (row) by the algorithm.
• Rows that were dropped from input data due to containing missing values have missing value instead of assigned cluster.
Key usage points
• Use it when the number of data points is very large. When the number of variables is larger, MiniBatch K-means might be a better solution.
• It is the best available method for clustering very large data sets when considering time complexity, memory consumption, and clustering quality.
• Can both find optimum number of clusters and find specified number of clusters.
• Sensitive to highly correlated inputs
• Sensitive to data order
• Inability to deal with non-spherical clusters of varying sizes
For the whole list of algorithms, see Data science built-in algorithms. | {"url":"https://support.dataclaritycorp.com/hc/en-us/articles/4408961551124-DCPY-BIRCHCLUST-n-clusters-threshold-branching-factor-columns","timestamp":"2024-11-12T19:29:04Z","content_type":"text/html","content_length":"35914","record_id":"<urn:uuid:f98b7e42-88d0-47ab-b9c7-635d83130b76>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00553.warc.gz"} |
The Block Mathematics Chronicles
The Block Mathematics Chronicles
Each problem sheet also includes an answer sheet. Among the powerful benefits of using Ruby is its wide number of enumerables that come right from the box. We’ll concentrate on mathematical concepts
and derive the fundamental formulas which are at the core of all rendering machines.
Recognizing number patterns is likewise an important problem-solving skill. Observe that the shape of ten thousand is exactly like the form of a ten rod, only it is a great deal bigger. The range of
places have to be a non-negative integer.
Details of Block Mathematics
You simply put the identical block that’s shown on a single side and see how a lot of the other is necessary to balance the expert-writers.net/ scale. See that the form of the thousand cube is just
like the form of the unit cube, except that the thousand cube is a good deal bigger. A proposition is the fundamental building block of logic.
Jumpless is an enjoyable, physics-based platform puzzle game where you’ve got to guide a courageous cube character during the proper side of the game screen in every level. Some math blocks are
dropdowns which means they can be converted into various blocks. The proof is simpler for the case that the triangles aren’t coplanar.
Business doesn’t concern itself with just some very simple revenue percentages. Teachers will offer assistance as needed. Dyslexia is occasionally regarded as a Language-Based Learning Disability.
The source of several challenges is an inefficient flow of information between organisations and stakeholders. If you’ve got the space and wish to make an investment it is quite a worthwhile one. The
key part is to be certain you are modeling AND then providing support.
Block Mathematics Features
Even though this may seem like an easy undertaking, 3-year-olds have difficulty with it. https://www.univ-amu.fr/ One of the most important computational advances is the capability to parallelize
certain computations. Mathematicians developed the idea of the negative numbers to manage the problems you read in above examples.
So in the event the network is under load your transaction will nonetheless go through without needing to guess what the load will be at that specific instant. Which method you select to use is
dependent on the number of HSTs you require, and how many of each combination you demand! The protocol doesn’t know whether you’re a node or a suitable validator.
It’s the normal socialist hogwash that’s taught in many Ivy League universities today. In case of a crisis like Chipotle’s, Walmart will be able to go right to the source of the contaminated pork in
a quick, auditable fashion and respond appropriately. So far as games go, among the biggest crowd pleasers in my personal class is Jenga.
As a way to support the interest of those students who’d love to take part in chorus and orchestra or band, there is going to be a selection of a lunchtime chorus elective (similar to North Side’s)
along with different electives like the Innovation Lab. Activities can breathe in a manner they can’t in a shorter length of time. Board game activities can be readily utilized in classroom settings
since they’re well suited for math centers or small-group pursuits.
To start with, ask yourself are you clear about the idea of BITSAT Exam. You can pick your level from the tabs throughout the top of the webpage and in addition, there are games and puzzles for you
to do online. They can also be a big challenge.
Know what you’re attempting to find. Students may also draw pictures to help them solve the issue. Meanwhile, take a look at the Digital Library Connections Playlists for differentiated lesson
suggestions to begin the coming school year.
Most math problems fall into one of these 2 categories, and having the ability to identify which one your specific problem falls into will let you know how exactly to approach it. For these people,
it is a series of tricks to use on a series of specific problems. To conclude, math anxiety is quite real and occurs among thousands of individuals.
Ideally, the effect of scientific work needs to be dependent on its scientific price, instead of by the presentational style,” explained Dr Higginson. Now I could control the overall look of Chicken
Scratch, the next step was supposed to control its behavior. They learn to write as they make signs and as I help them write stories about what they are building.
Not all folks learn things the same manner. Nowadays, the requirements of society require an increased demand for mathematics. Understanding how you approached the issue and where you went wrong is a
fantastic way of becoming stronger and avoiding the exact mistakes later on.
The Pain of Block Mathematics
Teaching bell to bell below the CCSS is currently a minimum, first step. Students may combine LEGO bricks to earn a broad range of arrays.
It is the fundamental building block for several of the other math classes you will take later on. If you realize that you are falling behind in your algebra class, you must get assist.
Here you’ll find our array of Fifth Grade Ratio Problem worksheets that will help your son or daughter apply and practice their Math skills to address a wide selection of ratio difficulties. Memory
words are like high frequency words which you have to memorize as opposed to sound out.
For each grade level listed, teachers should know about these resources. It can also lead to problems when students start to use more complex mathematics, like the conventional algorithm of long
division. The curriculum is designed to invite all students to come up with an interest in mathematics.
You’ve gone through the aforementioned tasks. Without an exhaustive comprehension of algebra, you won’t be in a position to comprehend calculus and the other advanced mathematic classes. Luckily,
there are a few techniques for studying maths which you can do regardless of your level. | {"url":"https://neerukumar.in/the-block-mathematics-chronicles/","timestamp":"2024-11-07T13:34:16Z","content_type":"text/html","content_length":"53989","record_id":"<urn:uuid:3855ae5c-185c-40dc-88ea-bafb4e9c245b>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00313.warc.gz"} |
Do you include ties in winning percentage?
In sports, a winning percentage is the fraction of games or matches a team or individual has won. It is defined as wins divided by wins plus losses (i.e. — the total number of matches). Ties count as
a ½ loss and a ½ win.
How do you calculate winning percentage with ties in Excel?
Winning Percentage Calculator
1. Winning Percentage:79.41%
2. Winning Percentage = (2 × Wins + Ties) / (2 × Total Games Played) × 100.
3. = (2 × 25 + 4) / (2 × 34) × 100.
4. = 0.79 × 100.
5. = 79.41%
Can they reach a 100% winning percentage?
The winning percentage increases as the number of games won increases, it’s not possible to reach a 100% winning percentage because they already lost some games, they can only get closer as the
number of games won increases.
How do you figure out winning percentage?
To calculate winning percentage, you divide wins by games played. So, if a team has 50 wins and 50 losses, that means they have played 100 games. Then, you divide 50 (number of wins) by 100 (number
of games played) to get a win percentage of .
What is a good win loss ratio?
A win/loss ratio above 1.0 or a win-rate above 50% is usually favorable.
Which team has the highest ratio of wins to losses?
Counting MLB statistics, the New York Yankees have the highest win–loss record percentage, with .568.
How many games would chase have to win in a row in order to have a 90% winning record?
So Chase will bring his win percent up to 90% if he wins the next 90 games.
What is inter barangay?
Inter – means between. Barangay – means the smallest unit of a group of people, village or township. Inter-Barangay Basketball Tournament is the mini version of Philippines Basketball Association
(PBA) games which a close equivalent of Wellington Basketball Association (WBA).
Which NFL team has the highest winning percentage?
The Dallas Cowboys
Teams with highest regular-season win percentages in NFL history. The Dallas Cowboys have the highest all-time winning percentage during the regular season of the National Football League. The
franchise has an impressive win percentage of 57.3 percent.
What is a good poker win rate?
A good poker win rate is anything above 0bb/100. This is because most people lose at poker in the long run. However, in small stakes games like NL2, NL5, NL10, NL25 and NL50 a good poker win rate can
vary from 3bb/100 to 30bb/100.
What is a good gain/loss ratio?
A profit/loss ratio refers to the size of the average profit compared to the size of the average loss per trade. Many trading books and “gurus” advocate a profit/loss ratio of at least 2:1 or 3:1,
which means that for every $200 or $300 you make per trade, your potential loss should be capped at $100.
How many games would chase have to win in a row in order to have a 75% wining record?
So Chase will bring his win percent up to 75% if he wins the next 18 games.
What is the winning percentage of a tie?
Winning Percentage = (52 / 64) × 100 = .8125 × 100 = 81.25% If some games were drawn (i.e., both teams achieved the same score), you would use a different formula that is based on the assumption that
a tie represents 0.5 of a win. In such a scenario, you would use the following formula:
What happens when the VLT of a lens is higher?
The higher the VLT percentage is, the lighter the lens tint will be. Higher VLT percentage lenses will allow for more light to travel through the lens that will then hit the eye. Alternatively,
lenses with a lower VLT percentage will have a darker tint and will block more light from coming through to the eye.
Which is better visible light transmission or VLT?
When deciding which kind of lenses to purchase, a good place to start is with the visible light transmission (VLT). The higher the VLT percentage is, the lighter the lens tint will be. Higher VLT
percentage lenses will allow for more light to travel through the lens that will then hit the eye.
How is the winning percentage of a basketball team calculated?
When you calculate a winning percentage, you are essentially approximating a ratio of wins versus total attempts. If there were no draws, you simply divide the total number of wins by the number of
games that were played, as follows: Let’s look at an example. Let’s say that a basketball team has played 64 games, of which they have won 52. | {"url":"https://diaridelsestudiants.com/do-you-include-ties-in-winning-percentage/","timestamp":"2024-11-15T01:34:25Z","content_type":"text/html","content_length":"48353","record_id":"<urn:uuid:c60ebc6b-9e04-4a4b-934d-260d8474e369>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00690.warc.gz"} |
PPT - References PowerPoint Presentation, free download - ID:4370073
1. References • Book: Chapter 5, Image Processing, Analysis, and Machine Vision, Sonka et al, latest edition (you may collect a copy of the relevant chapters from my office) • Papers: • Harris and
Stephens, 4th Alvey Vision Conference, 147-151, 1988. • Matas et al, Image Vision Geometry, 22:761-767, 2004
2. Topics • Line detection • Interest points • Corner Detection • Moravec detector • Differential approaches • Harris corner detection • Maximally stable extremal regions
3. Line detection • Useful in remote sensing, document processing etc. • Convolve with line detection kernels
4. ? Lines and corner for correspondence • Interest points for solving correspondence problems in time series data. • Corners are better than lines in solving the above problem due to the aperture
problem • Consider that we are looking at a moving problem through a small aperture • The only component of the motion vector that is orthogonal to the line can be seen • A vertex or corner
provides better correspondence
5. ? Corners • Challenges • Gradient computation is less reliable near a corner due to ambiguity of edge orientation • Corner detector are usually not very robust. This deficiency is overcome either
by manual intervention or large redundancies. The later approach leads to many more corners than needed to estimate transforms between two two images (e.g., RANSAC for correspondence building).
6. Corner detection • Moravec detector: detects corners as the pixels with locally maximal contrast • Differential approaches: • Beaudet’s approach: Corners are measured as the determinant of the
Hessian. Note that the determinant of a Hesian is equivalent to the product of the min&max Gaussian curvatures
7. Continued … • Kitchen and Rosenfeld approach: Corners are defined as the rate of change of gradient direction multiplied by the gradient magnitude which is equivalent to the second directional
derivative in the direction orthogonal to the gradient • Using a bi-cubic facet model
8. Harris corner detector • Key idea: Measure changes over a neighborhood due to a shift and then analyze its dependency on shift orientation • Orientation dependency of the response for lines Δ Δ Δ
Δ Δ High response for shifts along the edge direction; low responses for shifts toward orthogonal direction direction Anisotropic response Δ
9. Key idea: continued … • Orientation dependence of the shift response for corners Δ Δ Δ Δ Δ High response for shifts along all directions Isotropic response Δ
10. Harris corner: mathematical formulation • An image patch or neigborhood W is shifted by a shift vector Δ=[Δx, Δy]T. • A corner does not have the aperture problem and therefore should show high
shift response for all orientation of Δ. • The square intensity difference between the original and the shifted image over the neighborhood W is • Apply first-order Taylor expansion
13. Harris matrix • The matrix AW is called the Harris matrix and its symmetric and positive semi-definite. PCA of AW gives an eigen vector and eigen value (λ1, λ2) system of the orientation
dependency of partial derivative of local intensity function. Three distinct situations: • Both λ1 and λ2 are small no edge or corner; a flat region • λi is large but λji is small existence
of an edge; no corner • Both λ1 and λ2 are large existence of a corner • Harris response function • A value of κ between 0.04 and 0.15 has be used in literature.
14. Algorithm: Harris corner detection • Filter the image with a Gaussian • Estimate intensity gradient in two coordinate directions • For each pixel c and a neighborhood window W • Calculate the
local structure matrix A • Compute the response function R(A) • Choose the best candidates for corners by selecting thresholds on the response function R(A) • Apply non-maximal suppression
18. Maximally Stable Extremal Regions (MSER) • Key idea: Consider a movie generated by thresholding a gray level image (intensity values [0,1,…,255]) at all possible thresholds starting from 0 and
ending at 255. • Initially it’s an empty image and then some dots (local minima) appear and starts growing; new dots appear and starts growing and so on; from time to time two disconnected
regions get merged and finally all regions get merged into a single component. • BUT, the important observation here is that – starting from a tiny seed area (one or a few pixels), a region
continues growing till it fills the object containing the initial seed area and then remains (almost) unchanged for quite sometime in the threshold movie until it get merged with the bigger
(generally, parent) object to which it belongs • MSER intends to capture these stable regions
20. MSER: properties • Invariance under monotonic transforms M : II of image intensities • Invariance under homeomorphic transformations (adjacency preserving) T: CC of the image space • Stability:
only extremal regions remail virtually unchanged over a threshold range are selected • Multi-scale detection: Extremal regions of all scales are detected simultaneously • The set of all MSERs are
enumerated in O(n*log log n) time, where n is the number of image pixels
21. Algorithm: MSER enumeration • Input: Image I and the Δ parameter • Output: List of nested extremal regions • For all pixels shorted by intensity • Place a pixel in the image as its tern come •
Update the connected component structure • Update the area for the effected connected components • For all connected components • Detect regions with local minima w.r.t. rate of change of
connected component area with threshold; define each such region as a MSER | {"url":"https://fr.slideserve.com/toan/references","timestamp":"2024-11-13T01:02:42Z","content_type":"text/html","content_length":"89344","record_id":"<urn:uuid:12bffe26-ccea-4411-bf1d-7f6c488b3fbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00371.warc.gz"} |
how to use the capacitor energy storage formula
Capacitor Charge & Energy Calculator ⚡
Free online capacitor charge and capacitor energy calculator to calculate the energy & charge of any capacitor given its capacitance and voltage. Supports multiple measurement units (mv, V, kV, MV,
GV, mf, F, etc.) for inputs as well as output (J, kJ, MJ, Cal, kCal, eV, keV, C, kC, MC). Capacitor charge and energy formula and equations with calculation | {"url":"https://fiziopanzio.eu/11049_05_2019.html","timestamp":"2024-11-14T09:07:47Z","content_type":"text/html","content_length":"39230","record_id":"<urn:uuid:929a4dee-f62c-41e3-95b2-01e23ef4c4d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00740.warc.gz"} |
Implementing new operators
Implementing new operators#
Users are welcome to create new operators and add them to the PyProximal library.
In this tutorial, we will go through the key steps in the definition of an operator, using the pyproximal.L1 as an example. This is a very simple operator that represents the L1 norm and can be used
to compute the proximal and its dual for the L1 norm of a vector x.
Creating the operator#
The first thing we need to do is to create a new file with the name of the operator we would like to implement. Note that as the operator will be a class, we need to follow the UpperCaseCamelCase
convention both for the class itself and for the filename.
Once we have created the file, we will start by importing the modules that will be needed by the operator. While this varies from operator to operator, you will always need to import the
pyproximal.ProxOperator class, which will be used as parent class for any of our operators:
from pyproximal import ProxOperator
After that we define our new object:
followed by a numpydoc docstring (starting with r""" and ending with """) containing the documentation of the operator. Such docstring should contain at least a short description of the operator, a
Parameters section with a detailed description of the input parameters and a Notes section providing a mathematical explanation of the operator. Take a look at some of the core operators of
PyProximal to get a feeling of the level of details of the mathematical explanation.
We then need to create the __init__ method where the input parameters are passed and saved as members of our class. Input parameters change from operator to operator, however two of them are common
to most operators and can be passed directly to the super method that invokes the __init__ of the parent class. The first one, called Op can be used to equip the proximal operator with a PyLops
linear operator in case one is required. Moreover, if we can compute the gradient of the functional associated to the proximal operator that we want our class to represent, we can pass the hasgrad
flag and choose it to be True. If this is the case, we then need to implement the grad method where the gradient is computed and returned.
In our example, as no linear operator is required in our implementation of the proximal operator of a L1 norm we will pass None. We also pass False to the hasgrad flag as we know that the L1 norm is
non-differentiable (around zero). Two additional inputs, namely sigma and g, can also be provided by the user. The first one represents a scaler applied to the norm, whilst the second is a vector to
be subtracted to x inside the norm. Note that apart from storing sigma and g inside member variables we also define two additional variables gdual and box which will be needed to implement the dual
of the proximal operator.
def __init__(self, sigma=1., g=None):
super().__init__(None, False)
self.sigma = sigma
self.g = g
self.gdual = 0 if g is None else g
self.box = BoxProj(-sigma, sigma)
The first method that we will be required to implement is the __class__ method. This method can be called to evaluate the functional that our operator implements given an input vector x, in the case
the L1 norm.
def __call__(self, x):
return self.sigma * np.sum(np.abs(x))
We can then move onto writing the proximal operator in the method prox. Such method is always composed of the inputs (the object itself self, the input vector x, and the scalar coefficience model tau
). In this case the code to be added to the forward is very simple, as the proximal of the L1 norm is a soft-thresholding to be applied to each element of x. Such a tresholding could be implemented
directly here, but as it may be useful in other cases it is implemented by an external method that we call. We finally need to return the result of this operation:
def prox(self, x, tau):
x = _softthreshold(x, tau * self.sigma, self.g)
return x
Note the @_check_tau decorator. Such decorator should be added to every proximal and dual proximal methods. This ensures that if tau is zero or negative an error will be raised before any computation
is performed.
Finally we can also implement the dual of the proximal operator. Such a method is very useful and required by the so-called primal-dual solvers. However, it is not always easy to find an analytical
expression for the dual of the proximal operator of a functional. If that is the case, we can simply omit this method. If the user calls it, an indirect implementation of it will be triggered (by the
definition of proxdual) in the base class which is based on the so-called Moreau identity. In this case we have a closed form, which corresponds the orthogonal projection of a box from -sigma to
sigma as defined in the __init__ method. Our proxdual is therefore written as:
def proxdual(self, x, tau):
x = self.box(x - self.gdual)
return x
Testing the operator#
Being able to write an operator is not yet a guarantee of the fact that the operator is correct. Testing proximal operators is however not easy. Two different scenarios can be identified:
• a closed form is available for both the proximal and the dual proximal. In this case, we can directly implement both of them and use the Moreau identity (pyproximal.utils.moreau) to validate
their correctness:
\[\mathbf{x} = \prox_{\tau f} (\mathbf{x}) + \tau \prox_{\frac{1}{\tau} f^*} (\frac{\mathbf{x}}{\tau})\]
• a closed form is not available for either the proximal or the dual proximal. In this case, we cannot validate one implementation against the other and we need to rely on ad-hoc tests to validate
the implementation that we have of either of the method. Here it is suggested to consider some edges cases where we know the expected result of applying the proximal or dual proximal to a vector
and validate that we get the numbers that we expect.
Either way, all you need to do is create a new test within an existing test_*.py file in the pytests folder (or in a new file).
Generally a test file will start with a number of dictionaries containing different parameters we would like to use in the testing of one or more operators. The test itself starts with a decorator
that contains a list of all (or some) of dictionaries that will would like to use for our specific operator, followed by the definition of the test
@pytest.mark.parametrize("par", [(par1),(par2)])
def test_L1(par):
At this point we can first of all create the operator, compute the norm, and validate the proximal and dual proximal implementations via the pyproximal.utils.moreau preceded by the assert` command.
"""L1 norm and proximal/dual proximal
l1 = L1(sigma=par['sigma'])
# norm
x = np.random.normal(0., 1., par['nx']).astype(par['dtype'])
assert l1(x) == par['sigma'] * np.sum(np.abs(x))
# prox / dualprox
tau = 1
assert moreau(l1, x, tau)
Documenting the operator#
Once the operator has been created, we can add it to the documentation of PyProximal. To do so, simply add the name of the operator within the index.rst file in docs/source/api directory.
Moreover, in order to facilitate the user of your operator by other users, a simple example should be provided as part of the Sphinx-gallery of the documentation of the PyProximal library. The
directory examples contains several scripts that can be used as template.
Final checklist#
Before submitting your new operator for review, use the following checklist to ensure that your code adheres to the guidelines of PyLops:
• you have created a new file containing one or multiple classes and added to a new or existing directory within the pyproximal package.
• the new class contains at least __init__, __call__, and prox, and optionally proxdual and grad methods.
• the new class (or function) has a numpydoc docstring documenting at least the input Parameters and with a Notes section providing a mathematical explanation of the operator
• a new test has been added to an existing test_*.py file within the pytests folder.
• the new operator is used within at least one example (in examples directory) or one tutorial (in tutorials directory). | {"url":"https://pyproximal.readthedocs.io/en/latest/adding.html","timestamp":"2024-11-13T19:12:49Z","content_type":"text/html","content_length":"39346","record_id":"<urn:uuid:88ad94e5-4ece-4885-a1fc-896edbb4054a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00816.warc.gz"} |
The Edmund R. Michalik Distinguished Lecture Series in the Mathematical Sciences
This annual intellectual highlight commemorates Edmund R. Michalik’s long history with the University of Pittsburgh and the Department of Mathematics. He received his BA in education from Pitt in
1937 and went on to receive his MS in mathematics in 1940. After he graduated, Michalik joined the Navy to serve during WWII. The day after he was discharged in 1946, he met his future wife, Martha.
He came back to teach at Pitt until 1951. Over the next few years Michalik worked for a variety of organizations, including the Army, Atlantic Research Corporation and the Department of
Mathematics at the Mellon Institute, where he was the head of applied mathematics. In 1957 he worked for PPG as the head of the applied mathematics, and later as the senior engineer and head of
computer research, when he retired in 1980. Throughout this time Michalik volunteered his time and taught as an adjunct professor in the Department of Mathematics. He was dedicated to the study of
Assistant Professor, Department of Mathematics
Fordham University
October 25, 2024
125 Frick Fine Art Building
After decades of work by mathematicians called "theoretical computer scientists", computer systems for formalization -- expressing proofs as strict chains of logical reasoning, down to the axioms --
are now powerful and user-friendly enough for modern research mathematics. And indeed, instances of hot-off-the-press theorems being formalized are starting to accumulate, including Hales' proof of
the Kepler conjecture, Clausen-Scholze's work on condensed mathematics, and Gowers-Green-Manners-Tao's proof of the Polynomial Freiman-Ruzsa conjecture.
These developments suggest new paradigms for the interaction of mathematics and computation. Computer formalization systems (called ITPs, for "interactive theorem provers") can explain to us what
our "abstract nonsense" means, and force us to explain to them what we mean by "routine" proofs; in both cases an implicit algorithm is made explicit. ITPs are the ideal vehicle for very
computational proofs -- so much so that the line between "brute force" proofs and "principled" proofs begins to blur. And seemingly in contrast to this, ITPs may nudge us toward a different style of
mathematical discovery, less reasoned and more intuitive.
Professor, Department of Mathematics
University of California Los Angeles
November 3, 2023
University Club, Ballroom B
Mason Porter is a professor in the Department of Mathematics at UCLA. He also has a secondary appointment in UCLA's Department of Sociology and is an External Professor of Santa Fe Institute. Mason
earned his B.S. in Applied Mathematics at Caltech in 1998 and his Ph.D. from the Center for Applied Mathematics at Cornell University in 2002. After postdoctoral positions at Georgia
Tech, Mathematical Sciences Research Institute, and Caltech, Mason joined the faculty in the Mathematical Institute at University of Oxford in 2007. He moved to UCLA in 2016. Mason studies many
topics in complex systems, networks, and nonlinear systems. Thus far, twenty-six PhD students have completed their doctorates under Mason's mentorship; he has also mentored many postdoctoral scholars
and undergraduate students. In 2017, Mason received the Council on Undergraduate Research (CUR) Faculty Mentoring Award (Advanced Career Category) in the Mathematics and Computer Science Division.
Mason is a Fellow of the American Mathematical Society, the American Physical Society, and the Society for Industrial and Applied Mathematics.
Dr. Michael Harris
Professor, Department of Mathematics
Columbia University
The Frick Fine Arts Building Room 125
The Langlands program originally referred to a collection of conjectures proposed by Robert Langlands over 50 years ago to connect a wide range of mathematical concepts and structures, which has been
highly influential ever since. The talk will describe some of the mathematical problems that can be solved with the help of the Langlands program, or that fit under its umbrella. Most of these
problems arise in number theory, but the applications, as well as the methods involve geometry, representation theory, and harmonic analysis; there will even be a glimpse of quantum physics. The talk
will also give a scope of how the Langlands program is constantly expanding. If you leave the lecture with the impression that the Langlands program can be about anything at all, you won't be
entirely mistaken!
Professor, Department of Mathematics
Penn State University
October 15, 2020
University Club Ballroom B
Oceanographers in the 60s conducted an ambitious experiment in which they tracked waves that were generated by large storms near New Zealand across the Pacific Ocean until they hit the beaches at
Alaska. Paradoxically, at about the same time, mathematicians in the Soviet Union, the U.S., and England separately developed mathematical models that predicted such waves to be unstable, meaning
that they could not survive to be tracked all the way across the Pacific. In the 70s experimentalists conducted laboratory experiments on these types of waves. They generated waves with a given
frequency that propagated down a wavetank, but at the end of the wavetank, the waves had a slightly lower frequency. The mathematical model did not explain this observation. In this talk, we consider
these observations and more recent ones within the framework of the mathematical model, the scalar and vector nonlinear Schroedinger equations. These partial differential equations are examples of
integrable systems; they also model phenomena observed in optics and in plasmas. They have special mathematical properties, some of which we use to determine when they are adequate models of water
waves. We use the field and laboratory experiments to guide variations of the models with the goal of accurately predicting the observations.
Director, Max-Planck Institute for Mathematics in Sciences, Leipzig, Germany
Professor of Mathematics, Statistics, and Computer Science
University of California at Berkely
February 28, 2020
University Club, Ballroom B
Two fundamental pillars of mathematics, algebra and geometry come together in the study of properties of geometric objects. For example, counting the number of intersections of two solids gives rise
to a problem in algebra. Dr. Sturmfels works at this interface, both in developing theory and in using computer-based methods, and on applications that arise in an impressive variety of fields
including statistics and computational biology. In this lecture, he illustrates these methods in the study of a famous 19th century problem, sometimes known as “Steiner’s conic problem,” on the
tangency of conics. He also discusses applications to the statistical problem of maximum likelihood estimation, which relates to computing a solution that is most likely to fit given data.
Alexander vonHumbolt Professor of Applied Mathematics
Friedrich-Alexander-Universitat, Erlanger-Nurnberg
October 10, 2019
University Club, Ballroom A
Dr. Zuazua, is Alexander von Humboldt Foundation Professor at the Friedrich Alexander University, Erlangen-Nürnberg. He is also a Full Professor of Mathematics at the Universidad Autónoma of Madrid
(part time), and leads (part time) the Chair of Applied Mathematics at the University of Deusto in Bilbao, funded by the European Research Council (ERC). Previously he was Chair Research Professor at
the Basque Center for Applied Mathematics (BCAM), of which he was also the founding Director. Dr. Zuazua is the recipient of numerous awards and honors. He was an invited Speaker at the ICM2006 in
Madrid, received the national prize in Mathematics and Computer Science research in Spain in 2007, and received a Doctor Honoris Causa degree from the Université de Lorraine in France in 2014. He was
the leader of the project ERC Advanced Grant NUMERIWAVES, 2010-2014. He is currently the leader of the project ERC Advanced Grant DYCON: Dynamic Control, 2016-2021. His research areas are Applied
Mathematics, Partial Differential Equations, Control Theory and Numerical Analysis as well as their applications to concrete problems from industry.
Thanks to the generosity of the Michalik family and in honor of Edmund R. Michalik, the Department of Mathematics this semester brought Professor Martin Nowak to Pitt for two exciting events that
provided a mathematical connection with Pitt's "Year of the Healthy U" theme. Nowak is a Professor of Mathematics and Biology and the Director of the Program for Evolutionary Dynamics at Harvard
University. He is a world expert on evolutionary game theory (e.g, including the evolution of cooperation and other altruistic-seeming behaviors), population structures, cancer dynamics and
treatment, and a range of other topics. He has written several hundred articles and has published books on virology, evolutionary dynamics, and cooperation.
On Wednesday, March 21st, Dr. Nowak was joined by Dr. Gilles Clermont (Pitt Critical Care Medicine and long-time collaborator of several current Pitt Mathematics faculty), Dr. David Galloway (Pitt
Public Health Dynamics Lab), and Dr. Wilbert Van Panhuis (Pitt Epidemiology/Biomedical Informatics) for a panel on "Computational Methods for Fostering a Healthy Community". This event
John D. MacArthur Professor of Mathematics, University of Wisconsin at Madison
February 28, 2017
Ballroom, O'Hara Student Center
New York Times bestselling author Jordan Ellenberg has been writing for a general audience about math for more than fifteen years and his work has appeared in the New York Times, the Wall Street
Journal, the Washington Post, Wired, The Believer, and the Boston Globe. He is author of How Not to Be Wrong: The Power of Mathematical Thinking (Penguin Press, 2014).
Prof. J. Tinsley Oden
Foundations of Predictive Computational Science:Selection and Validation of Models of Complex Systems in the Presence of Uncertainty
September 23, 2016
Ballroom, O'Hara Student Center
J. Tinsley Oden is Associate Vice President for Research, Cockrell Family Regents' Chair in Engineering No. 2, Peter O'Donnell Jr. Centennial Chair in Computing Systems, and the founding Director of
the Institute for Computational Engineering and Sciences at The University of Texas at Austin. His research is on the mathematical theory and implementation of numerical methods applied to problems
in linear and nonlinear solid and fluid mechanics. Dr. Oden has authored over 600 scientific publications and has authored or edited 56 books. He is a recipient of numerous awards, including a
member of the U.S. National Academy of Engineering, a Fellow of the American Academy of Arts and Sciences, an Honorary Member of the American Society of Mechanical Engineers, the Theodore von Karman
Medal, the John von Neumann medal, and the Newton/Gauss Congress Medal.
Prof. Martin Hairer, University of Warwick
Taming infinities
December 2, 2015
Ballroom, O'Hara Student Center
Martin Hairer is an Austrian mathematician working in the field of stochastic analysis, in particular stochastic partial differential equations. He is Regius Professor of Mathematics at the
University of Warwick,^ having previously held a position at the Courant Institute of New York University.^ He was awarded the Fields Medal in 2014.
Prof. John Ball, Oxford University
Defects in Materials and their Mathematical Description
March 17, 2014 at 4 p.m.
Ballroom A, University Club at the University of Pittsburgh
John Ball is Sedleian Professor of Natural Philosophy at the University of Oxford and a Fellow of the Queen's College. He was President of the International Mathematical Union from 2003-06. His
research interests include elasticity, the mathematics of solid and liquid crystals, the calculus of variations, and infinite-dimensional dynamical systems. He is a foreign member of the French
Academy of Sciences and the Norwegian Academy of Science and Letters, and is a fellow of the Royal Societies of London and Edinburgh, and of the American Mathematical Society, among many other honors
and prizes.
Prof. Frank Morgan
Soap Bubbles, Tilings, and Other Partitioning Problems
March 22, 2013 at 4 p.m.
Ballroom B, University Club at the University of Pittsburgh
Abstract: The Ancient Greeks proved that the circle is the least-perimeter way to enclose given area.Similarly the round soap bubble provides the least-perimeter way to enclose a given volume of air,
although that was not proved until 1884 by Schwarz. Similarly the double bubble that forms when two soap bubbles come together is the least-perimeter way to enclose and separate two given volumes of
air, although that wasn't proved until 2000 by Hutchings, Morgan, Ritoré, and Ros.
Lord Kelvin sought the least-perimeter way to divide all of space into unit volumes, and his conjecture stood for 100 years, until Weaire and Phelan found a better way in 1994. Whether their new
candidate is best remains open today. Even the least-perimeter way to divide the plane into unit areas, using the bees' hexagonal honeycomb tiling, though conjectured by the Ancient Greeks, was not
proven until 1999 by Hales. The most efficient tiling by pentagons remains open.
In many simple nonEuclidean possible universes, even the ideal shape for a single soap bubble remains open.
Prof. Shing-Tung Yau
October 5, 2012
Shing-Tung Yau has made fundamental contributions to differential geometry which have influenced a widerange of scientific disciplines, including astronomy and theoretical physics. Yau’s first major
contribution to differential geometry was his proof of the Calabi conjecture, which concerns how volume and distance can be measured not in four, but in five or more dimensions. In 1979 Yau and
Richard Schoen proved Einstein’s positive mass conjecture by applying methods devised by Yau. The proof was based on their work with minimal surfaces. In 1982 Yau was awarded the Fields Medal, the
highest award in mathematics, and in 1994 he shared with Simon Donaldson of Oxford University the Crafoord Prize of the Royal Swedish Society, in recognition of his “development of nonlinear
techniques in differential geometry leading to the solution of several outstanding problems.” In 2010 Yau published the book The Shape of Inner Space.
Sir Roger Penrose
"Can We See Through the Big Bang into Another World?"
January 24, 2011
Prof. Sir Roger Penrose has made many contributions to the fields of Mathematics and Physics. He proved that singularities (such as black holes) could be formed from the gravitational collapse of
immense, dying stars and invented spin networks which later came to form the geometry of spacetime in loop quantum gravity. Prof. Penrose is also well known for his 1974 discovery of Penrose tilings,
which are formed from two tiles that can only tile the plane non-periodically, and are the first tilings to exhibit fivefold rotational symmetry. He is the recipient of many awards and honors,
including a Royal Medal from the Royal Society and a Wolf Prize, which he shares with Stephen Hawking. Prof. Penrose’s book "The Road to Reality" gives a comprehensive guide to the laws of physics.
His latest book is "Cycles of Time."
Prof. Luis A. Caffarelli
"Non linear, geometric homogenization"
March 26, 2010
Professor Luis A. Caffarelli holds the Sid Richardson Chair in Mathematics at the University of Texas at Austin.The focus of his research has been elliptic nonlinear partial differential equations
and their applications. Some of his most significant contributions are the regularity of free boundary problems and solutions to nonlinear
elliptic equations, optimal transportation theory and, more recently, results in the theory of homogenization.
Professor Caffarelli is a member of the National Academy of Sciences. He has been awarded Doctor Honoris Causa from l'Ecole Normale Superieure in Paris, Universidad Autónoma de Madrid, and
Universidad de la Plata in Argentina. He received the Bôcher Prize in 1984 and the prestigious Rolf Schock Prize in Mathematics of the Royal Swedish Academy of Sciences in 2005. He was recently
awarded the Leroy P. Steele Prize for Lifetime Achievement in Mathematics.
Tony F. Chan
"Images, PDEs and Wavelets"
March 20, 2009
Dr. Chan's research interests include mathematical image processing and computer vision, VLSI physical design and computational brain mapping. He is a Fellow of the American Association for the
Advancement of Science. Dr. Chan has published over 200 refereed papers and has mentored over 25 PhD students and 15 postdoctoral fellows. He is a co-founder of the Institute for Pure and Applied
Mathematics at UCLA, established to promote collaborations between the mathematical sciences and the general scientific and engineering disciplines. Dr. Chan currently serves as Assistant Director of
the Directorate for Mathematical and Physical Sciences at the National Science Foundation. The MPS is the largest Directorate at NSF with an annual budget of just over $1B.
View more information (PDF) »
Tony F. Chan's Home Page »
Neil J. A. Sloane
"The Online Encyclopedia of Integer Sequences: Solved and Unsolved Problems"
April 4, 2008
Neil Sloane is a Fellow at AT&T Shannon Labs in Florham Park, NJ. He
is a member of the National Academy of Engineering, an IEEE Fellow, and recipient of the IEEE Hamming Medal and the MAA Chauvenet Prize. He is the author or co-author of numerous books, including
“The Theory of Error-Correcting Codes” (with F. J. MacWilliams) and “Sphere Packing, Lattices and Groups” (with J. H. Conway).
View more information (PDF) »
Neil J. A. Sloane's Home Page »
Dr. Cathleen Morawetz
"Collisionless Shocks in Space"
April 6, 2007
Professor Cathleen Morawetz is a Fellow of the American Association for the Advancement of Science, the American Academy of Arts and Sciences, and National Academy of Sciences. She was Director of
the Courant Institute of Mathematical Sciences, and the President of the American Mathematical Society. She receieved the National Medal of Science in 1998, and the Lifetime Acheivement award from
the Americam Mathematical Society in 2004.
Robert F. Engle, Ph.D., Nobel Laureate
April 7, 2006
Dr. Engle was awarded the 2003 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel for "methods of analyzing economic time series with time-varying volatility (ARCH)."
Dr. Engle received his PhD in economics from Cornell University in 1966. His work is distinguished by exceptional creativity in the empirical modeling of dynamic economic and financial phenomena. He
is a renowned expert in Financial Economics, Time Series Analysis, Volitility and Risk Management and Empirical Market Microstructure. | {"url":"https://www.mathematics.pitt.edu/about/edmund-r-michalik-distinguished-lecture-series-mathematical-sciences","timestamp":"2024-11-08T05:14:10Z","content_type":"text/html","content_length":"119589","record_id":"<urn:uuid:e17dc109-66f5-45ca-ba72-cb132e7e9b3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00491.warc.gz"} |
HYPERSIM Documentation
The distributed parameters line (DPL) theory is used to represent a half of the classic constant parameter (CP) model [1]. Overall, the CP model assumes that the line parameters R, L and C are
independent of the frequency effects caused by the skin effect on phase conductors and on the ground. The model considers L and C to be distributed (ideal line) and R to be lumped at three places (R/
4 on both ends and R/2 in the middle). The shunt conductance G is taken as zero.
Two half CP components must be connected through transceiver elements (or real-time simulator I/Os) to represent a transmission line. Line parameters must be the same in both components. The
implementation of the half CP follows the same formulation from the standard CP line [1]. However, note that with the half CP components, the propagation delay of the line is the sum of what is
accounted in a buffer inside the component and what is caused by the actual delay outside the components. The latter is referred as "extra delay" in the form of the model. For the digital simulation
of the model on one simulator, the actual delay is caused by the transceiver. For the simulation on 2 simulators, the actual delay includes the delay in IO drive.
The single-phase version half CP, 1-ph is extended to the three-phase line by using a modal transformation to decouple the equations from phase domain to modal domain. The decoupled circuits are
solved separately and transformed back to the phase domain. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used. For an
untransposed line (3 distinct modes) an input matrix is used.
Name Description Unit Variable = {Possible Values}
Length The length of the line km Length = {'1e-12, 1e12'}
Transposition (Untransposed/Transposed)
Continuously transposed No {0} Untransposed line transp = { 0, 1}
Yes {1} Transposed line
R Per unit length resistance for each phase (mode) Ω/km Resistance = {'0 1e12'}
L Per unit length inductance for each phase (mode) H/km Inductance = {'1e-12, 1e12'}
C Per unit length capacitor for each phase (mode) F/km Capacitance = {'1e-12, 1e12'}
Transformation matrix between mode current and phase current ([Iphase] = [Ti] x [Imode]); not used in the case of transposed line.
Transformation matrix (Ti) Ti = { [-1e64, 1e64] }
For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used.
The extra delay variable is an integer that refers to the number of delays produced outside the half CP components. For the digital
Extra delay simulation of the model on one simulator, the extra delay is caused by the transceiver. For the simulation on two simulators, the extra extradelaynb = { [0, 200] }
delay includes the delay in IO drive.
Ports, Inputs, Outputs and Signals Available for Monitoring
This component supports a single-phase transmission line
Name Description
net(a,b,c) Power network connection of phases (a,b,c) of one side of the line
Name Description
hi_(a,b,c) Historic current of phases (a,b,c) FROM the other side of the line
Name Description
ho_(a,b,c) Historic current of phases (a,b,c) TO the other side of the line
Vt_(a,b,c) Terminal voltage of phases (a,b,c) in V
ols_(a,b,c) Terminal current of phases (a,b,c) in A
Single-phase model
The half CP parameters are calculated at a given frequency; thus, it is considered as a frequency independent line model. This model is less accurate than frequency-dependent line and cable models.
However, it can be successfully used to analyze cases with limited frequency dispersion.
The half CP model is based on the formulation of the classic CP line model, which neglects the frequency dependence of parameters and first assumes a lossless line. The losses are included at a later
stage. The following figure shows the equivalent circuit representation of the EMT-type transmission line model.
The lossless single-phase transmission line is described by the following main equations:
where and are history currents defined as:
where, and are the nodal voltages; and are the injected current at both ends of the line; is the characteristic impedance and is the propagation time delay, respectively defined as:
where is the length of the line, and and are the inductance and capacitance per unit length of the line, respectively.
Note that the time-domain model described above creates a decoupling effect on the interconnected network. It is mentioned that the equation system provides an exact solution only when the
propagation time is an integer multiple of the simulation time step , i.e., . Therefore, a linear interpolation is used when .
Inclusion of losses
To include the losses , the line is divided into two equal lossless models of halved propagation time. Then, the total is lumped at three places (line ends and line middle ) as shown in the following
The resulted lossy line equivalent model to be implemented is given by:
History current buffer
The half CP line is used to represent one Norton equivalent circuit of the lossy equivalent line. Therefore, two half CP model must be linked to each other to represent a transmission line. Both half
CP components must be set with the same line parameters and connected through transceiver components. This connection is used to interchange the history current information between both ends of the
line. That is,
Then, the output history current from the end becomes the input history current of the end and vice versa. In the implementation of the model, a history buffer is kept and rotated for calculating the
history current sources. The buffer length depends on the propagation delay and the simulation time step. The propagation delay must be greater to the integration time-step.
Note that the propagation delay of the line is the sum of what is accounted in the buffer and what is caused by the actual delay outside the components. The latter is referred as "extra delay", which
must be indicated in the form of the model. For the digital simulation of the model on one simulator, the actual delay is caused by the transceiver. For the simulation on 2 simulators, the actual
delay includes the delay in IO drive.
Three-phase model
The single-phase model is extended to the three-phase line by using a modal transformation to decouple the equations from phase domain to modal domain. The hatted vector are modal quantities:
where and are the series impedance and shunt admittance of the line, respectively. The transformation matrices are given from the following eigenvalue problem:
where is a diagonal matrix of the eigenvalues of the product . Also, the transformation matrices follow the relation:
For an untransposed line there are as many distinct modes as phases. In the half CP model, a constant and real transformation is used, which is calculated at a given model frequency. The LineData
model can be used to obtain this matrix. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used.
To find a solution in the three-phase system, the decoupled circuits are solved separately and transformed back to the phase domain. See the following figure:
Two half CP components must be connected through transceiver elements to represent a transmission line. Line parameters must be the same in both components. Next figures show how to build a
transmission line using 1 and 3 transceivers.
1. "H. W. Dommel, "Digital computer solution of electromagnetic transients in single and multiphase networks," IEEE Trans. Power App. Syst., vol. pas-88, pp. 388-99, 04/ 1969."
The distributed parameters line (DPL) theory is used to represent a half of the classic constant parameter (CP) model [1]. Overall, the CP model assumes that the line parameters R, L and C are
independent of the frequency effects caused by the skin effect on phase conductors and on the ground. The model considers L and C to be distributed (ideal line) and R to be lumped at three places (R/
4 on both ends and R/2 in the middle). The shunt conductance G is taken as zero.
Two half CP components must be connected through transceiver elements (or real-time simulator I/Os) to represent a transmission line. Line parameters must be the same in both components. The
implementation of the half CP follows the same formulation from the standard CP line [1]. However, note that with the half CP components, the propagation delay of the line is the sum of what is
accounted in a buffer inside the component and what is caused by the actual delay outside the components. The latter is referred as "extra delay" in the form of the model. For the digital simulation
of the model on one simulator, the actual delay is caused by the transceiver. For the simulation on 2 simulators, the actual delay includes the delay in IO drive.
The single-phase version half CP, 1-ph is extended to the three-phase line by using a modal transformation to decouple the equations from phase domain to modal domain. The decoupled circuits are
solved separately and transformed back to the phase domain. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used. For an
untransposed line (3 distinct modes) an input matrix is used.
Name Description Unit Variable = {Possible Values}
Length The length of the line km Length = {'1e-12, 1e12'}
Transposition (Untransposed/Transposed)
Continuously transposed No {0} Untransposed line transp = { 0, 1}
Yes {1} Transposed line
R Per unit length resistance for each phase (mode) Ω/km Resistance = {'0 1e12'}
L Per unit length inductance for each phase (mode) H/km Inductance = {'1e-12, 1e12'}
C Per unit length capacitor for each phase (mode) F/km Capacitance = {'1e-12, 1e12'}
Transformation matrix between mode current and phase current ([Iphase] = [Ti] x [Imode]); not used in the case of transposed line.
Transformation matrix (Ti) Ti = { [-1e64, 1e64] }
For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used.
The extra delay variable is an integer that refers to the number of delays produced outside the half CP components. For the digital
Extra delay simulation of the model on one simulator, the extra delay is caused by the transceiver. For the simulation on two simulators, the extra extradelaynb = { [0, 200] }
delay includes the delay in IO drive.
Ports, Inputs, Outputs and Signals Available for Monitoring
This component supports a single-phase transmission line
Name Description
net(a,b,c) Power network connection of phases (a,b,c) of one side of the line
Name Description
hi_(a,b,c) Historic current of phases (a,b,c) FROM the other side of the line
Name Description
ho_(a,b,c) Historic current of phases (a,b,c) TO the other side of the line
Vt_(a,b,c) Terminal voltage of phases (a,b,c) in V
ols_(a,b,c) Terminal current of phases (a,b,c) in A
Single-phase model
The half CP parameters are calculated at a given frequency; thus, it is considered as a frequency independent line model. This model is less accurate than frequency-dependent line and cable models.
However, it can be successfully used to analyze cases with limited frequency dispersion.
The half CP model is based on the formulation of the classic CP line model, which neglects the frequency dependence of parameters and first assumes a lossless line. The losses are included at a later
stage. The following figure shows the equivalent circuit representation of the EMT-type transmission line model.
The lossless single-phase transmission line is described by the following main equations:
where and are history currents defined as: | {"url":"https://opal-rt.atlassian.net/wiki/spaces/PDOCHS/pages/149620988/Half+CP+3-ph?atl_f=content-tree","timestamp":"2024-11-06T11:24:28Z","content_type":"text/html","content_length":"1050365","record_id":"<urn:uuid:9858e587-6ef9-4e65-86e7-7b77080246f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00243.warc.gz"} |
Putting Consciousness into the Equations: A Scientific REvolution
Many years ago, when I was a PhD candidate at Johns Hopkins University in Baltimore, as part of course work I read Thomas Kuhn's works. He made the point that science advances in two ways: Most of
the time, it is by steady growth, bit by bit, as many scientists toil away digging out details. But occasionally, there is a revolution, a new sweeping insight, when one or two scientists see
something no one else has noticed, something outside the box. The history of science is studded with a precious few of these revolutions: Archimedes, Pythagoras, Descartes, Newton, Leibniz, Einstein,
Heisenberg and Bohr, the names of those who created such revolutions are well known. New mathematics and new theories blossomed from a few exceptional minds. But there have been no scientific
revolutions since 1935 when Relativity and Quantum Mechanics came on the scene. When scientists talk about a "breakthrough" these days, most often it is just the finding of something like the Higgs
Boson, an accomplishment, to be sure, but hardly a scientific revolution, just confirmation of something the Standard Model predicted; nothing really new.
Now there IS new mathematics and a new theory! In TDVP, Dr. Neppe and I have put consciousness, mind and Spirit into the equations! Here are some details:
What we know about the elements of the Periodic Table is almost entirely based on experimental data from investigations of just four distinct elementary entities, known as the photon, the electron,
the up-quark, and the down-quark. Table One lists these elementary distinctions with two measurable parameters that characterize them: charge and mass. In addition, the charge and mass data for the
proton, composed of two up-quarks and a down-quark, and the neutron, composed of one up-quark and two down quarks, as basic components of atomic structure, are included in the table.
TABLE ONE
Elementary Particle Charge & Mass
│ Particle │Symbol│ Charge │ Mass (m) │
│ │ │ │ │
│ │ │(Coulomb)│ (MeV/c^2) │
│ Photon │ Ɣ │ 0 │ 0 │
│ electron │ e │ -1 │ 0.511 │
│ up quark │ u │ +^2⁄[3] │ 1.5–3.3 │
│down quark │ d │ −^1⁄[3] │ 3.5–6.0 │
│ proton │ P^+ │ +1 │6.5 – 12.6 │
│ Neutron │ N^0 │ 0 │8.5 – 15.3 │
Except for the electron, the data for the mass of the particles in Table One are presented as ranges of values. The masses of elementary particles, i.e. the up and down quarks, are indirectly
determined as energy equivalents from particle collider detector and collector data. They are non-integer decimal fractions because the standards for the units of measurement (MeV/c^2) do not happen
to be integer multiples of the smallest possible mass. In the ultimately smallest possible quantized units, the numerical values of the actual masses of these particles must be integers, so we are
justified in normalizing the data for purposes of comparison and calculation.
The normalized values in Table Two are obtained by simply taking the smallest elementary mass, that of the electron, as unitary. Then to convert the mass of the up quark and down quark into multiples
of that standard unit, we divide the average of each particle’s range of empirical data by 0.511and round the results to the nearest whole number. Even if the basic unit we have derived in this way
is not be the actual smallest possible unit, it will be a multiple of the real quantum unit, and the normalized values will reflect the relative proportionality of the actual masses of the particles.
Because charge is a product of spin, we have also normalized it to avoid fractions by simply taking the charge of the electron as - 3. This normalizes the charge of the up quark to + 2 and the down
quark to - 1. Again, as with mass, we are justified in normalizing the measure of charge because the standard unit, the Coulomb, is not necessarily an integral multiple of the actual quantum unit of
charge, which may be either positive or negative. The balance, or zero sum of + and – charges in the decay process (called parity) reflects the operation of the Law of Conservation of Energy.
TABLE TWO
Normalized Units
│ Particle │Symbol│Charge│Mass│
│ Photon │ │ 0 │ 0 │
│ │ Ɣ │ │ │
│ electron │ e │ - 3 │ 1 │
│ up quark │ u │ + 2 │ 5 │
│down quark │ d │ − 1 │ 9 │
│ proton │ P^+ │ + 3 │ 19 │
│ Neutron │ N^0 │ 0 │ 23 │
How do these different particles form?
Consider a state, perhaps in the early, high-energy, high-temperature universe, when there was a great abundance of free quarks and decaying and decayed quarks, - i.e. less massive quarks and
electrons. There would also have been other short-lived energetic by-products of quark collisions and decay like photons and neutrinos, radiating off to infinity, and perhaps there were other types
of free particles like those created in experimental particle colliders, but we are focusing here on the decayed and decaying quarks destined to form Hydrogen atoms and the other atoms of the
periodic table.We have presented relevant evidence and arguments for the following ideas:
1. Elementary particles are created as the result of the interaction of the three universal processes of expansion, contraction and rotation or spin. Their cause is thus triadic.
2. Reality exists within at least nine finite, sequentially-nested existential dimensions.
3. We are only partially aware of five of these dimensions through our physical senses: three of space, one of time, and one of consciousness.
4. The processes affecting the creation and combination of elementary particles to form meaningful structures are rooted in the dimensionometric forms of nine finite dimensions and one or more
transfinite dimensions. The mathematical expression of this dimensionometric form is Σ^n[i=1] (X[n])^m = Z^m (to be explored in more detail below).
5. These particles are triadic in nature, comprised of a universal substance which manifests as matter, energy and consciousness interacting in the nine-dimensional domain.
The Origin of the Particles that form the Elements of the Periodic Table
How do particles form in the first place? Consider an elementary contraction in the substance of reality characterized as the drawing of a distinction. Call that distinction D[1]. We have posited
that it is comprised of mass, energy and consciousness. As demonstrated in our discussion of intrinsic spin and the Cabibbo angle, this elementary distinction is rapidly spinning. With no external
influence, and therefore no preferred reference frame, the distinction spins in nine dimensions, and each plane of rotation will cause it to resist movement like a spinning gyroscope. This resistance
to movement, or inertia, is interpreted as mass. Since energy is quantized, the spin in each of the nine planes of rotation contributes one unit of inertia, and the particle will possess nine units
of inertia. If we accept that these units are equivalent to units of mass, and normalized as we’ve done above, then such a particle is equivalent to the down quark in Table Two.
Under the entropic expansive action characterized by the Second Law of Thermodynamics, down quarks decay into up quarks, releasing a photon and a neutrino. This process, documented in many
experiments, conforms to the Law of Conservation of Mass and Energy. Nothing is lost or destroyed in the process; a portion of the substance of the particle of rotational inertia simply changes from
one form (mass) to another: energy. Table Three below illustrates how an elementary distinction, D[1], recognized as the down quark, d, in experimental observations, decays to form other less massive
particles. These new particles formed by natural entropic decay have the exact normalized inertial masses that are characteristic of the other two elementary particles, the down quark and the
electron, the particles that make up all of the elements of the Periodic Table.
TABLE THREE
Natural Decay Path of Elementary Particle Distinctions
│Elementary │Mass in Normalized Units│Units Emitted as Energy^*│New Mass in Normalized Units │New Entity│
│ │ │ │ │ │
│Distinction│ │ │ │ │
│ D[1 =] d │ 9 │ 4^ │ 5 │ u │
│ D[2 =] u │ 5 │ 4 │ 1 │ e │
│ D[3 =] e │ 1 │ 1 │ 0 │ Ɣ │
^* Energy emitted is in the form of photons, Ɣ, and neutrinos. v[e][ ], one photon plus one neutrino = 4 normalized units. The energy of neutrinos and photons can vary, but, since energy is
quantized, their energy can only consist of integer multiples of normalized units. There are some indications that neutrinos may have a miniscule amount of mass, so the neutrino emitted may have 1
unit of mass and one or two units of energy. The photon’s mass is zero, and its energy is proportional to its wave length, so the photons emitted in the decay process may have a wave length
reflecting the energy of one or two units.
Notice that the four units emitted, identified as mass in the down quark, are either mass or energy in the photon and neutrino. This indicates an equivalence between normalized mass units and energy,
and more importantly, a transformation of the measurable aspect of the universal substance from mass to energy. We know that the transformation relationship between mass and energy is E = mc^2. Since
c^2 is a constant, we may normalize the units of energy into units equivalent with our normalized mass units very easily as follows: The standard unit used to measure the energy of elementary
particles is the MeV(one million electron volts) the standard unit used in measuring the mass of the particles in Table One is one million electron volts divided by the speed of light squared (MeV/c^
2), equivalent to 1.782662×10^−36 kg., a very small fraction of a kilogram. So X units of mass in our normalized units in Table Two = X ∙ Mev/c^2. That occurs as a result of the Substituting m = X ∙
Mev/c^2 into and E = mc^2, we get E = X ∙ (Mev/c^2 )^ ∙ c^2^ = X ∙ Mev. Thus the units in Table Three are normalized equivalence units measuring both mass and energy. Any given number, X, of
normalized units of mass in this table^ is equivalent to X normalized units of energy.
We have also posited that the substance of reality is not just mass and energy, i.e. binary in mode, but triadic, existing in three forms: mass, energy and consciousness. Since mass and energy are
measurable in normalized units, it is reasonable to expect that consciousness might also be. If, e.g., like mass and energy, consciousness is quantized, and each unit of consciousness is equivalent
to a constant multiple of energy units, then consciousness can also be also be measured in multiples of these normalized units of equivalence, and the processes that form the elements of the Periodic
Table can be described and analyzed using them. Even though we have not yet defined what a unit of consciousness might consist of, we may be able to define it indirectly relative to energy and mass
in terms of the equivalence units. Extending the logic of E = mc^2, the mathematical relationship between mass, energy and consciousness, C, is probably of the form: C = E(∆t/∆ C)^I where ∆t/∆ C is
the ratio of the minimum increments at the border of the T and C domains just as c is the ratio at the border of the S and T domains, and I is > 2. Substituting, we have: C = m^2 (∆t/∆ C)^I.
Notice that no units of consciousness appear in the decay process depicted in Table Three because it is a natural entropic process. So why don’t all elementary distinctions simply decay in these
three quick steps into photons and neutrinos that expand to infinity, resulting in a swift return to a state of maximum entropy, a state where there are no distinctions in the substance of reality?
Regardless of how particles originate, something happens to counteract the action of the Second Law of Thermodynamics. What happens to perpetuate negative entropy? The answer lies in the conveyance
of the logic of the C-substrate (dimensions 7, 8 and 9 of the nine dimensional domain of reality) into the 3S-1T domain of observation, by the intrinsic form of the dimensionometric domains
represented mathematically by the equation Σ^n[i=1] (X[n])^m = Z^m and by the action of one or more units of consciousness to organize mass and energy into stable structures reflecting the logic of
the conscious substrate. In order for quarks to combine to form the stable sub-atomic particles we call protons and neutrons observable in the 3S-1t domain, they must meet the requirements of the
Conveyance Expression equation when m = 3 and n = 2 as integer multiples of normalized units of mass, energy and consciousness. This is where Fermat’s Last Theorem enters the picture.
Fermat’s Last Theorem and the Combination of Quantum Particles
Consider the combination of two elementary particles to form a new particle. This may be modeled by the Conveyance Expression when n = 2 and m = 3.
With n = 2 and m = 3, the expression Σ^n[i=1] (X[n])^m = Z^m yields the equation (X[1])^3 + (X[2])^3 = Z^3.
X[1] and X[2] represent the number of normalized units making up the particles, i.e. quarks, which combine to form the proton, P^+ and the neutron, N^0. (X[1])^3 and (X[2])^3 represent the volumes of
two combining particles and Z^3 represents the volume of the particle formed in the combination. In nine dimensions, at the sub-quark level, whether mass, energy or consciousness, the numerical
measures of the spinning entities in normalized equivalence units are integers and dimensionometrically equivalent. They are therefore called Triadic Rotational Units of Equivalence (TRUE).
Triadic Rotational Units of Equivalence, or TRUE units, for short, are the Calculus of Distinction equivalents of the infinitesimals of the Calculus of Newton and Leibniz. The difference, and it is a
very important one, is that TRUE units are finite and integral. While the value of the differentiation variable of a function in Newtonian Calculus may approach zero infinitely closely, the Calculus
of Distinctions numerical values of both content and extent variables of the finite distinctions of mass, energy and consciousness are quantized and thus cannot be smaller than one TRUE unit. Thus
the TRUE unit is the bottom, or limit of infinite descent for all variables. Because elementary particles are rotating extremely rapidly, regardless of the probabilistic distribution of density (such
as that demonstrated in our analysis of the electron)^6, a TRUE unit occupies a perfectly symmetrical, or spherical volume.
Using the axioms presented above, and TRUE units, we will proceed to describe the processes that lead to the formation of the Hydrogen atom and the other elements of the Periodic Table. The values of
the mass of the elementary entities, in multiples of the TRUE unit, are determined by normalization of experimental data as described above; the values of the energy of the entities, also in
multiples of the TRUE unit, are calculated using the established mathematical relationship between mass and energy (E = mc^2); and the values of the measures of the consciousness of the elementary
entities in multiples of the TRUE units are determined by application of the Conveyance Equation and the assumption that a mathematical relationship, analogous to E = mc^2 exists between energy and
At the quantum level, to be stable quantum particles, existing as finite three-dimensional distinctions, each of these volumes must be equivalent to either the volume of a TRUE unit, or multiples of
the volume of the TRUE unit. This means that X[1], Y[2] and Z must be integers.
Fermat’s Last Theorem tells us there are no integer solutions for this equation, which means that no two particles consisting of TRUE units, or integral multiples of TRUE units, can combine to form a
new symmetrical entity. Such asymmetrical combinations of rapidly spinning entities will tumble or spiral, especially under the influence any external force, and will thus be far less stable than
symmetric forms.
However, when n = m =3, the expression yields the equation
(X[1])^3 + (X[2])^3 + (X[3])^3= Z^3,
which does have integer solutions. The first one (with the smallest integers) is
3^3 + 4^3 + 5^3 = 6^3
It is important to recognize that the equations produced by Σ^n[i=1] (X[n])^m = Z^m when n and m, and the X[i] and Z are integers are exact expressions of the form of the logical structure of the
C-substrate as it is conveyed to the 3S-1t domain. For this reason, we will call this expression the Conveyance Expression. This expression, generalizing the summation of n finite m-dimensional
distinctions, and the equations it generates when all variables are integers, including the equations of the Pythagorean Theorem and Fermat’s Last Theorem, prove to be indispensably useful in the
mathematical analysis of the combination of elementary particles.
The simplest symmetric form in three-dimensional space is the sphere, and as noted above, we can assume that the TRUE unit of substance is spherical. If the particles are also spherical, their
volumes are
4/3 π r[1]^3
4/3 π r[2]^3,
4/3 π r[3]^3
, where
r[1], r[2]
are the radii of the particles. But, since
the volumes of the particles are integral multiples of
TRUE unit, r[1]
must be integer multiples of the radius of the TRUE unit. So let
r[1]= X[1]R[T]
r[2 ]= X[2]R[T]
, and
r[3 ]= X[3]R[T]
X[1], X[2 ]
are integers and
is the radius of the TRUE unit.
The Conveyance Equation representation of the combination of the three particles becomes:
4/3 π (X[1]R[T]) ^3 + 4/3 π (X[2] R[T]) ^3 + 4/3 π (X[3]R[T]) ^3
The newly combined particle is also spherical, represented by the expression 4/3 π (ZR[T]) ^3, where, Z is necessarily an integer, and we have:
4/3 π (X[1]R[T]) ^3 + 4/3 π (X[2] R[T]) ^3 + 4/3 π (X[3]R[T]) ^3 = 4/3 π (ZR[T]) ^3
Dividing both sides of the equation by all of the common constant factors: 4/3, π and (R[T])^3, we have:
(X[1])^3 + (X[2])^3 + (X[3])^3= Z^3, where the X[i] and Z are integers representing the number of elementary particles in each term.
Since spinning elementary particles are symmetric, and multiples of TRUE units, which are also symmetric, the fact that this equation has integer solutions, while the equation (X[1])^3 + (X[2])^3 = Z
^3 does not, tells us that for the elementary particles in Table Three to combine to form the most stable, symmetric compound distinctions, three particles, not two, must combine.
Note that this conclusion is independent of the actual shape of the combining particles and is even independent of the size and substance of the TRUE unit. As long as the particles have the same
symmetrical form, the shape factor, in this case, 4/3 π (R[T] )^3, cancels out. They could, e.g., be any of the regular polyhedrons like tetrahedrons, with four equilateral triangular sides,
hexahedrons (better known as cubes), with six square sides, octahedrons, etc.
Just as the intrinsic structure of dimensional domains of three or more dimensions causes the combination of two elementary particles to be asymmetric, it allows the combination of three particles to
be symmetric and very stable.
We will limit the scope of the remainder of this mathematical description to the five particles that make up the elements of the Periodic Table, namely the electron, the up quark, the down quark, and
the two composite particles, the proton and the neutron. This set of particles form a finite physical system in the four dimensional domain of space-time. This means that these elementary particles
are subject to the Second Law of Thermodynamics, which is expressed through the process of universal expansion toward maximum entropy; but this tendency toward maximum entropy is opposed by the
processes of contraction of the substance of reality into finite distinctions and high-velocity rotation and spin. We postulate that this substance of reality is triadic in nature, composed of mass,
energy and consciousness. With the introduction of the TRUE unit as the Calculus of Distinctions basic unitary distinction, and nine-dimensional spin as the dynamic nature of distinctions of matter
and energy supporting the logic of consciousness, we are re-integrating our understanding of physical reality with the awareness of the conscious substrate as the mathematically logical matrix from
which physical reality originates.
Albert Einstein and Hermann Minkowski began this re-integration of the scientific description of reality with consciousness by recognizing the geometric nature of space and time as dimensional,
establishing the concept of space-time^4, and the nature of matter and energy as two different forms of the same substance. We are extending this integration to include the five additional finite
dimensions indicated by the mathematics of Dimensional Extrapolation, and including consciousness as the third aspect of the substance of reality. Even though Einstein coined the new term
“space-time” describing the new concept of a four-dimensional geometric domain, and established the mathematical equivalence of matter and energy with E = mc^2, he introduced no new terminology for
the generic substance of reality. We will use the terms “essential substance” or “essence” to indicate the substance of reality manifesting triadically as matter, energy and consciousness.
Consistent with decay from the strange quark, stabilization through the Conveyance Equation and the participation of TRUE units of consciousness, the following tables describe the electron, up
quarks, down quarks, protons, and neutrons. The elements of the Periodic Table, derived from these tables tell us a lot about how the elements are formed, how consciousness is involved, and how
living, conscious entities are an integral part of the reality we experience.
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Logic theories and deductive inference (syllogism) as Artificial Intelligence tools
Georgios M. Milis
PHOEBE Research and Innovation Ltd & EUROCY Innovations Ltd, Nicosia, Cyprus
Email: info@phoebeinnovations.com , info@eurocyinnovations.com
1. Introduction
The study of "Logic" has its roots in the Aristotelian theory of syllogism (deduction) [1]. According to Aristotle, a syllogism is "speech (logos) in which certain things have been supposed,
different from those supposed results of necessity because of their being so". The "things supposed" comprise the premise (protasis), while the "results of necessity" comprise the conclusion
(sumperasma), that is, the logical consequence. For instance, a statement $C$ is the consequence (result of necessity) of the fact that $A$ and $B$ being true (they are supposed), if its logical
truthfulness cannot be challenged given the truthfulness of the things supposed.
The Aristotelian theory of logic had remained dominant until the mid 19th century, followed by approximately one hundred years of research that led to the conception of "modern logic". During that
transformation time, G.E. Moore and B. Russel had put effort in formalising logic by referring to "common sense" [2][3], then they started adding bounds and discussing the "set theory" as the
foundations for mathematics. During early twentieth century, the bounds became narrower following the realisation of the "paradoxes of logic" and the work of B. Russel on "Types theory". These led to
the establishment of "First-Order Logic (FOL)", otherwise called the "modern logic", which had not happened before the mid 20th century [4].
2. The road towards modern logic
During the second half of the 19th century, logic was studied within the framework of the classical philosophy, as well as in relation with the science of mathematics; algebra, axiomatics and
analysis. Since the late 19th century the theory of logic started becoming a blend of the propositional and predicate logic (http://www.iep.utm.edu/prop-log/) on one hand and the theory of sets and
relations on the other hand. At that time, “thinking” was considered the ability of mind to relate between “things”, i.e., derive conclusions about properties of certain things based on knowledge
about properties of other things. This process of “thinking”, performed by the mind, is essentially a mapping between pre-stored knowledge about things. Based on this understanding, B. Russell [5]
formed the “symbolic logic” within the areas of “calculus of propositions”, “calculus of classes” and “calculus of relations”. B. Russel, together with Peano and Frege agreed in their separate works
that mathematics were actually “symbolic logic”, i.e., the symbols are used to help expressing logical propositions and relations between things.
More specifically, at the beginning of the 20th century, the work of Russell, Peano and Frege considered the logical systems within the bounds of the theory of sets. They had believed in the
principle of “comprehension”, meaning that a thing/object could be fully described by certain attributes, properties, qualities, etc. However, later this theory faced certain contradictions (or
paradoxes or antinomies), with the most significant being the “Russell’s paradox”, which was formulated in terms of notions of the set theory itself, i.e. the notion of negation and membership. The
discovery of the paradoxes, led to the revision of the foundations of set-theory and its effect on logic and mathematics.
In order to resolve the issues revealed by the paradoxes, Russell developed his “Theory of Types”, which was incorporated into his work on “Principia Mathematica”, co-authored with Whitehead [5]. The
theory of types [6] introduced the notion of “types” of elements of a set. This way, the notion of “comprehension” was preserved. According to van der Waerden, in his Moderne Algebra [7], the “Theory
of Types” was then the most important system of logic. However, in 1931, Godel announced his first “incompleteness theorem” [8]. During that time (1930), the early work of Skolem [9] on the Zermelo
system and the work of Von Neumann on his logic system [10], led to the appearance of “First Order Logic (FOL)” as the basic system for logic. Hilbert & Ackermann’s “Grundzuge der theoretischen
Logik” [11] shows that FOL was studied as a separate system of logic by around 1928. This book shows also that the use of higher-order logic would be possible and necessary when one would require to
analyse meta-concepts of mathematics.
FOL was adopted as the natural system of logic by 1950, adhering to:
i) The principle of deduction (conditions of arguments to be correct following analysis of their validity);
ii) The principle of universality (logical sentences formed independently of specific topic);
iii) The “Kant’s principle” (study the formulation of arguments and deductions using logical variables and not instantiations within a domain of discourse;
iv) The Leibnizian ideal (compute logical deductions with machine algorithms).
The latter was first achieved by Boole with his algebra of logic. Then, Frege made an impressive step forward and Godel followed by establishing certain implementation limitations.
The concept “domain of discourse” was defined by G. Boole in 1854 in his work “Laws of Thought”, as:
“In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its
operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with
those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only
under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of
the field within which all the objects of our discourse are found, that field may properly be termed the domain of discourse...”
In summary, it can be deduced that around 1900 logic was conceived as a theory of sentences, sets and relations; almost until 1930 the established logic system was (simple) type theory, while by 1950
FOL became the paradigm logical system.
3. First Order Logic
In general, a first-order language system assigns variable symbols to logical constants. The language also defines the "domain of discourse", which specifies the range of the variables, in line with
the Russel's Theory of Types. Sentences written using such languages have clear semantics. For instance, assume the domain of discourse $\mathcal{D}$, being a set of objects of a certain type. An
example of a first-order statement within this domain could be the $ x,a(x)$, stating that some logical sentence $x$ is true and at the same time it's logical transformation through the predicate $a
(x)$ is also true. If a domain of discourse is not clearly defined, the logical derivations may not be always feasible. For instance, the sentence $\forall x, x > y$ cannot be logically interpreted
without knowing the domains of the logical variables $x$ and $y$. So, if $x$ and $y$ are real numbers, the statement is false because there are real numbers that can be greater than other real
numbers. But, if $x$ is a positive number and $y$ is a negative number, the statement becomes always true since all positive numbers are greater than any negative number. Moreover, assuming $\mathcal
{D}$ being the domain of building zones, we can define the adjacency of two building zones as a binary predicate $a(\cdot)$, taking two building zones are arguments and returning $\top$ if the two
zones are indeed adjacent or $\bot$ if they are not.
If the validity or not of any given sentence can always be deduced through some language system, then this language is said to be "complete". According to Godel, the FOL language system is complete,
although mathematical systems cannot guarantee in general syntactic and semantic completeness [8]. This understanding made FOL a sufficient language for codifying mathematical proofs. Moreover, the
tools utilised in mathematical proofs (theorems, sentences, etc.) can be explicitly supported by quantification expressions such as the: "given any", "there is", "for any", "for each". For instance,
one can write a sentence like: $\forall{x}(a(x) \mapsto b(x))$ .
Summarizing, to guarantee completeness, a FOL language always needs to define a "domain of discourse" $\mathcal{D}$, also defining the classes $\Omega$ of all terms used in the language. For
instance, an $n$-ary relation $r$ is essentially a mapping of the form $\Omega(r)$: $\mathcal{D}^{n} \mapsto \{ \mathrm {true,false} \}$. That is, a statement is true if it can be inferred by some
logical deduction within the domain $\mathcal{D}$. In mathematics, a formula is logically valid only if it is true no matter the instantiation of the variables in the defined domain of discourse.
Certain "axioms" can be also defined within a domain of discourse, as being true anyway and helping the inference.
4. Syllogism - Deductive Inference Systems
Basic systems of "Logic" deal with certain logical "statements" or "sentences" being true or false. The logical statements under consideration are typically called "propositions" (see also the terms
"propositional logic" or "sentential logic" or even "zero-order logic"). Logical statements can be combined using "logical connectives". For instance, the "and" (conjunction), the "or" (disjunction),
the "not" (negation) and the "if" (denoting condition of existence) are logical connectives of the English language. Moreover, "logical axioms" can be pre-defined, stating things that are believed to
be true by design. A third element of logic systems are the "inference rules", which specify given knowledge about the truthfulness of combinations of propositions through logical connectives. In
logic, an "inference rule" is a logical $n$-ary function that takes as input certain statements, analyzes their syntax and returns a conclusion. For example, in the "modus ponens" inference system,
an inference rule takes two inputs, one statement in the form "if $a$ then $b$" and another in the form "$a$", and returns the conclusion "$b$". A fourth element are the "quantifiers", which are
operators that map propositions (their symbolic representation) to a specific domain of discourse.
It has been mentioned earlier, that according to the Kant's principle, only the form of the statements should matter and not the instances of the logical variables. We take an example of a syllogism
from Aristotle: All men are speakers. No oyster is a speaker. Therefore, no oyster is a man. This example can be transformed to the first-order sentence: $\forall a,b(a)$. $\not b(c)$. Therefore, $a
\ne c$. Another example would be:
• Statement 1: Presence of people in a closed room without any other sources of CO2, causes CO2 to increase.
• Statement 2: There are no people in the room.
• Deduction: The CO2 value cannot be high.
The given statements are what we know is true (knowledge facts). Other knowledge can be inferred by the given statements after applying some inference mechanism. The knowledge facts and the inference
rules are considered known a-priori in deductive inference, either because they are part of the expert knowledge or because they can be derived analytically using a-priori knowledge of the system.
Therefore, the concept "Deductive Inference" is concerned with checking whether a statement is true given the truthfulness of another statement or a combination of statements through logical
connectives. There are many such systems for first-order logic, including the "Hilbert-style deductive systems", the "natural deduction", the "sequent calculus", the "tableaux method", and the
"resolution". It is noted that derivations of proofs in "proof theory" are essentially deductions. A deductive inference system is considered "sound" if any statement that can be derived in the
system as true is logically valid. Conversely, a deductive inference system is "complete" if every logically valid statement can be derived through the system. All above mentioned deductive inference
systems are both sound and complete. The conclusions in these deductive inference systems do not typically consider the semantic interpretations of the statements in a domain of discourse. On the
other hand, in FOL, deductive inference is only semi-decidable, that is, if $a$ logically implies $b$ then this can be deduced by a deductive inference system. However, if $a$ does not logically
imply $b$, this does not mean that $a$ logically implies $\not b$.
In general, the deductive inference systems use several logical and non-logical symbols from the alphabet, such as:
• The quantifier symbols $\forall$ and $\exists$
• The logical connectives: $\wedge$ for conjunction, $\vee$ for disjunction, $\Rightarrow$ for implication, $\Leftrightarrow$ or $\equiv$ for biconditional, $\neg$ or $\tilde{}$ for negation,
• Parentheses, brackets, and other punctuation symbols.
• An infinite set of variables, often denoted by lower-case letters at the end of the alphabet $x,y,z,\cdot$. Variables are often distinguished by sub-scripts: $x_0,x_1,x_2,\cdot$.
• An equality symbol, e.g., $=$
• The truth constants are included, e.g., $\top$ for "true" and $\bot$ for "false".
• Additional logical connectives that may be required, such as "NAND" and "exclusive OR".
On the other hand, non-logical symbols represent semantic relations, i.e. mappings of knowledge objects within a domain of discourse. In the past, FOL considered a single fixed set of non-logical
symbols to apply deductive inference. The current practice, however, is to define a different set of non-logical symbols within the bounds of an application. Such definition of logical symbols can be
made through "ontology languages". Ontology languages are essentially collections of a finite number of $n$-ary predicates/symbols, representing relations between $n$ objects. For example, $\text
{man}(x)$ is an example of a predicate of arity 1, which defines that "$x$ is a man"; $\text{father}(x,y)$ is a predicate of arity 2, interpreted as "$x$ is the father of $y$".
For completeness, we note here that there are also other types of inference:
i) Inductive inference, where the rules are not considered known a-priori; possible rules are derived from the propositions and the observations/experiences, as if the logic system is "identified"
from inputs and outputs;
ii) Abductive inference, where the observations/experiences as well as the rules are considered known, and the logically valid truths are inferred accordingly. E.g., I observe something (fault
detection) and knowing how the system works I try to infer what might have caused it (fault isolation).
5. Ontology languages and Description Logic
Ontology languages allow the encoding of pre-existing or acquired data/information about a specific domain of discourse. They typically include also inference (or "reasoning") rules that help with
making logical conclusions above the modeled data/information. Ontology languages are usually declarative languages and are commonly based either on FOL or on "Description Logic" (DL) [12]. DL is a
family of formal knowledge representation languages, which are typically subsets of FOL (less expressive than FOL). They are used to represent a domain of discourse in a structured way, offering
clear decide-ability with no gaps due to adding constraints. DLs usually use binary predicated (two-variable logic). There are enough good-quality reasoning mechanisms (decision procedures) designed
and implemented for DLs.
DLs have been used as the logical formalism for "ontologies" and the "Semantic Web". For instance, the Web Ontology Language (OWL) and its profile is based on DLs [13]. In DLs usually a "unary
predicate" of FOL, is called a "class", a "binary predicate" is called a "property" and a "constant" is called an "individual". DLs pose certain relationships with other logics as well. For instance,
Fuzzy description logic combines fuzzy logic with DLs. Fuzzy logic deals with the notions of vagueness and uncertainty about the classes of certain logical variables. These properties are common in
intelligent systems, where concepts do not have clear boundaries. Fuzzy logic therefore generalises the description logic to deal with vague concepts. In addition, the "Temporal Description Logic"
allows reasoning about time dependent concepts and can be the combination of DL with a modal temporal logic such as "Linear Temporal Logic". Concluding, DLs are a very good tool for representing data
and information models and inference rules, and subsequently inferring mappings and values from known facts, i.e. generating knowledge.
We make use of ontologies and inference rules in our innovative SEMIoTICS architecture which is part of our $\text{Domognostics}^{TM}$ product in the smart buildings domain.
Schematic Summary
[1] S. Robin, "Aristotle’s Logic," in The Stanford Encyclopedia of Philosophy, winter 2016 ed., E. N. Zalta, Ed. Metaphysics Research Lab, Stanford University, 2016.
[2] G. E. Moore, "A Defence of Common Sense," in Contemporary British Philosophy (2nd series), J. H. Muirhead, Ed. Allen and Unwin, London, 1925, pp. 192–233.
[3] B. Russell, Common Sense and Nuclear Warfare, 1st ed. Routledge, Oxford, UK, May 2001.
[4] J. Ferreiros, "The road to modern logic-an interpretation," Bulletin of Symbolic Logic, vol. 7, no. 4, pp. 441–484, 12 2001. [Online]. Available: http://projecteuclid.org/euclid.bsl/1182353823
[5] B. Russell, Ed., The principles of mathematics. Cambridge University Press, 1903, (2nd edition 1937). Reprint London, Allen and Unwin, 1948.
[6] B. Russel, "Mathematical logic as based on the theory of types," American Journal of Mathematics, vol. 30, pp. 222–262, 1908.
[7] B. L. V. der Waerden, Ed., Moderne Algebra. Springer, Berlin, 1930.
[8] K. Godel, "Über formal unentscheidbare sätze der principia mathematica und verwandter systeme i," Monatshefte für Math. u. Physik, vol. 38, pp. 173–198,
[9] T. Skolem, "Einige bemerkungen zur axiomatischen begriindung der mengenlehre," Dem Femte skandinaviska mathematikerkongressen, Akademiska Bokhandeln, 1923, helsinki.
[10] J. V. Neumann, "Eine axiomatisierung der mengenlehre," Journal fur die reineund angewandte Mathematik, vol. 154, pp. 219–240, 1925, reprint in Collected Works, vol. 1, Oxford, Pergamon, 1961.
[11] W. A. David Hilbert, Ed., Grundzuge der theoretischen Logik. Springer, Berlin, 1928, (2nd edition 1937). Reprint London, Allen and Unwin, 1948.
[12] F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, and P. Patel-Schneider, Eds., The Description Logic Handbook. Cambridge University Press, 2003.
[13] P. P.-S. I. Horrocks, "Reducing owl entailment to description logic satisfiability," In Proc. of the 2nd International SemanticWeb Conference (ISWC), 2003, http://www.cs.man.ac.uk/ horrocks/ | {"url":"https://public.phoebeinnovations.com/node/23","timestamp":"2024-11-06T02:42:50Z","content_type":"text/html","content_length":"74456","record_id":"<urn:uuid:0e53389a-fb28-43f4-ab82-fe7745b4d3a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00710.warc.gz"} |
PEMDAS: Remembering Math's Order of Operations
PEMDAS is the tried-and-true method that gives us the order to work when solving mathematical problems. HowStuffWorks
Nearly every middle school in the U.S. teaches its students to remember this simple phrase: "Please excuse my dear Aunt Sally." But why are we apologizing for her behavior? Did she wear white after
Labor Day or something?
The world may never know. In all seriousness, "Please Excuse My Dear Aunt Sally," or PEMDAS, is just a mnemonic. It's a tool educators use to help us memorize information through a catchy rhyme,
phrase or acronym. Now let's explore how to use this tool to solve equations.
Is PEMDAS wrong?
In the U.S., PEMDAS is more common where we first calculate Parentheses, then Exponents, then Multiplication and Division, and Addition and Subtraction at the end. However, most of the world uses
BODMAS — Brackets, Orders, Division, Multiplication, Addition and Subtraction.
Why is PEMDAS in that order?
PEMDAS basically creates a pyramid for different functions in an equation. For example, the first priority is given to the parentheses — and for good reason. Not only does this give order to
equations but also drives more accurate results.
What is the formula of PEMDAS?
According to PEMDAS, it is important that the equation is simplified before it is calculated. This means squaring any roots off on both sides, any canceling effects and more. After that, the
parentheses, exponents, multiplication, division, addition and subtraction order must be followed, solving each element from left to right.
What is better BODMAS or PEMDAS?
There has been a long debate about whether BODMAS or PEMDAS is better but the difference between them primarily arises from regional terminology and preferences. Some say there is no difference
between the two since they suggest that multiplication and division must be done from left to right, regardless of what comes first, while others prefer to follow the BODMAS mnemonic. | {"url":"https://science.howstuffworks.com/math-concepts/PEMDAS.htm","timestamp":"2024-11-04T01:44:28Z","content_type":"text/html","content_length":"189795","record_id":"<urn:uuid:89d17018-0814-4ec6-8676-dd9c20f2fe01>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00080.warc.gz"} |
Interview with Brad Guy: Part 1 of 3
In my post titled Demolition, Deconstruction, and Discussion, I mentioned DfD expert Brad Guy. Brad has devoted his career to research on DfD, and it shows; if you look up "design for deconstruction"
on the internet, chances are you will quickly come across references to his many papers and interviews.
I have actually been in contact with Brad since last year, when I invited him to speak at a green buildings panel for the 2020 Cornell Business Impact Symposium. Rather than simply continuing to read
about him, I invited him to chat for an informal interview for this blog, and he agreed! The interview was 90 minutes long, so after condensing it and editing for clarity, I have decided to post it
in three parts.
Without further ado, Part 1!
Luna: I see your name everywhere when I'm reading about DfD. So I was wondering— how you ended up becoming the spokesperson for DfD in the US?
Brad: I started in deconstruction around ‘95 or ’96. Salvage has been going on since there have been buildings, but we [deconstruction researchers] were trying to make it more research-oriented, more
quantifiable, more rigorous, rather than just ad-hoc anecdotal. The US EPA sponsored us, so I produced several papers on deconstruction.
I was also a member of an international research group called CIB [1]. They have working groups and task groups of academics internationally, and we started a sub-group for deconstruction. The goal
of that organization is to help people meet peers, produce work, get it published, have conferences, get in proceedings, and things like that. Proceedings are very helpful for any academic to just
get their stuff out and get it referenced. So maybe that's why, just because papers and formal research on deconstruction was extremely rare.
I also advised on the paper for the Chartwell School DfD project in California [2]. The architecture firm EHDD did the actual project, and that helped it gain some traction. And people look towards
the West Coast. It’s almost like anything you can do in California, Oregon, or Washington gets more attention, because people are interested in sustainability there. There's more of a— sympathetic
population that puts the word out, and it spreads.
Yeah, and then I guess the last thing is just talking about it a lot [laughs]. Giving presentations, some teaching, and just being an advocate.
Luna: Makes sense. Some say that we should focus more on reducing operational carbon than on reducing embodied carbon because operational carbon is responsible for a larger share of total emissions
in the US [3]. Have you come across a scenario in which the two are in conflict? For example, a design decision that requires prioritizing one over the other?
Brad: I was actually going to tell you that I think there are good synergies there.
If you look at the time spans of most building components, your flooring is expected to last like 5 years, most appliances and electrical devices could be anywhere from 5 to 15 years... that sort of
thing. If all that stuff is only lasting 5 to 15 years, that’s stuff that you really want to be able to upgrade.
And windows, refrigerators, heat pumps too— they're constantly being innovated. Which means that if I want to maintain operational efficiency, I already know I have to be able to change building
components over time. If I'm operating a building 20 years in the past in terms of efficiency, but the cost for me to upgrade operational components (like heat pumps) is prohibitive... You can't be
efficient if you let buildings become obsolete like that. Clearly, making it easier to upgrade will keep you operationally efficient.
Luna: Well, for example, concrete is hard to reuse, correct? But some people argue that concrete is good in terms of operational carbon because [of its property as a thermal mass]. There seems to be
a tradeoff between the embodied carbon aspect and the operational carbon aspect there, because if the building has a short lifespan the embodied carbon may account for a larger proportion of total
building emissions, and if the building has a long lifespan the operational carbon may matter more.
Brad: But it would depend on which parts of it you’d build, right? Like, how many alternatives are there to concrete footings? Basically zero. Unless you did a wood piling system, which is like
ancient technology.
As we get more energy-efficient with the buildings, the percentage of embodied energy will go up over the hundred-year life of the building. So when we have more energy-efficient buildings it really
matters that making the material is more energy-efficient.
A presentation I got in trouble for— people were really unhappy with me, but I did a lifecycle assessment of an office building in D.C. that was built in the 1980s and was about to be renovated [4].
It’s like wow, it’s only 30-40 years old, and they're completely gutting this thing and totally renovating it. They changed all the windows and all the mechanical only after 40 years! A pretty nice
office building, not degraded, it was being used up to the day...
They added photovoltaics (PV) to the roof; they were trying to get it to net zero energy. It’s like the Bullitt Center in Seattle, so the whole roof was completely covered and the PV was even
overhanging the edges.
I said, well okay, here's an example where they reused the building, they saved all that embodied energy, and there's a huge benefit to that. It’s super energy efficient; like 60% more
energy-efficient than it was. But they wanted to get that last percentage, so they added photovoltaics, which are extremely energy-intensive materials.
So is that really a good idea? In theory they’re going to net zero operational, but they added so much to the embodied energy with the PV. And people have different methods to calculate the embodied
energy payback of PV, but the way I did it, it would take like 10 years to pay back that investment in the PV saved through the avoidance of operational electricity.
And then they got really upset with me because typically you want to put PV everywhere, all the time, right? It's like save energy now and go to net zero now, but people don't necessarily think,
“well, what about the investment in additional energy put into the PV itself?” You can't really say that it’s free [laughs.] It's not free; it took some pretty precious metals and all kinds of stuff
to make the PV. So if the climate crisis is... if we only have 10 years, then maybe you shouldn’t add PV, you should just renovate the building and leave it at that.
Points are (1) these things are a little bit complicated, (2) I think it's more or less worth it to reuse everything you can imagine unless it's toxic, or it's truly obsolete and operating in an
extremely inefficient way, in which case you need to do that upgrade. But I would say there's a limit to how much more material you should start adding to something to increase its energy efficiency.
If the carbon crisis tipping point is in 10 years or so, a 50-year life cycle is starting to be kind of long, and a 100-year life cycle may be out of scope, because most buildings don't even last
that long anyway.
[3] Embodied carbon refers to emissions associated with building materials and the construction process. Operational carbon refers to emissions associated with building energy use (emissions after
the building has been constructed).
[4] With this question, I was trying to play devil's advocate and see if Brad could come up with a scenario in which embodied carbon was lowered, but as a result operational carbon increased.
Instead, he came up with a scenario in which operational carbon was lowered, but as a result embodied carbon increased. | {"url":"https://www.lunaoiwa.com/post/interview-with-brad-guy-part-1","timestamp":"2024-11-14T18:20:33Z","content_type":"text/html","content_length":"923464","record_id":"<urn:uuid:bb4fa9c4-2d46-46d0-941c-0cea4b0b81f0>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00162.warc.gz"} |
Stage 1-6 Maths resources - what is there?
This year I have been working on my maths resources for Stage 1-6 in the New Zealand classrooms. There are lots that have been created to give you a comprehensive way to confidently teach maths in
your classroom.
Above shows a picture of all the different Stage 1-6 Maths resources that are created to fit together. These include:
• Stage 1-6 Teaching Packs: these are the activities and items that can be used in guided maths, individual work or even as follow ups. These cover number knowledge and number strategy for each
stage. Purchase separately or as a bundle.
• Stage 1-6 Exit Slips: these can be used in modelling books as formative or summative assessment; or just as end of lesson checks. Purchase separately or as a bundle.
• Stage 1-6 Maths Strategy Follow Ups: these are follow up worksheet books that give you more opportunities for students to reinforce their learning. These are in three packs (stage 1,2; stage
3,4; stage 5,6) or as a bundle.
• Stage 3-6 basic facts: these are basic facts practice that align with key concepts for basic facts in each stage. Purchase separately or as a bundle.
• Stage 1-6 Strategy Anchor Chart Posters: these show students an example of the strategy with information on what the strategy means.
• Stage 1-6 Wall trackers and progressions: these are for your classroom wall as a tracker for visible learning and unpacking. The progressions can be used in modelling books as a tracker also.
• Stage 1-5 Google forms: these are a digital google form test or check of the learning for each strategy. These are free to access and have not got the whole set completed yet.
Purchase these below to add these to your classroom resources!
Stage 4 Maths
Stage 4 Maths
This resource is a fabulous resource for those of you with students working in Stage 4. It includes activities, games and ideas that can be used in an independent maths programme, or alongside your
teaching sessions. This bundle is perfect for beginning teachers looking to set up their maths section, or a teacher looking for more ideas.
It includes a learning progression sheet that highlights the areas for number knowledge and number strategy in Stage 4. The resources follow the learning progression and are noted in the top right
hand corner with what progression they match too.
Stage 4 includes:
• I can read all my numbers 0-100
• I can skip count forwards and backwards in 2’s, 5’s and 10’s to 100
• I can count forwards over 10’s between 0-100
• I can count backwards over 10’s between 0-100
• I can say the number before up to 100
• I can say the number between up to 100
• I can say the number after up to 100
• I can order numbers between 0-100
• I can order numbers backwards from 100-0
• I can read the fraction symbol for ½, ¼, â…“, â…•
• I can solve addition up to 100 by counting on
• I can solve subtraction up to 100 by counting back
• I can solve addition and subtraction using groups of tens and ones
• I can solve multiplication problems by skip counting in 2’s, 5’s and 10’s
• I can solve division problems by using materials to share equal sets of 1, 2 and 5
It also comes with decorations for your box to keep track of your resources.
This resource pack is best laminated to ensure durability and can be used with whiteboard pens as wipe sheets.
These align with NZ Curriculum pink book learning progressions.
This resource has 187 pages in it.
Stage 3 Maths
Stage 3 Maths
This resource is a fabulous resource for those of you with students working in Stage 3. It includes activities, games and ideas that can be used in an independent maths programme, or alongside your
teaching sessions. This bundle is perfect for beginning teachers looking to set up their maths section, or a teacher looking for more ideas.
It includes a learning progression sheet that highlights the areas for number knowledge and number strategy in Stage 3. The resources follow the learning progression and are noted in the top right
hand corner with what progression they match too.
Stage 3 includes:
• I know the difference between ty and teen numbers
• I can skip count forwards in 2’s to 20
• I can skip count backwards in 2’s to 20
• I can skip count forwards in 5’s to 20
• I can skip count backwards in 5’s to 20
• I know addition and subtraction facts up to 5
• I know my addition groupings with 5
• I know all the doubles up to 10
• I can add two groups up to 20
• I can take away from a group up to 20
• I can takeaway groups of 10 in my head
• I can add groups of 10 to find the answer
It also comes with decorations for your box to keep track of your resources.
This resource pack is best laminated to ensure durability and can be used with whiteboard pens as wipe sheets.
These align with NZ Curriculum pink book learning progressions.
There are 89 pages in this resource.
Stage 2 Maths
Stage 2 Maths
This resource is a fabulous resource for those of you with students working in Stage 2. It includes activities, games and ideas that can be used in an independent maths programme, or alongside your
teaching sessions. This bundle is perfect for beginning teachers looking to set up their maths section, or a teacher looking for more ideas.
It includes a learning progression sheet that highlights the areas for number knowledge and number strategy in Stage 2. The resources follow the learning progression and are noted in the top right
hand corner with what progression they match too.
Stage 2 includes:
• I can read numbers 0-20
• I can count forwards from 0-20
• I can count backwards from 0-20
• I can say the number after between 0-20
• I can say the number between 0-20
• I can say the number before 0-20
• I can order numbers from 0-20
• I know my finger patterns between 1-10
• I know 10s frame patterns from numbers 1-10
• I can add two groups of materials together up to 20
• I can add materials to find the missing number
• I can take materials away to find whats left up to 20
• I can take materials away to find the missing number
• I can make groups of ten with materials to 50
It also comes with decorations for your box to keep track of your resources.
This resource pack is best laminated to ensure durability and can be used with whiteboard pens as wipe sheets.
These align with NZ Curriculum pink book learning progressions.
There are 97 pages in this resource
Stage 1 Maths
Stage 1 Maths
This resource is a fabulous resource for those of you with students working in Stage 1. It includes activities, games and ideas that can be used in an independent maths programme, or alongside your
teaching sessions. This bundle is perfect for beginning teachers looking to set up their maths section, or a teacher looking for more ideas.
It includes a learning progression sheet that highlights the areas for number knowledge and number strategy in Stage 1. The resources follow the learning progression and are noted in the top right
hand corner with what progression they match too.
Stage 1 includes:
• I can read numbers 0-10
• I can count forwards from 0-10
• I can count backwards from 0-10
• I can say the number after between 0-10
• I can say the number between 0-10
• I can say the number before 0-10
• I can order numbers from 0-10
• I know my finger patterns between 1-5
• I know 10s frame patterns from numbers 1-5
• I can count a set up to 10
• I can make a set up to 10
• I can count objects up to 5
It also comes with decorations for your box to keep track of your resources.
This resource pack is best laminated to ensure durability and can be used with whiteboard pens as wipe sheets.
These align with NZ Curriculum pink book learning progressions.
There are 68 pages in this resource.
Stage 5 Maths
Stage 5 Maths
This resource is a fabulous resource for those of you with students working in Stage 5. It includes activities, games and ideas that can be used in an independent maths programme, or alongside your
teaching sessions. This bundle is perfect for beginning teachers looking to set up their maths section, or a teacher looking for more ideas.
It includes a learning progression sheet that highlights the areas for number knowledge and number strategy in Stage 5. The resources follow the learning progression and are noted in the top right
hand corner with what progression they match too.
Stage 5 includes:
• I can read all my numbers 0-1000
• I can order numbers to 1000
• Count in 1's, 10's and 100's to 1000
• I can say the number 1 more/less, 10 more/less, 100 more/less from 1-1000
• I know how many 10s are in a 3 digit number
• I know groupings of 2 that are in numbers to 20
• I know groupings of 5 that are in numbers to 50
• I know the number of hundreds in centuries and thousands
• I can round a 3 digit whole number to the nearest 10 or 100
• I can skip count up to 100 in 3's (forwards and backwards)
• I can read fractions 1/2 to 1/10
• I can order fractions with like denominators
• I can use place value to solve my problems (ones, tens and hundreds)
• I can solve addition or subtraction problems by reversing the sign
• I can use compatible numbers to solve the problem
• I can solve numbers by using tidy numbers
• I can solve problems using a number line
• I can solve problems using doubles
It also comes with decorations for your box to keep track of your resources.
This resource pack is best laminated to ensure durability and can be used with whiteboard pens as wipe sheets.
These align with NZ Curriculum pink book learning progressions.
There are 177 pages in this resource.
On Sale
On Sale
Stage 1-6 Maths Bundle
NZ$48.00 NZ$38.40
Stage 1-6 Maths Bundle
This is a bundle pack of my Stage 1-6 Maths. Grab them all individually or in one cheaper bundle.
These resources are fabulous for those of you with students working in Stage 1-6. It includes activities, games and ideas that can be used in an independent maths programme, or alongside your
teaching sessions. This bundle is perfect for beginning teachers looking to set up their maths section, or a teacher looking for more ideas.
It includes a learning progression sheet that highlights the areas for number knowledge and number strategy in Stage 1- Stage 6. The resources follow the learning progression and are noted in the
top right hand corner with what progression they match too.
Stage 1 Maths
Stage 2 Maths
Stage 3 Maths
Stage 4 Maths
Stage 5 Maths
Stage 6 Maths
It also comes with decorations for your box to keep track of your resources.
This resource pack is best laminated to ensure durability and can be used with whiteboard pens as wipe sheets.
These align with NZ Curriculum pink book learning progressions.
**This will be downloaded as a zip file so please make sure you have access to be able to unzip to access the files.
Stage 1-7 Maths assessment progressions and wall trackers
Stage 1-7 Maths assessment progressions and wall trackers
This is a free addition to the Stage 1-7 Maths Packs resources that I have created for Stage 1-7 in New Zealand.
Find these resources here.
This freebie has an assessment progression sheet for each stage. This can be used in modelling books, with assessments or as a tracker. Print one out per student or modelling book - whichever your
This freebie also includes wall progressions. These can be used for students to view the progressions in each stage, or as an interactive tracker by printing the calculators off and writing the name
and learning intention for the child. Children can use this to visibly see what they are working on and what they are aiming for with the rest of the progressions.
Stage 1 Exit Slips
Stage 1 Exit Slips
These are a great reflection, formative assessment or follow up exit slip that can be used to support the Stage 1 resource pack. This is aligned to the NZ Curriculum Maths stages, with this one
covering Stage 1.
To purchase the Stage 1 resource please find it here.
To purchase the Stage 1-5 bundle resource please find it here.
The stage 1 exit slips have 4 exit slips for each learning intention:
• I can read numbers 0-10
• I can count forwards from 0-10
• I can count backwards from 0-10
• I can say the number after between 0-10
• I can say the number between 0-10
• I can say the number before 0-10
• I can order numbers from 0-10
• I know my finger patterns between 1-5
• I know 10s frame patterns from numbers 1-5
• I can count a set up to 10
• I can make a set up to 10
• I can count objects up to 5
Print off these exit slips and have them handy for your students.
**Recently updated with new font and 4 per page. If you have purchased these previously and want the updated version; send [email protected] a message with your order number.
Stage 2 Exit Slips
Stage 2 Exit Slips
These are a great reflection, formative assessment or follow up exit slip that can be used to support the Stage 2 resource pack. This is aligned to the NZ Curriculum Maths stages, with this one
covering Stage 2.
To purchase the Stage 2 resource please find it here.
To purchase the Stage 1-5 bundle resource please find it here.
The stage 2 exit slips have 4 exit slips for each learning intention:
• I can read numbers 0-20
• I can count forwards from 0-20
• I can count backwards from 0-20
• I can say the number after between 0-20
• I can say the number between 0-20
• I can say the number before 0-20
• I can order numbers from 0-20
• I know my finger patterns between 1-10
• I know 10s frame patterns from numbers 1-10
• I can add two groups of materials together up to 20
• I can add materials to find the missing number
• I can take materials away to find whats left up to 20
• I can take materials away to find the missing number
• I can make groups of ten with materials to 50
Print off these exit slips and have them handy for your students.
**Recently updated with new font and 4 per page. If you have purchased these previously and want the updated version; send [email protected] a message with your order number.
Stage 3 Exit Slips
Stage 3 Exit Slips
These are a great reflection, formative assessment or follow up exit slip that can be used to support the Stage 3 resource pack. This is aligned to the NZ Curriculum Maths stages, with this one
covering Stage 3.
To purchase the Stage 3 resource please find it here.
To purchase the Stage 1-5 bundle resource please find it here.
The stage 3 exit slips have 4 exit slips for each learning intention:
• I know the difference between ty and teen numbers
• I can skip count forwards in 2’s to 20
• I can skip count backwards in 2’s to 20
• I can skip count forwards in 5’s to 20
• I can skip count backwards in 5’s to 20
• I know addition and subtraction facts up to 5
• I know my addition groupings with 5
• I know all the doubles up to 10
• I can add two groups up to 20
• I can take away from a group up to 20
• I can takeaway groups of 10 in my head
• I can add groups of 10 to find the answer
Print off these exit slips and have them handy for your students.
**Recently updated with new font and 4 per page. If you have purchased these previously and want the updated version; send [email protected] a message with your order number.
Stage 4 Exit Slips
Stage 4 Exit Slips
These are a great reflection, formative assessment or follow up exit slip that can be used to support the Stage 4 resource pack. This is aligned to the NZ Curriculum Maths stages, with this one
covering Stage 4.
To purchase the Stage 4 resource please find it here.
To purchase the Stage 1-5 bundle resource please find it here.
The stage 4 exit slips have 4 exit slips for each learning intention:
• I can read all my numbers 0-100
• I can skip count forwards and backwards in 2’s, 5’s and 10’s to 100
• I can count forwards over 10’s between 0-100
• I can count backwards over 10’s between 0-100
• I can say the number before up to 100
• I can say the number between up to 100
• I can say the number after up to 100
• I can order numbers between 0-100
• I can order numbers backwards from 100-0
• I can read the fraction symbol for ½, ¼, â…“, â…•
• I can solve addition up to 100 by counting on
• I can solve subtraction up to 100 by counting back
• I can solve addition and subtraction using groups of tens and ones
• I can solve multiplication problems by skip counting in 2’s, 5’s and 10’s
• I can solve division problems by using materials to share equal sets of 1, 2 and 5
Print off these exit slips and have them handy for your students.
**Recently updated with new font and 4 per page. If you have purchased these previously and want the updated version; send [email protected] a message with your order number.
Stage 5 Exit Slips
Stage 5 Exit Slips
These are a great reflection, formative assessment or follow up exit slip that can be used to support the Stage 5 resource pack. This is aligned to the NZ Curriculum Maths stages, with this one
covering Stage 5.
To purchase the Stage 5 resource please find it here.
To purchase the Stage 1-5 bundle resource please find it here.
The stage 5 exit slips have 4 exit slips for each learning intention:
• I can read all my numbers 0-1000
• I can order numbers to 1000
• I can count in 1's, 10's and 100's to 1000
• I can say the number 1 more/less, 10 more/less, 100 more/less from 1-1000
• I know how many 10s are in a 3 digit number
• I know groupings of 2 that are in numbers to 20
• I know groupings of 5 that are in numbers to 50
• I know the number of hundreds in centuries and thousands
• I can round a 3 digit whole number to the nearest 10 or 100
• I can skip count up to 100 in 3's (forwards and backwards)
• I can read fractions 1/2 to 1/10
• I can order fractions with like denominators
• I can use place value to solve my problems (ones, tens and hundreds)
• I can solve addition or subtraction problems by reversing the sign
• I can use compatible numbers to solve the problem
• I can solve numbers by using tidy numbers
• I can solve problems using a number line
• I can solve problems using doubles
Print off these exit slips and have them handy for your students.
**Recently updated with new font and 4 per page. If you have purchased these previously and want the updated version; send [email protected] a message with your order number.
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Measuring the Achievement Elephant
It's an old story. A group of blind people want to know what an elephant looks like. One feels the elephant's trunk, another a leg, and another the tail. The first concludes that the elephant is like
a snake, the second like a tree, and the third like a rope. It's impossible to get an accurate image of the whole elephant by examining only a few of its parts.
The story illustrates the problem of getting a fix on student achievement. Like the elephant, the subject of student achievement is big. A few pieces of data can give an incomplete picture—or worse,
a misleading one.
To illustrate this point, let's look at a few examples that represent starting places for thinking about some little-recognized aspects of student achievement data. With a better understanding of the
whole elephant, school leaders can not only make better use of test score data but also convey the meaning of these data more effectively to school personnel, students, and parents.
The Problem with Cut Points
A common approach in this era of test-based accountability is to measure student achievement in terms of how many students score at or above some predetermined "proficiency" level, which
statisticians call a cut point. State and federal accountability systems, with a few exceptions, are nearly always set up this way.
Focusing on the cut point has at least one major drawback: It provides no information about changes in the achievement of students who remain above or below this point. Further, in a high-stakes
environment, the use of a single cut point can have negative consequences, encouraging teachers to focus most of their attention on those students who are just below the cut point in an effort to
boost them over the line so the school can show "improvement." Meanwhile, what's happening to students who are well above this cut point? What's happening to students who are so far below the cut
point that there seems to be only a remote possibility of getting them above it before the next round of testing?
We need to consider measures that yield a broader understanding of student achievement. These measures could result in a better approach to accountability and a more equitable and effective
distribution of precious instructional resources.
Looking at the Whole Distribution
One way of moving beyond the cut point approach is to examine the whole distribution of scores by percentiles. Figure 1 (p. 32), taken from the long-term trend series of the National Assessment of
Educational Progress (NAEP), illustrates some of the insights that we can gain from this approach.
Figure 1. Percentile Distributions of NAEP Reading Scores by Age and Racial/Ethnic Group, 1990 and 2004
* Indicates a statistically significant difference 1990 to 2004.
Source: Data from the National Assessment of Educational Progress analyzed by Educational Testing Service.
This figure shows average NAEP reading scores at selected percentiles for 1990 and 2004. We can see that 9-year-old students at the 50th, 25th, and 10th percentiles improved significantly, and
students at the 90th and 75th percentiles decreased or did not improve significantly. When we look at different racial/ethnic groups, we see that black students showed significant gains at all
percentiles and Hispanic students made significant gains at all but the 90th percentile during this period.
Now look at older students' reading scores. The contrast is striking. The total group of 13-year-old students showed no significant improvement at any percentile. Seventeen-year-old white, black, and
Hispanic students showed declines at every level; and when all three racial/ethnic groups are added together, the sample is large enough to disclose that these declines were statistically significant
at the 75th, 25th, and 10th percentiles.
The news was better in mathematics, where gains were made throughout most of the score distribution for 9- and 13-year-olds. But the mystery of the disappearing achievement gains has been evident
during the last few decades. The student achievement gains we've seen at ages 9 and 13 typically disappear at age 17. Looking at achievement trends at different percentiles and at different ages can
inform policymakers about where changes may or may not be occurring—where students are being helped and where they may be falling behind.
Looking at Quartiles
We can tell a more concise story by examining quartiles, calculating average scores for each quartile, and tracking changes. Ever since NAEP made such comparisons possible, the public has widely
recognized that achievement gaps by race and ethnicity exist. But it may come as a surprise that the largest and only reduction in the minority achievement gap for 17-year-olds in reading, looking at
black and Hispanic students combined, occurred from 1975 to 1990. From 1990 through 2004 (the most recent data available from the long-term NAEP), there has been no reduction in the gap. The
reduction from 1975 to 1990 was large—the gap was nearly halved—and it happened across the board in all four quartiles (Barton & Coley, 2008).
Many people consider the period of the 1990s and early 2000s to be the time of the flowering of education reform, including the implementation of standards-based reform and test-based accountability.
Why the reduction in the minority achievement gap for older students stopped during this period—and why the large gap reduction between 1975 and 1990 occurred in the first place—is unknown. We should
seek the answer as policymakers craft new programs to raise overall levels of student achievement and to close the achievement gap.
End-of-Year Comparisons vs. Gains
There has been considerable debate in the United States, particularly since the passage of No Child Left Behind (NCLB), about how to use test scores to set standards for accountability. NCLB uses
end-of-year test scores to determine how many students meet a set level of proficiency, thus comparing different cohorts of students each year. We have been among those arguing that we would get more
useful information by measuring how much students gain in knowledge during the school year. A considerable number of studies have shown that schools found to be "failing" on one measure are not
"failing" on the other. That is, there is a low correlation between the results obtained by the two different measures (Barton, 2008).
Data from the National Assessment of Educational Progress illustrate this discrepancy. NAEP reports results in terms of end-of-year scores: for example, by comparing 8th graders in 1996 with 8th
graders in 2000. In contrast, to obtain a view of "growth," we would calculate how much the scores of students who were 4th graders in 1996 grew by the time they were in 8th grade in 2000. (For a
discussion of the statistical and measurement challenges inherent in this latter approach and a comparison of how the various states did on each of the two measures, see Coley, 2003).
The differences in state rankings in student achievement on NAEP using these two measures are large. For example, at the end of grade 8, Maine ranked number one in "level of knowledge," with an
average score of 273 on the 0–500 scale. However, it placed fourth from the bottom in terms of the gain in scale points from 4th to 8th grade (Barton & Coley, 2008).
These two methods will produce similar disparities in the rankings of individual schools. A school that does not make adequate yearly progress as measured by end-of-year comparisons may actually be
doing well in terms of student gains during the year, whereas a school showing high end-of-year test scores may be doing poorly if we look at how much its students are gaining during the year.
A Panoramic View of Achievement Inequality
The purpose of disaggregating achievement test scores is to gain insight about inequality and achievement gaps. The most informative view of these gaps is seen by looking at the full distribution of
achievement scores from top to bottom, as well as looking at scores of students of different ages and grades side by side. NAEP long-term trend data permit such analysis on an age basis, as we saw in
Figure 1. The degree of overlap in the score distributions of 9-year-old students, 13-year-old students, and 17-year-old students is substantial. When we display such a chart during speeches, the
gasp from audiences is sometimes audible.
The bottom line: In reading, about the bottom one-fourth of 17-year-olds score at about the same level as do the top one-tenth of 9-year-olds. This wide range of achievement levels occurs within each
racial/ethnic group to varying degrees. However, overlaps also shed light on the differences between racial/ethnic groups. For example, the distribution of scores for 17-year-old black and Hispanic
students looks similar to that for 13-year-old white students.
When we see such huge disparities in achievement among students of the same age and grade, it is hard to understand what the frequently used phrase being on grade level means. Across the United
States, students in every grade fall at different points in an achievement range that starts very high and ends very low; there is nothing "level" about it.
Achievement Gap Misunderstandings
No Child Left Behind requires states to "close the achievement gap" and bring all racial and ethnic subgroups to the same level—or so it has been widely declared. However, the law is precise in what
it says, and although its successful operation might well narrow achievement gaps, it does not require that they be closed.
NCLB requires only that all defined population subgroups reach the "proficient" level a state has established. But even if all the subgroups increase their scores to above the cut point, the gaps
between average scores of different groups may remain. In addition to tracking the gap in the percentage of subgroups reaching a particular cut point, we need to measure and compare the average
scores in each subgroup to identify whether gaps are being reduced or closed (Holland, 2002).
The difference between using a cut point standard and an average score is seen in the 2007 NAEP mathematics data for 8th graders. The gap between white and black students varies depending on whether
we compare the difference in average scores or the percentage reaching the basic or proficient level. West Virginia, for example, has a 21-point gap in average scores, a 32-point gap in the
percentage of students reaching the basic level, and only a 15-point gap in the percentage of students reaching the proficient level. Massachusetts, on the other hand, has a 40-point gap in average
scores, a 37-point gap in the percentage of students reaching the basic level, and a 45-point gap in the percentage of students reaching the proficient level (Barton & Coley, 2008).
If ever the circumstance arose, some states might be shocked to find they have met the required proficiency levels for each subgroup while maintaining exactly the same gaps in average scores that
they had before.
Giving More Meaning to Test Scores
• A student with a scale score of 259 would probably be able to recognize misrepresented data.
• A student with a score of 305 could probably identify fractions listed in ascending order.
• A student with a score of 355 would most likely be able to estimate the side length of a square, given the area.
From the standpoint of the student, the parent, the public, and the teacher, a "scale score" on a standardized test can be abstract and uninformative. Item mapping provides better information about
what students can and cannot do. And reviewing this information for subgroups of students can help policymakers better comprehend the meaning of the achievement gap.
The Goal: Better Understanding
Test score data are abstract—and important. Here, we have described different methods for clarifying the meaning of test scores, using data available from the National Assessment of Educational
Progress. When education leaders apply similar methods to clarify meaning of test scores at the state and local level, they are rewarded by a better understanding of student achievement, greater
public acceptance of important decisions that affect students, and greater success in using tests to improve teaching and learning. | {"url":"https://ascd.org:443/el/articles/measuring-the-achievement-elephant","timestamp":"2024-11-05T21:42:56Z","content_type":"application/xhtml+xml","content_length":"221648","record_id":"<urn:uuid:46507103-a969-42a5-b7cd-178ab5f3fb44>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00260.warc.gz"} |
The Integer Ambiguity
Source: GPS for Land Surveyors
The solution of the integer ambiguity, the number of whole cycles on the path from satellite to receiver, would be more difficult if it was not preceded by pseudoranges, or code phase measurements in
most receivers. This allows the centering of the subsequent double-difference solution. In other words, a pseudorange solution provides an initial estimate of the candidates for the integer ambiguity
within a smaller range than would otherwise be the case, and, as more measurements become available, it can reduce them even further. After the code-phase measurements narrows the field, there are
several methods used to solve the integer ambiguity. In the geometric method, the carrier phase data from multiple epochs are processed, and the constantly changing satellite geometry is used to find
an estimate of the actual position of the receiver. This approach is also used to show the error in the estimate by calculating how its results hold up as the geometry of the constellation changes.
This strategy requires a significant amount of satellite motion to succeed, and, therefore, takes time to converge on a solution. It works pretty well, but requires satellite motion and takes time to
converge. Another approach to solving the integer ambiguity is filtering. Independent measurements are averaged to find the estimated position with the lowest noise level. A third uses a search
through the range of possible integer ambiguity combinations from which it calculates the one with the lowest residuals. These approaches can't assess the correctness of the particular answer, but
they can provide the probability with certain conditions, that the answer is within given limits. Most GPS receivers use a combination of methods. Nearly all narrow the field by beginning with an
initial position established by the code phase measurements. They then use one or more of the methods in combination to come up with the most probable value for the solution of the integer ambiguity,
the N, the number of full wave cycles between the receiver and the satellite at lock on, the key to carrier phase observations.
Signal Squaring
There is a method that does not use the codes carried by the satellite’s signal. It is called codeless tracking, or signal squaring. It was first used in the earliest civilian GPS receivers,
supplanting proposals for a TRANSIT-like Doppler solution. It makes no use of pseudoranging and relies exclusively on the carrier phase observable. Like other methods, it also depends on the creation
of an intermediate or beat frequency. But with signal squaring, the beat frequency is created by multiplying the incoming carrier by itself. The result has double the frequency and half the
wavelength of the original. It is squared. There are some drawbacks to the method. For example, in the process of squaring the carrier, it is stripped of all its codes. The chips of the P(Y) code,
the C/A code, and the Navigation message normally modulated onto the carrier by 180° phase shifts are eliminated entirely. As discussed earlier, the signals broadcast by the satellites have phase
shifts called code states that change from +1 to –1 and vice versa, but squaring the carrier converts them all to exactly 1. The result is that the codes themselves are wiped out. Therefore, this
method must acquire information such as almanac data and clock corrections from other sources. Other drawbacks of squaring the carrier include the deterioration of the signal-to-noise ratio, because
when the carrier is squared, the background noise is squared, too. And cycle slips occur at twice the original carrier frequency. But signal squaring has its up-side as well. It reduces
susceptibility to multipath. It has no dependence on PRN codes and is not hindered by the encryption of the P code. The technique works as well on L2 as it does on L1 or L5, and that facilitates
ionospheric delay correction. Therefore, signal squaring can provide high accuracy over long baselines. So, there is a cursory look at some of the different techniques used to process the signal in
the RF section. Now let's look at the microprocessor of the receiver. | {"url":"https://www.e-education.psu.edu/geog862/node/1787","timestamp":"2024-11-06T11:57:12Z","content_type":"text/html","content_length":"39356","record_id":"<urn:uuid:c90a6dcc-7a03-4c77-98bf-b9529401db02>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00735.warc.gz"} |
Does the lagrangian contain all the information about the representations of the fields in QFT?
2332 views
Given the Lagrangian density of a theory, are the representations on which the various fields transform uniquely determined?
For example, given the Lagrangian for a real scalar field $$ \mathscr{L} = \frac{1}{2} \partial_\mu \varphi \partial^\mu \varphi - \frac{1}{2} m^2 \varphi^2 \tag{1}$$ with $(+,-,-,-)$ Minkowski sign
convention, is $\varphi$ somehow constrained to be a scalar, by the sole fact that it appears in this particular form in the Lagrangian?
As another example: consider the Lagrangian $$ \mathscr{L}_{1} = -\frac{1}{2} \partial_\nu A_\mu \partial^\nu A^\mu + \frac{1}{2} m^2 A_\mu A^\mu,\tag{2}$$ which can also be cast in the form $$ \
mathscr{L}_{1} = \left( \frac{1}{2} \partial_\mu A^i \partial^\mu A^i - \frac{1}{2} m^2 A^i A^i \right) - \left( \frac{1}{2} \partial_\mu A^0 \partial^\mu A^0 - \frac{1}{2} m^2 A^0 A^0 \right). \tag
{3}$$ I've heard$^{[1]}$ that this is the Lagrangian for four massive scalar fields and not that for a massive spin-1 field. Why is that? I understand that it produces a Klein-Gordon equation for
each component of the field: $$ ( \square + m^2 ) A^\mu = 0, \tag{4}$$ but why does this prevents me from considering $A^\mu$ a spin-1 massive field?
[1]: From Matthew D. Schwartz's Quantum Field Theory and the Standard Model, p.114:
A natural guess for the Lagrangian for a massive spin-1 field is $$ \mathcal{L} = - \frac{1}{2} \partial_\nu A_\mu \partial_\nu A_\mu + \frac{1}{2} m^2 A_\mu^2,$$ where $A_\mu^2 = A_\mu A^\mu$.
Then the equations of motion are $$ ( \square + m^2) A_\mu = 0,$$ which has four propagating modes. In fact, this Lagrangian is not the Lagrangian for a amassive spin-1 field, but the Lagrangian
for four massive scalar fields, $A_0, A_1, A_2$ and $A_3$. That is, we have reduced $4 = 1 \oplus 1 \oplus 1 \oplus 1$, which is not what we wanted.
This post imported from StackExchange Physics at 2014-11-27 10:37 (UTC), posted by SE-user glance
Why would writing out the Lagrangian for each component of a vector field prevent you from viewing the vector field as a vector field? I think whatever you heard about not being able to do so is
This post imported from StackExchange Physics at 2014-11-27 10:37 (UTC), posted by SE-user bechira
Also re the title, at least within the scope of what you're asking, the Lagrangian specifies the representation by the virtue that it is written in terms of a field in some specific rep, e.g. a
scalar field Lagrangian specifies dynamics of a scalar field not a vector one. But of course that says nothing about not being able to view components of a vector field as scalar fields
This post imported from StackExchange Physics at 2014-11-27 10:37 (UTC), posted by SE-user bechira
If each component of A satisfies the Klein-Gordon equation, that doesn't necessarily mean that the components of A transform like a vector under Lorentz transformations.
This post imported from StackExchange Physics at 2014-11-27 10:37 (UTC), posted by SE-user jabirali
For the actual Lagrangians describing a vector field, google the Maxwell Lagrangian (massless spin-1 field) and Proca Lagrangian (massive spin-1 field).
This post imported from StackExchange Physics at 2014-11-27 10:37 (UTC), posted by SE-user jabirali
@glance i see i misundersyood the question a bit the first time. the author is saying that the lagrangian constructed is not in the desired 3+1 rep, as jabirali pointed out here
This post imported from StackExchange Physics at 2014-11-27 10:37 (UTC), posted by SE-user bechira
Comment to the question (v5): As M. Schwartz mentions on top of p. 115, the energy density for the Lagrangian (2) is not bounded from below because the kinetic term of $A_0$ field has the wrong sign,
and hence the theory is not physical in the first place. Therefore the discussion of possible representations and interpretations of (2) seems somewhat academic. On the other hand, if $A_0$ did not
have the wrong sign, then $A_{\mu}$ could not be viewed as a 4-covector, but could only be interpreted as 4 scalars.
This post imported from StackExchange Physics at 2014-11-27 10:37 (UTC), posted by SE-user Qmechanic
Yes I understand that. However I'm trying to understand if there are also reasons/consistency arguments from the group-theoretical point of view. For example: why can't I say (or can I?) that $A^\mu$
is a spin-1 field for that choice of the Lagrangian? (despite the fact of it being unphysical for independent reasons). This is also addressed on that same page (p.115), and my question actually
arises from that argumentation which I'm not sure I get.
This post imported from StackExchange Physics at 2014-11-27 10:38 (UTC), posted by SE-user glance
Field $\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}$ with a given spin and mass (i.e. field which transforms under irrep of the Poincare group) must satisfy some determined conditions called
irreducibility conditions: $$ \tag 1 \hat{W}^{2}\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}} = -m^{2}\frac{n + m}{2}\left(\frac{n + m}{2} + 1\right)\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}, $$
$$ \tag 2 \hat{P}^{2}\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}} = m^{2}\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}. $$ Here $\hat{W}$ is Pauli-Lubanski operator and $\hat{P}$ is translation
operator. Representation with equal quantity $\frac{n + m}{2}$ are equivalent.
If you construct lagrangian which leads to $(1), (2)$, you will uniquely determine transformation properties of field with a given mass and spin.
This post imported from StackExchange Physics at 2014-11-27 10:38 (UTC), posted by SE-user Andrew McAddams | {"url":"https://physicsoverflow.org/25161/lagrangian-contain-information-about-representations-fields","timestamp":"2024-11-15T01:40:12Z","content_type":"text/html","content_length":"149312","record_id":"<urn:uuid:3fe7e2a4-de56-44dd-9eed-34b7ac3b2721>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00017.warc.gz"} |
Linear Hashing with l<sub>8</sub>guarantees and two-sided Kakeya bounds
We show that a randomly chosen linear map over a finite field gives a good hash function in the l8 sense. More concretely, consider a set S ? Fqn and a randomly chosen linear L: Fqn Fqt with qt taken
to be sufficiently smaller than |S|. Let US denote a random variable distributed uniformly on S. Our main theorem shows that, with high probability over the choice of L, the random variable L(US) is
close to uniform in the l8 norm. In other words, every element in the range Fqt has about the same number of elements in S mapped to it. This complements the widely-used Leftover Hash Lemma (LHL)
which proves the analog statement under the statistical, or l1, distance (for a richer class of functions) as well as prior work on the expected largest 'bucket size' in linear hash functions [1]. By
known bounds from the load balancing literature [2], our results are tight and show that linear functions hash as well as truly random function up to a constant factor in the entropy loss. Our proof
leverages a connection between linear hashing and the finite field Kakeya problem and extends some of the tools developed in this area, in particular the polynomial method.
Original language English (US)
Title of host publication Proceedings - 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022
Publisher IEEE Computer Society
Pages 419-428
Number of pages 10
ISBN (Electronic) 9781665455190
State Published - 2022
Externally published Yes
Event 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022 - Denver, United States
Duration: Oct 31 2022 → Nov 3 2022
Publication series
Name Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume 2022-October
ISSN (Print) 0272-5428
Conference 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022
Country/Territory United States
City Denver
Period 10/31/22 → 11/3/22
All Science Journal Classification (ASJC) codes
• Hashing
• Kakeya
• Leftover Hash Lemma
• Polynomial Method
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This pattern is an oscillator.
This pattern is periodic with period 15.
This pattern runs in standard life (b3s23).
The population fluctuates between 32 and 62.
This evolutionary sequence works in multiple rules, from b3-cknys23aeiy through to b34cjkqtyz5-e6-ik7cs234cjkqyz5-aiqy6-c.
Pattern RLE
Glider synthesis
#C [[ GRID MAXGRIDSIZE 14 THEME Catagolue ]]
#CSYNTH xp15_075777757z4s4zdddz9f9 costs 6 gliders (true).
#CLL state-numbering golly
x = 59, y = 14, rule = B3/S23
Sample occurrences
There are 20 sample soups in the Catagolue:
Official symmetries
Unofficial symmetries
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Usage examples
Minimal example
The main function is fit_cumhist() that takes a data frame with time-series as a first argument. In addition, you need to specify the name of the column that codes the perceptual state (state
argument) and a column that holds either dominance phase duration (duration) or its onset (onset). The code below fits data using Gamma distribution (default family) for a single run of a single
participant. By default, the function fits cumulative history time constant but uses default fixed mixed state value (mixed_state = 0.5) and initial history values (history_init = 0).
gamma_fit <- fit_cumhist(br_singleblock,
Alternatively, you specify onset of individual dominance phases that will be used to compute their duration.
You can look at the fitted value for history time constant using history_tau()
#> # A tibble: 1 x 3
#> Estimate `5.5%` `94.5%`
#> <dbl> <dbl> <dbl>
#> 1 1.00 0.787 1.25
and main effect of history for both parameters of gamma distribution
#> # A tibble: 2 x 4
#> DistributionParameter Estimate `5.5%` `94.5%`
#> <fct> <dbl> <dbl> <dbl>
#> 1 shape 1.07 0.139 2.00
#> 2 scale 0.242 -0.751 1.30
The following model is fitted for the example above, see also companion vignette for details on cumulative history computation. \[Duration[i] \sim Gamma(shape[i], rate[i]) \\ log(shape[i]) = \alpha^
{shape} + \beta^{shape}_H \cdot \Delta h[i] \\ log(rate[i]) = \alpha^{rate} + \beta^{rate}_H \cdot \Delta h[i] \\ \Delta h[i] = \text{cumulative_history}(\tau, \text{history_init})\\ \alpha^{shape},
\alpha^{rate} \sim Normal(log(3), 5) \\ \beta^{shape}_H, \beta^{rate}_H \sim Normal(0, 1) \\ \tau \sim Normal(log(1), 0.15)\]
Passing Stan control parameters
You can pass Stan control parameters via control argument, e.g.,
gamma_fit <- fit_cumhist(br_singleblock,
control=list(max_treedepth = 15,
adapt_delta = 0.99))
See Stan documentation for details (Carpenter et al. 2017).
By default, fit_cumhist() function assumes that the time-series represent a single run, so that history states are initialized only once at the very beginning. You can use run argument to pass the
name of a column that specifies individual runs. In this case, history is initialized at the beginning of every run to avoid spill-over effects.
Experimental session
Experimental session specifies which time-series were measured together and is used to compute an average dominance phase duration that, in turn, is used when computing cumulative history: \(\tau_H =
\tau \cdot <D>\), where \(\tau\) is normalized time constant and \(<D>\) is the mean dominance phase duration. This can be used to account for changes in overall alternation rate between different
sessions (days), as, for example, participants new to the stimuli tend to “speed up” over the course of days (Suzuki and Grabowecky 2007). If you do not specify session parameter then a single mean
dominance phase duration is computed for all runs of a single subject.
Random effect
The random_effect argument allows you to specify a name of the column that codes for a random effect, e.g., participant identity, bistable display (if different displays were used for a single
participant), etc. If specified, it is used to fit a hierarchical model with random slopes for the history effect (\(\beta_H\)). Note that we if random independent intercepts are used as prior
research suggest large differences in overall alternation rate between participants (Brascamp et al. 2019).
Here, is the R code that specifies participants as random effect
gamma_fit <- fit_cumhist(kde_two_observers,
And here is the corresponding model, specified for the shape parameter only as identical formulas are used for the rate parameter as well. Here, \(R_i\) codes for a random effect level (participant
identity) and a non-centered parametrization is used for the pooled random slopes.
\[Duration[i] \sim Gamma(shape[i], rate[i]) \\ log(shape[i]) = \alpha[R_i] + \beta_H[R_i] \cdot \Delta h[i] \\ \Delta H[i] = \text{cumulative_history}(\tau, \text{history_init})\\ \alpha[R_i] \sim
Normal(log(3), 5) \\ \beta_H[R_i] = \beta^{pop}_H + \beta^{z}_H[R_i] \cdot \sigma^{pop}_H\\ \beta^{pop}_H \sim Normal(0, 1) \\ \beta^{z}_H[R_i] \sim Normal(0, 1) \\ \sigma^{pop}_H \sim Exponential(1)
\\ \tau \sim Normal(log(1), 0.15)\]
Identical approach is take for \(\tau\), if tau=' "1|random"' was specified and same holds for mixed_state=' "1|random"' argument, see below.
Fixed effects
fit_cumhist() functions allows you to specify multiple fixed effect terms as a vector of strings. The implementation is restricted to:
• Only continuous (metric) independent variables should be used.
• A single value is fitted for each main effect, irrespective of whether a random effect was specified.
• You cannot specify an interaction either between fixed effects or between a fixed effect and cumulative history variable.
Although this limits usability of the fixed effects, these restrictions allowed for both a simpler model specification and a simpler underlying code. If you do require more complex models, please
refer to companion vignette that provides an example on writing model using Stan directly.
You can specify custom priors (a mean and a standard deviation of a prior normal distribution) via history_effect_prior and fixed_effects_priors arguments. The former accepts a vector with mean and
standard deviation, whereas the latter takes a named list in format .
Once fitted, you can use fixef() function to extract a posterior distribution or its summary for each effect.
Cumulative history parameters
fit_cumhist() function takes three parameters for cumulative history computation (see also companion vignette):
• tau : a normalized time constant in units of mean dominance phase duration.
• mixed_state : value used for mixed/transition state phases, defaults to 0.5.
• history_init : an initial value for cumulative history at the onset of each run. Defaults to 0.
Note that although history_init accepts only fixed values either a single value used for both states or a vector of two. In contrast, both fixed and fitted values can be used for the other three
parameters. Here are possible function argument values
• a single positive number for tau or single number within [0, 1] range for mixed_state. In this case, the value is used directly for the cumulative history computation, which is default option for
• NULL : a single value is fitted and used for all participants and runs. This is a default for tau.
• 'random' : an independent tau is fitted for each random cluster (participant, displays, etc.). random_effect argument must be specified.
• '1|random' : values for individual random cluster are sampled from a fitted population distribution (pooled values). random_effect argument must be specified.
You can specify custom priors for each cumulative history parameter via history_priors argument by specifying mean and standard deviation of a prior normal distribution. The history_priors argument
must be a named list, , e.g., history_priors = list("tau"=c(1, 0.15)).
Once fitted, you can use history_tau() and history_mixed_state()functions to obtain a posterior distribution or its summary for each parameter.
Distribution family
fit_cumhist currently supports three distributions: 'gamma', 'lognormal', and 'normal'.
\[Duration[i] \sim Gamma(shape[i], rate[i])\] For Gamma distribution independent linear models with a log link function are fitted for both shape and rate parameter. Priors for intercepts for both
parameters are \(\alpha ~ Normal(log(3), 5)\).
\[Duration[i] \sim LogNormal(\mu[i], \sigma)\] The \(\mu\) parameter is computed via a linear model with a log link function. Priors for the intercept are \(\alpha ~ Normal(log(3), 5)\). Prior for \
(\sigma\) was \(\sigma \sim Exponential(1)\).
\[Duration[i] \sim Normal(\mu[i], \sigma)\] The \(\mu\) parameter is computed via a linear model. Priors for the intercept are \(\alpha ~ Normal(3, 5)\). Prior for \(\sigma\) was \(\sigma \sim
Model comparison
Models fits can be compared via information criteria. Specifically, the log likelihood is stored in a log_lik parameter that can be directly using loo::extract_log_lik() function (see package (@ loo?
)) or used to compute either a leave-one-out cross-validation (via loo() convenience function) or WAIC (via waic()). These are information criteria that can be used for model comparison the same way
as Akaike (AIC), Bayesian (BIC), or deviance (DIC) information criteria. The latter can also be computed from log likelihood, however, WAIC and LOOCV are both preferred for multi-level models, see
(Vehtari, Gelman, and Gabry 2017). The model comparison itself can be performed via loo::loo_compare() function of the loo package.
Predicted values
You can predict durations for individual dominance phases via predict() function. You have an option of getting a summary (an average expected duration plus an optional credible interval) or
computing predicted durations for every sample.
For summary statistics with 89% credible interval.
Predictions for every sample for full length time-series (invalid samples are filled with NA):
Predictions for every sample only for valid samples:
Computing and using cumulative history
If you are interested in the cumulative history itself, you can extract from the fitted object via predict_history() function. Note that there are five different history types you can extract:
• "1": cumulative history for the first perceptual state, i.e., state with index of 1.
• "2": cumulative history for the second perceptual state, i.e., state with index of 2.
• "dominant": for the state that is dominant during the following phase.
• "suppressed": for the state that is suppressed during the following phase.
• "difference": difference between cumulative histories (\(\Delta h = h_{suppressed} - h{dominant}\)), which is used in linear models.
Alternatively, you can skip fitting and compute history directly using predefined values via compute_history().
Brascamp, Jan W., Cheng Stella Qian, David Z. Hambrick, and Mark W. Becker. 2019.
“Individual differences point to two separate processes involved in the resolution of binocular rivalry.” Journal of Vision
19 (12): 15.
Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. 2017.
“Stan : A Probabilistic Programming Language.” Journal of Statistical Software
76 (1).
Suzuki, Satoru, and Marcia Grabowecky. 2007.
“Long-term speeding in perceptual switches mediated by attention-dependent plasticity in cortical visual processing.” Neuron
56 (4): 741–53.
Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017.
“Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.” Statistics and Computing
27 (5): 1413–32. | {"url":"https://cran.r-project.org/web/packages/bistablehistory/vignettes/usage-examples.html","timestamp":"2024-11-03T07:33:35Z","content_type":"text/html","content_length":"32765","record_id":"<urn:uuid:b570d6af-0cd3-4af2-bf76-9c0949f96dca>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00519.warc.gz"} |
To design Series RL circuit and find out the current flowing thorugh each component.
Resistor, Inductor, AC power source, ammeter, voltmeter, connection wire etc..
When we apply an ac voltage to a series RL circuit as shown below, the circuit behaves in some ways the same as the series RC circuit, and in[R] is still in phase with I, and V[L] is still
90° out of phase with I. However, this time V[L] leads I  it is at +90° instead of -90°.
For this circuit, we will assign experimental values as follows: R = 25Ω, L = 1 H and V[AC] = 10 V[rms]. We build the circuit and measure 9.29 V across L, and 3.7 V across R. As we might have
expected, this exceeds the source voltage by a substantial amount and the phase shift is the reason for it.
The vectors for this example circuit are shown to the right. This time the composite phase angle is positive instead of negative, because VL leads IL . But to determine just what that phase angle is,
we must start by determining XL and then calculating the rest of the circuit parameters.
This can be easily verified using the simulator, by creating the above mentioned circuit and measuring the current and voltages across the resistor and inductor.
These circuits exhibit important types of behaviour so they are fundamental to analogue electronics. It has wide applications in Electronic filter topology and Piezo electric shunt damping system. | {"url":"https://vlab.amrita.edu/?sub=1&brch=75&sim=332&cnt=1","timestamp":"2024-11-08T14:21:15Z","content_type":"text/html","content_length":"21743","record_id":"<urn:uuid:e1868eeb-d72b-4fb3-a483-da0bd138813d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00218.warc.gz"} |
Sets and Functions MCQs [PDF] Quiz Questions Answers | Sets and Functions MCQ App Download & e-Book: Test 1
Class 10 Math MCQs - Chapter 13
Sets and Functions Multiple Choice Questions (MCQs) PDF Download - 1
The Sets and functions Multiple Choice Questions (MCQs) with Answers PDF (sets and functions MCQs PDF e-Book) download Ch. 13-1 to study Grade 10 Math Course. Learn Sets and Functions Quiz Questions
and Answers to study online classes courses. The Sets and Functions MCQs App Download: Free learning app for recognize of operations on sets, binary relation, sets test prep for online high school
The MCQ: If 2 sets A and B are given, then the set consisting of all the elements which are either in A or in B or in both is called; "Sets and Functions" App Download (Free) with answers: Union of A
and B; Intersection of A and B; Complement of A; Complement of B; to study online classes courses. Solve Recognize of Operations on Sets MCQ Questions, download Google eBook (Free Sample) for online
school and college.
Sets and Functions MCQs with Answers PDF Download: Quiz 1
MCQ 1:
If 2 sets A and B are given, then the set consisting of all the elements which are either in A or in B or in both is called
1. Intersection of A and B
2. union of A and B
3. Complement of A
4. Complement of B
MCQ 2:
The set consisting of all the first elements of each ordered pair in the relation is called
1. subset
2. domain of relation
3. range of relation
4. complement of a set
MCQ 3:
If U = {1,2,3,4,5} and A = {2,4} then A' should be
1. {2,4,5}
2. {2,4}
3. {1,2,3,4,5}
4. {1,3,5}
MCQ 4:
The number of elements in power set {1, 2, 3} are
1. 5
2. 6
3. 7
4. 8
MCQ 5:
The range of R = {(0, 2), (2, 4), (3, 4), (4, 5)} is
1. {0, 2, 4, 5}
2. {0, 2, 3, 4}
3. {2, 4, 5}
4. {3, 4, 5}
Sets and Functions Learning App: Free Download Android & iOS
The App: Sets and Functions MCQs App to learn Sets and Functions Textbook, 10th Grade Math MCQ App, and College Math MCQs App. The "Sets and Functions" App to free download iOS & Android Apps
includes complete analytics with interactive assessments. Download App Store & Play Store learning Apps & enjoy 100% functionality with subscriptions! | {"url":"https://mcqlearn.com/grade10/math/sets-and-functions.php","timestamp":"2024-11-08T17:36:53Z","content_type":"text/html","content_length":"72346","record_id":"<urn:uuid:a62bbf24-5398-49c6-bfa2-c22d9cb49c8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00174.warc.gz"} |
Definable groups of partial automorphisms
The motivation for this paper is to extend the known model-the-oretic treatment of differential Galois theory to the case of linear difference equations (where the derivative is replaced by an
automorphism). The model-theoretic difficulties in this case arise from the fact that the corresponding theory ACFA does not eliminate quantifiers. We therefore study groups of restricted
automorphisms, preserving only part of the structure. We give con-ditions for such a group to be (infinitely) definable, and when these conditions are satisfied we describe the definition of the
group and the action explicitly. We then examine the special case when the theory in question is ob-tained by enriching a stable theory with a generic automorphism. Finally, we interpret the results
in the case of ACFA, and explain the connection of our construction with the algebraic theory of Picard-Vessiot extensions. The only model-theoretic background assumed is the notion of a defin-able
• ACFA
• Definable Galois groups
• Difference equations
• Picard-Vessiot theory
ASJC Scopus subject areas
• General Mathematics
• General Physics and Astronomy
Dive into the research topics of 'Definable groups of partial automorphisms'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/definable-groups-of-partial-automorphisms","timestamp":"2024-11-06T01:50:27Z","content_type":"text/html","content_length":"55095","record_id":"<urn:uuid:29d8d977-391b-4faa-960c-b6a73471b2c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00722.warc.gz"} |
Systems of Equations and Augmented Matrices (examples, solutions, videos, worksheets, games, activities)
Related Topics:
Common Core (Algebra) Common Core for Mathematics
Examples, solutions, videos, and lessons to help High School students learn how to represent a system of linear equations as a single matrix equation in a vector variable.
Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater)
Suggested Learning Targets
• Write a system of linear equations as a single matrix equation.
• Find the inverse of the coefficient matrix in the equation, if it exits. Use the inverse of the coefficient matrix to solve the system. Use technology for matrices with dimensions 3 by 3 or
Common Core: HSA-REI.C.8, HSA-REI.C.9
Introduction to Augmented Matrices
This video introduces augmented matrices for the purpose of solving systems of equations. It also introduces row echelon and reduced row echelon form.
Row Echelon Form
Augmented Matrices: Row Echelon Form
This video shows how to transform and augmented matrix to row echelon form to solve a system of equations.
Ex 1: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)
This video provides an example of how to solve a system of two linear equations with two unknowns by writing an augmented matrix in row echelon form. This example has one solution.
Ex 2: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)
This video provides an example of how to solve a system of two linear equations with two unknowns by writing an augmented matrix in row echelon form. This example has no solution.
Ex 3: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)
This video provides an example of how to solve a system of two linear equations with two unknowns by writing an augmented matrix in row echelon form. This example has infinite solutions.
Reduced Row Echelon Form
Augmented Matrices: Reduced Row Echelon Form
This video shows how to transform and augmented matrix to reduced row echelon form to solve a system of equations.
Ex 1: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
This video explains how to solve a system of equations by writing an augmented matrix in reduced row echelon form. This example has one solution.
Ex 2: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
This video explains how to solve a system of equations by writing an augmented matrix in reduced row echelon form. This example has no solution.
Ex 3: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
This video explains how to solve a system of equations by writing an augmented matrix in reduced row echelon form. This example has an infinite number of solutions.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/systems-equation-augmented-matrices-hsa-rei8.html","timestamp":"2024-11-05T00:53:44Z","content_type":"text/html","content_length":"43138","record_id":"<urn:uuid:be21a4ce-355c-4f79-815d-5ed0a9181806>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00490.warc.gz"} |
60 Free 4 Digit by 1 Digit Multiplication Word Problems with Answers
Get 150+ Free Math Worksheets!
These 4 digit by 1 digit multiplication word problems worksheets will help to visualize and understand methods of solving multiplication word problems. In this article, 1st & 2nd-grade students will
learn basic multiplication methods and can improve their basic math skills with our free printable worksheets.
5 Worksheets on 4 Digit by 1 Digit Multiplication Word Problems
In this article, I will provide you with a number of word problems for your kiddos. Download the worksheet and practice.
4 Digit by 1 Digit Multiplication: Don’t Worry We Are Here
To calculate 4 digit by 1 digit multiplication word problems, let’s consider a 4 digit by 1 digit multiplication problem here. Suppose, 4295 is the 4 digit multiplicand and 6 is the multiplier. So,
the problem will be 4295 x 6 =? So first, multiply the one’s place number by 6.
In this problem, the one’s place number is 5. This will be multiplied by 6. 5 x 6 = 30.
Here 0 will be written in the one’s place of the product and 3 is carry.
Then multiply the ten’s place by 6. Here the te’s place number is 9. So, 9 x 6 = 54. You have a carry 3. Now, add the carry with 54. So, 54 + 3 = 57.
Write down the 7 in ten’s place of the product and 5 is carry. In this way, multiply the other numbers by 6 and add the carry if there is any.
At last, the result will be 4295 x 6 = 25770. Did this result match your kiddo’s product?
If the answer is yes, then congratulate him or her for getting the basic idea of 4 digit by one digit multiplication. If he or she failed to solve this problem, help him or her to solve the problem.
2 Interactive Word Problems for 4 Digit by 1 Digit Multiplication
These worksheets help your young champion create a strong foundation by teaching them the fundamentals of mathematical operation learning. Download the worksheets and practice.
Element-Based Word Problems of 4 Digit by 1 Digit Multiplication
In this portion of this article, the word problems are element-based. You can use any kind of element to make a visualization of the numbers and operations. If you don’t get any kind of element, then
use colorful dots.
4 Digit by 1 Digit Money-Related Multiplication Word Problems
This is another activity for 4 digit by 1 digit multiplication word problems. Hopefully, this problem will help your kiddos to grow more capability in multiplication.
4 Digit by 2 Digit Multiplication Word Problems
The kiddos who can perfectly complete the word problems of 4 digit by 1 digit multiplication word problems worksheet, you can proceed to 4 digit to 2 digit multiplication with them. Word problems
containing 4 digit by 2 digit multiplication will be found in the pdf.
Download Free Printable Worksheet
Please download the following worksheets for 3rd-grade, 4th-grade, and 5th-grade students.
So today, we’ve discussed division with 4-digit by 1-digit multiplication word problems worksheets using the concepts element-based problems, and money-related word problems. Download our free
worksheets, and after practicing these worksheets, students will surely improve their mathematical skills and have a better understanding of multiplication.
Hi there! This is Souptik Roy, a graduate of the Bangladesh University of Engineering and Technology, working as a Content Developer for the You Have Got This Math project of SOFTEKO. I am a person
with a curious and creative mind. After finishing my Engineering degree, I want to explore different fields. This is why I am working here as a content developer. I have a massive interest in
creative content writing. When I find that someone can learn something from my articles, this gives a lot of inspiration. hopefully, you will find interest in my article, if you have a child and want
to teach them math with fun. | {"url":"https://youvegotthismath.com/4-digit-by-1-digit-multiplication-word-problems/","timestamp":"2024-11-06T17:42:30Z","content_type":"text/html","content_length":"350860","record_id":"<urn:uuid:8cf153d7-abbe-4e34-acd2-45c2a5947285>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00478.warc.gz"} |
%change Year over Year
Hi all,
I am trying to make a card where I will have bars for total premium by year, and one line which will show % change betwen years (current to previous, previous to two years ago etc.). Do you have any
advice how to do it, I tried to find solution on Dojo and Domo support but they point me to PoP knowledge base page.
Thank you in advance.
• I built a walkthrough for something similar on another post. Check out this link:
Obviously instead of doing one by month, you'll need to tweak it to show years if thats what you're wanting.
For your % change calculation (starting with "Current Period / Previous Period - 1" as your Variance calculation) you'll want to build out a Case When for whatever X period you decide.
So if you're showing X1 as 2016 vs 2015 and X2 is 2016 vs 2017 you can do this:
CASE WHEN year = 2016
THEN SUM(2016 metric) / SUM (2015 metric) - 1
WHEN year = 2017
THEN SUM(2017 metric / SUM (2015 metric) - 1
But instead of defining a specific year, you can do DATE_SUB(CURRENT_DATE(), INTERVAL 1 year) for 2017.
I know that's a lot of info to digest, just let me know if you have any questions.
**Please mark "Accept as Solution" if this post solves your problem
**Say "Thanks" by clicking the "heart" in the post that helped you.
• Good morning Valiant,
I am trying to implement the code you sent me into my function.
First I made this:
CASE WHEN `Type` = 'Current'
THEN YEAR(CURDATE())
WHEN `Type` = 'Previous_Yr'
THEN YEAR(DATE_ADD(CURDATE(), INTERVAL - 1 YEAR))
WHEN `Type` = '2_Yrs_Ago'
THEN YEAR(DATE_ADD(CURDATE(), INTERVAL - 2 YEAR))
WHEN `Type` = '3_Yrs_Ago'
THEN YEAR(DATE_ADD(CURDATE(), INTERVAL - 3 YEAR))
WHEN `Type` = '4_Yrs_Ago'
THEN YEAR(DATE_ADD(CURDATE(), INTERVAL - 4 YEAR))
WHEN `Type` = '5_Yrs_Ago'
THEN YEAR(DATE_ADD(CURDATE(), INTERVAL - 5 YEAR))
And then:
CASE WHEN `Type` = 'Current'
THEN (SUM(`Premium`) WHERE `Type` = 'Current') / (SUM(`Premium`) WHERE `Type` = 'Previous_Yr') - 1
WHEN `Type` = 'Previous_Yr'
THEN (SUM(`Premium`) WHERE `Type` = 'Previous_Yr') / (SUM(`Premium`) WHERE `Type` = '2_Yrs_Ago') - 1
But I am getting an error:
This calculation contained a syntax error.
Am I close? ?
Thank you,
• The second statement is incorrect because of the WHERE clauses.
Can you give me a few lines of example data using your available column names? That should help me better understand what you're working with and I can tailor my answer to suit.
• I am trying to get Premium of every single row where Type is certain year. Of course, let me send you example.
Thank you,
• Ok, so while we could go down a long path of lots of sql transform and beast modes to get this answer, let's see if the simple solution works first.
In your data, can you edit the data (either using an ETL or SQL transform) to convert each year (ie, 2013) to something like this (2013-01-01). The end result being that we want to convert that
column to a recognized date column type. Once you have that, you can go to the card, choose the Period over Period type and select 'Variance bar line'. Using the Year as your X axis and the
Premium as your Y, adjust the time selection to match the following:
Once you have that, you'll have bars for each year compared to the previous year and a variance (% change) line for your chart.
Let me know if that will work for your needs,
• I already have made this one, but we would like to have one bar for each year and one line showing % change between current and previous year. That's why I am trying to find a solution for this.
Thank you very much for your help,
• ok, so in that case, you're going to need to create to do the following steps via either ETL or SQL transforms:
1. Sum your premiums by year
SELECT `Year`, SUM(`Premium`) as 'PremiumTotal', `Year`-1 AS 'LastYear'
FROM dataset
GROUP BY `Year`
2. Add a 'Last Years Premiums' to your dataset
SELECT a.*,
b.`PremiumTotal` AS 'LastYearPremium'
FROM transform1 AS a LEFT JOIN transform1 AS b ON a.`Year` = b.`LastYear`
With that result you'll use Year as your X axis and your calc for YoY % change is
`PremiumTotal` / `LastYearPremium` - 1
Hope that helps,
• I tried to make a new dataset but output shows ammount of next year into last year field instead of last year amount. Confused!
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Trigonometry in CSS and JavaScript: Beyond Triangles
Original Source: http://feedproxy.google.com/~r/tympanus/~3/vR1zyMHn0Ck/
In the previous article we looked at how to clip an equilateral triangle with trigonometry, but what about some even more interesting geometric shapes?
This article is the 3rd part in a series on Trigonometry in CSS and JavaScript:
Introduction to TrigonometryGetting Creative with Trigonometric FunctionsBeyond Triangles (this article)
Plotting regular polygons
A regular polygon is a polygon with all equal sides and all equal angles. An equilateral triangle is one, so too is a pentagon, hexagon, decagon, and any number of others that meet the criteria. We
can use trigonometry to plot the points of a regular polygon by visualizing each set of coordinates as points of a triangle.
Polar coordinates
If we visualize a circle on an x/y axis, draw a line from the center to any point on the outer edge, then connect that point to the horizontal axis, we get a triangle.
If we repeatedly rotated the line at equal intervals six times around the circle, we could plot the points of a hexagon.
But how do we get the x and y coordinates for each point? These are known as cartesian coordinates, whereas polar coordinates tell us the distance and angle from a particular point. Essentially, the
radius of the circle and the angle of the line. Drawing a line from the center to the edge gives us a triangle where hypotenuse is equal to the circle’s radius.
We can get the angle in degrees by diving 360 by the number of vertices our polygon has, or in radians by diving 2pi radians. For a hexagon with a radius of 100, the polar coordinates of the
uppermost point of the triangle in the diagram would be written (100, 1.0472rad) (r, θ).
An infinite number of points would enable us to plot a circle.
Polar to cartesian coordinates
We need to plot the points of our polygon as cartesian coordinates – their position on the x and y axis.
As we know the radius and the angle, we need to calculate the adjacent side length for the x position, and the opposite side length for the y position.
Therefore we need Cosine for the former and Sine for the latter:
adjacent = cos(angle) * hypotenuse
opposite = sin(angle) * hypotenuse
We can write a JS function that returns an array of coordinates:
const plotPoints = (radius, numberOfPoints) => {
/* step used to place each point at equal distances */
const angleStep = (Math.PI * 2) / numberOfPoints
const points = []
for (let i = 1; i <= numberOfPoints; i++) {
/* x & y coordinates of the current point */
const x = Math.cos(i * angleStep) * radius
const y = Math.sin(i * angleStep) * radius
/* push the point to the points array */
points.push({ x, y })
return points
We could then convert each array item into a string with the x and y coordinates in pixels, then use the join() method to join them into a string for use in a clip path:
const polygonCoordinates = plotPoints(100, 6).map(({ x, y }) => {
return `${x}px ${y}px`
shape.style.clipPath = `polygon(${polygonCoordinates})`
See the Pen Clip-path polygon by Michelle Barker (@michellebarker) on CodePen.dark
This clips a polygon, but you’ll notice we can only see one quarter of it. The clip path is positioned in the top left corner, with the center of the polygon in the corner. This is because at some
points, calculating the cartesian coordinates from the polar coordinates is going to result in negative values. The area we’re clipping is outside of the element’s bounding box.
To position the clip path centrally, we need to add half of the width and height respectively to our calculations:
const xPosition = shape.clientWidth / 2
const yPosition = shape.clientHeight / 2
const x = xPosition + Math.cos(i * angleStep) * radius
const y = yPosition + Math.sin(i * angleStep) * radius
Let’s modify our function:
const plotPoints = (radius, numberOfPoints) => {
const xPosition = shape.clientWidth / 2
const yPosition = shape.clientHeight / 2
const angleStep = (Math.PI * 2) / numberOfPoints
const points = []
for (let i = 1; i <= numberOfPoints; i++) {
const x = xPosition + Math.cos(i * angleStep) * radius
const y = yPosition + Math.sin(i * angleStep) * radius
points.push({ x, y })
return points
Our clip path is now positioned in the center.
See the Pen Clip-path polygon by Michelle Barker (@michellebarker) on CodePen.dark
Star polygons
The types of polygons we’ve plotted so far are known as convex polygons. We can also plot star polygons by modifying our code in the plotPoints() function ever so slightly. For every other point, we
could change the radius value to be 50% of the original value:
/* Set every other point’s radius to be 50% */
const radiusAtPoint = i % 2 === 0 ? radius * 0.5 : radius
/* x & y coordinates of the current point */
const x = xPosition + Math.cos(i * angleStep) * radiusAtPoint
const y = yPosition + Math.sin(i * angleStep) * radiusAtPoint
See the Pen Clip-path star polygon by Michelle Barker (@michellebarker) on CodePen.dark
Here’s an interactive example. Try adjusting the values for the number of points and the inner radius to see the different shapes that can be made.
See the Pen Clip-path adjustable polygon by Michelle Barker (@michellebarker) on CodePen.dark
Drawing with the Canvas API
So far we’ve plotted values to use in CSS, but trigonometry has plenty of applications beyond that. For instance, we can plot points in exactly the same way to draw on a <canvas> with Javascript. In
this function, we’re using the same function as before (plotPoints()) to create an array of polygon points, then we draw a line from one point to the next:
const canvas = document.getElementById(‘canvas’)
const ctx = canvas.getContext(‘2d’)
const draw = () => {
/* Create the array of points */
const points = plotPoints()
/* Move to starting position and plot the path */
ctx.moveTo(points[0].x, points[0].y)
points.forEach(({ x, y }) => {
ctx.lineTo(x, y)
/* Draw the line */
See the Pen Canvas polygon (simple) by Michelle Barker (@michellebarker) on CodePen.dark
We don’t even have to stick with polygons. With some small tweaks to our code, we can even create spiral patterns. We need to change two things here: First of all, a spiral requires multiple
rotations around the point, not just one. To get the angle for each step, we can multiply pi by 10 (for example), instead of two, and divide that by the number of points. That will result in five
rotations of the spiral (as 10pi divided by two is five).
const angleStep = (Math.PI * 10) / numberOfPoints
Secondly, instead of an equal radius for every point, we’ll need to increase this with every step. We can multiply it by a number of our choosing to determine how far apart the lines of our spiral
are rendered:
const multiplier = 2
const radius = i * multiplier
const x = xPosition + Math.cos(i * angleStep) * radius
const y = yPosition + Math.sin(i * angleStep) * radius
Putting it all together, our adjusted function to plot the points is as follows:
const plotPoints = (numberOfPoints) => {
const angleStep = (Math.PI * 10) / numberOfPoints
const xPosition = canvas.width / 2
const yPosition = canvas.height / 2
const points = []
for (let i = 1; i <= numberOfPoints; i++) {
const radius = i * 2 // multiply the radius to get the spiral
const x = xPosition + Math.cos(i * angleStep) * radius
const y = yPosition + Math.sin(i * angleStep) * radius
points.push({ x, y })
return points
See the Pen Canvas spiral – simple by Michelle Barker (@michellebarker) on CodePen.dark
At the moment the lines of our spiral are at equal distance from each other, but we could increase the radius exponentially to get a more pleasing spiral. By using the Math.pow() function, we can
increase the radius by a larger number for each iteration. By the golden ratio, for example:
const radius = Math.pow(i, 1.618)
const x = xPosition + Math.cos(i * angleStep) * radius
const y = yPosition + Math.sin(i * angleStep) * radius
See the Pen Canvas spiral by Michelle Barker (@michellebarker) on CodePen.dark
We could also rotate the spiral, using (using requestAnimationFrame). We’ll set a rotation variable to 0, then on every frame increment or decrement it by a small amount. In this case I’m
decrementing the rotation, to rotate the spiral anti-clockwise
let rotation = 0
const draw = () => {
const { width, height } = canvas
/* Create points */
const points = plotPoints(400, rotation)
/* Clear canvas and redraw */
ctx.clearRect(0, 0, width, height)
ctx.fillStyle = ‘#ffffff’
ctx.fillRect(0, 0, width, height)
/* Move to beginning position */
ctx.moveTo(points[0].x, points[0].y)
/* Plot lines */
points.forEach((point, i) => {
ctx.lineTo(point.x, point.y)
/* Draw the stroke */
ctx.strokeStyle = ‘#000000’
/* Decrement the rotation */
rotation -= 0.01
We’ll also need to modify our plotPoints() function to take the rotation value as an argument. We’ll use this to increment the x and y position of each point on every frame:
const x = xPosition + Math.cos(i * angleStep + rotation) * radius
const y = yPosition + Math.sin(i * angleStep + rotation) * radius
This is how our plotPoints() function looks now:
const plotPoints = (numberOfPoints, rotation) => {
/* 6 rotations of the spiral divided by number of points */
const angleStep = (Math.PI * 12) / numberOfPoints
/* Center the spiral */
const xPosition = canvas.width / 2
const yPosition = canvas.height / 2
const points = []
for (let i = 1; i <= numberOfPoints; i++) {
const r = Math.pow(i, 1.3)
const x = xPosition + Math.cos(i * angleStep + rotation) * r
const y = yPosition + Math.sin(i * angleStep + rotation) * r
points.push({ x, y, r })
return points
See the Pen Canvas spiral by Michelle Barker (@michellebarker) on CodePen.dark
Wrapping up
I hope this series of articles has given you a few ideas for how to get creative with trigonometry and code. I’ll leave you with one more creative example to delve into, using the spiral method
detailed above. Instead of plotting points from an array, I’m drawing circles at a new position on each iteration (using requestAnimationFrame).
See the Pen Canvas spiral IIII by Michelle Barker (@michellebarker) on CodePen.dark
Special thanks to George Francis and Liam Egan, whose wonderful creative work inspired me to delve deeper into this topic!
The post Trigonometry in CSS and JavaScript: Beyond Triangles appeared first on Codrops.
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Hypersudoku - Sudopedia Mirror (2024)
Mirrored from Sudopedia, the Free Sudoku Reference Guide
• 1 Hypersudoku
• 2 How to Play Hypersudoku
• 3 Hypersudoku Logic
□ 3.1 New Game
□ 3.2 Example1
□ 3.3 Example 2
□ 3.4 Example 3
□ 3.5 Example 4
□ 3.6 Example 5
□ 3.7 Example 6
□ 3.8 Example 7
□ 3.9 Example 8
• 4 Hidden Constraints
• 5 Solving Techniques
• 6 Minimum Number of Givens
• 7 External Link
• 8 See Also
Hypersudoku, Windoku, NRC-Sudoku, or Four-square Sudoku is one of the Sudoku Variations with additional constraints. This format was first introduced in the Dutch newspaper NRC Handelsblad by Peter
Ritmeester and the first playable online version by Chris McCusker.
How to Play Hypersudoku
This game requires no mathematics skills. It is purely a logic game with the only prerequisite to be able to count to nine.
Hypersudoku Rules
The rules for playing hypersudoku are very simple.
1. All rows must have all the numbers from 1 - 9 in them (none can be repeated). There are 9 rows in the game.
2. All columns must have all the numbers from 1 - 9 in them (none can be repeated). There are 9 columns in the game.
3. All 3x3 squares must have all the numbers from 1 - 9 in them (none can be repeated). There are 13 squares in the game. There are the 9 underlaying 3x3 squares (divided by the dark blue lines) and
the 4 overlaying (shown in light blue in the following diagrams).
Hypersudoku Logic
New Game
Hypersudoku is a game of logic. The following is a brief explanation of the logic necessary to develop your strategies for the game of hypersudoku.
As you will see in the following diagrams there are many starting points for the game in figure 1. This is a fairly typical layout for an easy level hypersudoku game. These are only a few of the
many starting points in this particular game.
Looking at the center square we can see that column 4 has a 9 at the bottom(circled in red). This means that a nine can not appear again in this column.
Column 5 also has a 9 in it. This time up in the top middle square.
This leaves column 6 as the only option for a 9 to appear in the center square. As 6 and 4 take up 2 of the 3 squares available, there is only one square left. This is the one with the blue tick in
it. This square will contain the 9.
Example 2
In figure 3 if we look at the top right square, we can see that rows 2 and 3 already have a 5 (circled in red) in them.
This leaves the square with the blue tick as the only possibility for a 5 in this square.
Example 3
Again the same starting game, but this time looking at the middle right square. Rows 4 and 6 contain a 4 already, leaving only row 3.
As the other 2 squares already contain numbers, the square with the blue tick is the only square that can contain a 4.
Example 4
In figure 5 we can see that the top row and third row already have a 6 in them. The sixth column also has a 6 in it.
This leaves the square with the blue tick as the only square that could contain a 6 in this block.
Example 5
If we look at the bottom center square we can see that columns 5 and 6 both contain a 5 (circled in red).
This leaves only one square in this block that can contain a 5.(square with blue tick)
Example 6
In figure 7 we see that column 1 and 2 have a 2 in them already. Also note that row 5 has a 2 in it. (circled in red)
Again this only leaves the square with the blue tick in it as the only possible square in this block that could contain a 2.
Example 7
If we look at the top right blue square, it has to have a nine in it. The top corner square (in red) that overlaps the blue square already has a nine in it (circled in red). The middle top square
that also overlaps the blue square has a nine already in it (circled in red). The middle right-hand square also has a nine in it which only leaves the square with the blue tick as the only possible
square for nine to go in in the top right blue square.
As you can see the logic of the overlapping squares is quite helpful for solving these puzzles.
Example 8
In this example if we look at the bottom left blue square, it has to have a six in it. The bottom corner square (in red) that overlaps the blue square already has a six in it (circled in red). This
six is also the only six that can exist in this column so that eliminates the top middle blue square (marked with red cross) .The middle bottom square that also overlaps the blue square has a six
already in it (circled in red). This only leaves the square with the blue tick as the only possible square for six to go in in the bottom right blue square.
As you can see the logic of the overlapping squares is quite helpful for solving these puzzles.
There are many possible first moves to the start of the hypersudoku game. These are certainly by no means the only starting moves for this particular game.
Hidden Constraints
Although there are only 4 additional constraints mentioned by the publisher, the position of these 4 windows indirectly creates 5 additional constraints. These are shown in the following picture.
Solving Techniques
The solving techniques for a Windoku are similar to those of regular Sudoku. The extra constraints are placed in such a way that there are many more intersections to deal with. Each additional
constraint can contain subsets and strong links which could be used in coloring techniques.
There are fewer Unique Rectangles in the grid, because not all 468 possible unique rectangles for a standard Sudoku are located in exactly 2 additional houses.
Minimum Number of Givens
Valid Windokus with 11 givens exist. However, it is not known whether this is the minimum number of givens. See this page for examples for such Windokus.
External Link
See Also
• Sudoku-X
• Center Dot
• Asterisk
This page was last modified 03:09, 27 May 2008. | {"url":"https://cckdj.com/article/hypersudoku-sudopedia-mirror","timestamp":"2024-11-09T22:14:04Z","content_type":"text/html","content_length":"73525","record_id":"<urn:uuid:704273fc-7f74-4453-bf94-6195c02a10fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00384.warc.gz"} |
Excel Formula Python Lookup Value Across Sheets
In this tutorial, you will learn how to write an Excel formula in Python that allows you to look up a word and find the corresponding value across multiple sheets. This can be achieved using the
VLOOKUP function, which is a powerful tool for searching and retrieving data in Excel.
To perform the lookup, you will need to specify the range of sheets to search, which in this case is Jan to Dec. The formula will search for the value of the word in cell E7 and return the value from
the specified column in the range.
The VLOOKUP function takes four arguments: the value to be looked up, the range of cells to search, the column index from which to return the value, and a flag indicating whether an exact match is
To illustrate how the formula works, let's consider an example. Suppose we have data in the sheets Jan to Dec, where each sheet contains a list of words in column A and their corresponding values in
column B. If the value in cell E7 is 'Dog', the formula will search for 'Dog' in the first column of the range Jan:Dec!A:B and return the corresponding value from the second column.
By using this formula, you can easily retrieve values from multiple sheets based on a specific word or value. This can be useful for analyzing data across different time periods or categories.
In conclusion, the Excel formula for Python that looks up a word in E7 and finds the corresponding value across sheets Jan:Dec is a powerful tool for data analysis and retrieval. By understanding how
to use the VLOOKUP function and specifying the appropriate range, you can efficiently search and retrieve data from multiple sheets in Excel using Python.
An Excel formula
Formula Explanation
This formula uses the VLOOKUP function to look up the value of the word in cell E7 across the sheets Jan to Dec.
Step-by-step explanation
1. The VLOOKUP function searches for a value in the first column of a range (Jan:Dec!A:B) and returns a value in the same row from a specified column (column 2 in this case).
2. The value to be looked up is specified as E7, which is the cell containing the word to be searched.
3. The range Jan:Dec!A:B represents the range of cells to search for the value. It includes columns A and B in the sheets Jan to Dec.
4. The number 2 represents the column index from which to return the value. In this case, it is the second column (column B) of the range Jan:Dec!A:B.
5. The FALSE argument at the end of the formula indicates an exact match is required. This means that the formula will only return a value if an exact match is found in the first column of the
For example, let's say we have the following data in the sheets Jan to Dec:
Sheet Jan:
| A | B |
| | |
| Cat | 10 |
| Dog | 15 |
| Bird | 20 |
Sheet Feb:
| A | B |
| | |
| Cat | 12 |
| Dog | 18 |
| Bird | 25 |
Sheet Mar:
| A | B |
| | |
| Cat | 11 |
| Dog | 17 |
| Bird | 22 |
If the value in cell E7 is "Dog", the formula =VLOOKUP(E7,Jan:Dec!A:B,2,FALSE) would return the value 15, which is the value of "Dog" in the second column of the range Jan:Dec!A:B.
Similarly, if the value in cell E7 is "Bird", the formula would return the value 20 from the sheet Jan, 25 from the sheet Feb, and 22 from the sheet Mar, depending on which sheet contains the
matching value. | {"url":"https://codepal.ai/excel-formula-generator/query/31O7FHWU/excel-formula-python-lookup-value-across-sheets","timestamp":"2024-11-08T09:34:07Z","content_type":"text/html","content_length":"93905","record_id":"<urn:uuid:23b477a1-0719-4000-824e-cfc00ef4ea9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00805.warc.gz"} |
Differential forms on Carnot groups
Tripaldi, Francesca
Differential forms on Carnot groups.
[Laurea magistrale], Università di Bologna, Corso di Studio in
Matematica [LM-DM270]
, Documento ad accesso riservato.
Documenti full-text disponibili:
Documento PDF
Full-text non accessibile
Download (780kB) | Contatta l'autore
The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a
construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in
different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in
complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights,
which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex
also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear
in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's
spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an
alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which
will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key
examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the
spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight
filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.
The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a
construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in
different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in
complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights,
which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex
also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear
in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's
spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an
alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which
will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key
examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the
spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight
filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.
Altri metadati | {"url":"https://amslaurea.unibo.it/5707/","timestamp":"2024-11-13T02:30:01Z","content_type":"application/xhtml+xml","content_length":"39187","record_id":"<urn:uuid:73da669e-3d43-4132-9cc4-d93ff484e376>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00388.warc.gz"} |
Next: MINIMUM TEST COLLECTION Up: Covering, Hitting, and Splitting Previous: MINIMUM SET COVER   Index
• INSTANCE: Collection C of subsets of a finite set S.
• SOLUTION: A set cover for S, i.e., a subset S belongs to at least one member of C'.
• MEASURE: Sum of cardinalities of the subsets in the set cover, i.e.,
• Good News: Approximable within 276].
• Bad News: As hard to approximate as MINIMUM SET COVER [365].
• Comment: Transformation from MINIMUM SET COVER. The only difference between MINIMUM SET COVER and MINIMUM EXACT COVER is the definition of the objective function.
Viggo Kann | {"url":"https://www.csc.kth.se/~viggo/wwwcompendium/node147.html","timestamp":"2024-11-11T07:51:15Z","content_type":"text/html","content_length":"4054","record_id":"<urn:uuid:ea770319-f56b-4870-9b86-8f931f1f71dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00123.warc.gz"} |
RLC Impedance Calculator
Unravel the Z-Mystery with a Z-Solution!
# RLC Impedance Formula
Z = √(R^2 + (Xl - Xc)^2)
Greetings, impedance explorer! Calculating RLC impedance can be as puzzling as solving a Rubik’s Cube in the dark. Fear not! Our RLC Impedance Calculator is here to shed light on this electrifying
Categories of RLC Impedances
Discover different categories, types, and ranges of RLC Impedances, along with their interpretations, in this table:
Category Type Range RLC Impedance Calculation Interpretation
Electronics AC Circuits 1 Ω – 100 kΩ Z = √(R^2 + (Xl – Xc)^2) Evaluating impedance in electrical circuits
Electrical Inductive 100 Ω – 10 MΩ Z = √(R^2 + (Xl – Xc)^2) Measuring impedance in inductive elements
Power Systems High Voltage 1 kΩ – 1 MΩ Z = √(R^2 + (Xl – Xc)^2) Analyzing impedance in power transmission
RLC Impedance Calculation Methods
Explore various methods to calculate RLC Impedance, along with their advantages, disadvantages, and accuracy, in this table:
Method Advantages Disadvantages Accuracy
Formula-based Simple and widely applicable Requires knowledge of impedance Moderate
Circuit Analysis Precise modeling of complex circuits Requires advanced circuit analysis High
Simulation Software Accurate for complex systems Requires specialized software High
Evolution of RLC Impedance Calculation
Witness the evolution of RLC Impedance calculation over time in this table:
Era Key Developments
1800s Introduction of complex impedance in AC circuit theory.
1960s Advancements in computer-based circuit simulation tools.
2000s Integration of RLC impedance analysis in electronic design software.
Limitations of RLC Impedance Calculation Accuracy
1. Ideal Components: Assumes ideal components in calculations.
2. Frequency Dependent: Impedance varies with frequency.
3. Complex Circuits: Challenging for complex circuits.
Alternative Methods for Measuring RLC Impedance
Discover alternative methods for measuring RLC Impedance, along with their pros and cons, in this table:
Method Pros Cons
LCR Meter Measurement Quick and practical Limited to available LCR meters
Network Analyzer High precision and wide frequency range Expensive equipment
Impedance Bridge Highly accurate for specific impedance values Complex setup and calibration required
FAQs on RLC Impedance Calculator
1. What is RLC impedance?
□ It’s the opposition to the flow of alternating current in RLC circuits.
2. How is RLC impedance different from resistance?
□ RLC impedance considers both resistance and reactance.
3. What is reactance in RLC circuits?
□ It’s the opposition to AC flow due to inductance (Xl) and capacitance (Xc).
4. How do I calculate RLC impedance?
□ Use the formula: Z = √(R^2 + (Xl – Xc)^2).
5. What is the significance of RLC impedance in electronics?
□ It determines how components interact in AC circuits.
6. Can RLC impedance change with frequency?
□ Yes, it’s frequency-dependent.
7. What’s the unit of RLC impedance?
□ It’s measured in ohms (Ω).
8. How do I measure RLC impedance practically?
□ Use an LCR meter or network analyzer.
9. What are some common applications of RLC impedance analysis?
□ Filter design, circuit tuning, and antenna matching.
10. Why is impedance important in power transmission?
□ It helps optimize energy transfer and minimize losses.
Resources on RLC Impedance Calculations
1. All About Circuits – Impedance – In-depth guide on impedance in AC circuits.
2. MIT OpenCourseWare – Electric Circuits – MIT’s course on electric circuits. | {"url":"https://calculator.dev/physics/rlc-impedance-calculator/","timestamp":"2024-11-08T14:25:32Z","content_type":"text/html","content_length":"115350","record_id":"<urn:uuid:beff8440-b0e0-4b89-8b8c-5368092768ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00328.warc.gz"} |
T Test MCQ Quiz With Answers
Questions and Answers
How much do you know about the T-test? Think you can pass a quiz? Here, we present to you a T-Test MCQ Quiz with answers. A T-test is a very useful inferential statistic used that is to determine if
there is a noticeable difference between the means of two groups. Do you think you can pass this test? Why not try now and get to know yourself? Without wasting a moment, we should directly jump into
the quiz—best of luck to you before you start.
• 1.
A t-test is a significance test that assesses
□ A.
The means of two independent groups
□ B.
The medians of two dependent groups
□ C.
The modes of two independent variables
□ D.
The standard deviation of three independent variables
Correct Answer
A. The means of two independent groups
A t-test is a statistical test used to compare the means of two independent groups. It is used to determine if there is a significant difference between the means of the two groups, indicating
that there is a real effect or relationship. The t-test calculates a t-value, which is then compared to a critical value to determine if the difference between the means is statistically
significant. Therefore, the given answer correctly identifies that a t-test is used to assess the means of two independent groups.
• 2.
To use a t-test, the dependent variable must have
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Interval or ratio data
A t-test is a statistical test used to determine if there is a significant difference between the means of two groups. It is appropriate to use a t-test when the dependent variable has interval
or ratio data. Interval data is numerical data where the difference between any two values is meaningful, and ratio data is similar but includes a true zero point. These types of data allow for
meaningful calculations of means and differences between groups, which is necessary for conducting a t-test.
• 3.
Statistical significance or the probability of finding statistical significance is also known as
□ A.
□ B.
□ C.
□ D.
A constant source of frustration!
Correct Answer
B. p-value
The correct answer is p-value. Statistical significance or the probability of finding statistical significance is commonly referred to as the p-value. The p-value is a measure of the strength of
evidence against the null hypothesis and indicates the likelihood of obtaining the observed results by chance alone. It is used in hypothesis testing to determine whether the results of a study
are statistically significant or not.
• 4.
T-tests and other significance tests are frequently criticized. Over-representation of statistical significance in research may result in:
□ A.
□ B.
□ C.
□ D.
Confused graduate students
Correct Answer
A. Publication bias
The over-representation of statistical significance in research can lead to publication bias. This occurs when studies with statistically significant results are more likely to be published,
while studies with non-significant results are often overlooked or not published. This bias can distort the overall body of research, leading to an inaccurate understanding of the true effects of
a particular phenomenon.
• 5.
The three types of t-tests are
□ A.
□ B.
□ C.
Independent sample t-tests
□ D.
□ E.
Correct Answer(s)
A. One-sample t-tests
C. Independent sample t-tests
D. Paired samples t-tests;
The correct answer is a list of three types of t-tests: one-sample t-tests, independent sample t-tests, and paired samples t-tests. These three types of t-tests are commonly used in statistical
analysis to compare means between different groups or conditions. One-sample t-tests are used when comparing a sample mean to a known population mean. Independent sample t-tests are used when
comparing the means of two independent groups. Paired samples t-tests are used when comparing the means of two related or paired groups. Variable t-tests mentioned in the question are not a
recognized type of t-test.
• 6.
Into how many types can we classify measures of dispersion?
Correct Answer
C. 5
Measures of dispersion can be classified into five types. These types include range, quartiles, variance, standard deviation, and coefficient of variation. Range measures the spread between the
highest and lowest values in a dataset. Quartiles divide the data into four equal parts. Variance measures how spread out the data is from the mean. Standard deviation is the square root of
variance and provides a more intuitive understanding of the spread. Lastly, the coefficient of variation measures the relative variability of a dataset by comparing the standard deviation to the
• 7.
This is the mode value for the data set: 0,3,4,5,7,7,7,7,7,8,10,10
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 7
The mode value is the value that appears most frequently in a data set. In this case, the number 7 appears 5 times, which is more than any other number in the set. Therefore, the mode value for
this data set is 7.
• 8.
Which of these is a type of T-test?
□ A.
□ B.
Independent two-sample t-test.
□ C.
□ D.
Correct Answer
D. All of these
All of these options are types of T-tests. A T-test is a statistical test used to determine if there is a significant difference between the means of two groups. The one sample t-test is used to
compare the mean of a single sample to a known population mean. The independent two-sample t-test compares the means of two independent groups. The paired sample t-test compares the means of two
related groups, such as before and after measurements. Therefore, all three options mentioned are valid types of T-tests.
• 9.
The T-test is not a reliable test.
Correct Answer
B. False
The given statement suggests that the T-test is not a reliable test. However, the correct answer is False, which means that the T-test is indeed a reliable test. This implies that the T-test can
be trusted and is a valid statistical method for analyzing data and making inferences about population parameters.
• 10.
The T-test tells you about the significant difference between the two groups.
Correct Answer
A. True
The T-test is a statistical test that is used to determine if there is a significant difference between the means of two groups. Therefore, the statement that the T-test tells you about the
significant difference between the two groups is true. The T-test calculates a T-value and compares it to a critical value to determine if the difference between the groups is statistically
significant. If the calculated T-value is greater than the critical value, it indicates that there is a significant difference between the two groups. | {"url":"http://jclamkin.com/story8687.html","timestamp":"2024-11-02T01:43:04Z","content_type":"text/html","content_length":"414231","record_id":"<urn:uuid:69d340ea-4220-4c76-89c5-5a5afce9a023>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00281.warc.gz"} |
The Casimir force on a piston in the spacetime with extra compactified dimensions PDF
The Casimir force on a piston in the spacetime with extra compactified dimensions 8 Hongbo Cheng 0 ∗ 0 2 Department of Physics, East China University of Science and Technology, n Shanghai 200237,
China a J 8 1 Abstract ] h Aone-dimensionalCasimirpistonformasslessscalarfieldsobeyingDirichletbound- t - p ary conditions in high-dimensionalspacetimes withinthe frameof Kaluza-Klein theory e h is
analyzed. We derive and calculate the exact expression for the Casimir force on the [ piston. We also computetheCasimir forceinthelimit thatoneouter plateis moved to 1 v the extremely distant place
to show that the reduced force is associated with the prop- 0 erties of additional spatial dimensions. The more dimensionality the spacetime has, 1 8 the stronger the extra-dimension influence is. The
Casimir force for the piston in the 2 . model excluding one plate under the background with extra compactified dimensions 1 0 always keeps attractive. Further we find that when the limit is taken the
Casimir force 8 0 between one plate and thepiston will change to bethe same form as thecorresponding : v force for the standard system consisting of two parallel plates in the four-dimensional i X
spacetimes if the ratio of the plate-piston distance and extra dimensions size is large r a enough. PACS number(s): 03.70.+k; 11.10Kk ∗ E-mail address: hbcheng@public4.sta.net.cn 1 In 1948 a
remarkable macroscopic quantum effect describing the attractive force between two conducting and neutral parallel plates was predicted by Casimir [1]. The Casimir effect appears due to the disturbance
of the vacuum of the electromagnetic field induced by the presence of boundary. Twenty years later Boyer researched on the Casimir effect for a conducting spherical shell to find that this kind of
Casimir force is repulsive [2]. This effect is more complicated than we thought. Afterwards more efforts have been paid for the problem and related topics. All results brought attention to the fact
that whether the Casimir force is attractive or repulsive depends on the geometry of the configuration strongly [3]. However, there are several reasons to be suspicious of the analysis of the Casimir
effect problems. Maybe their results are not perfect. For example, we always investigated a massless scalar field in a confined region such as parallel plates, rectangular box and so on to find the
vacuum energy while we let the field satisfy the Dirichlet boundary conditions at the borders of the region [3-8]. Having regularized the vacuum energy, we obtain the Casimirenergy.
CertainlytheCasimirforcecanbereceivedbymeansofderivativeofCasimirenergy with respect to the distance between two edges. Here it should be pointed out that these former considerations on the topic
have not involved the contribution to the vacuum energy from the area outside the confined region which depends on its dimensions while we discard the divergent terms related to the boundary also
depending on the geometry and dimensions during the regularization process. In order to ignore the flaws mentioned above, a slightly different model called piston was put forward [9]. The system is a
single rectangular box with dimensions L b divided into two × parts with dimensions a b and (L a) b respectively by a piston which is an idealized plate × − × that is free to move along a rectangular
shaft. In ref. [9] the author calculated the Casimir force on a two-dimensional piston as a consequence of fluctuations of a scalar field obeying Dirichlet boundaryconditions on all surfaces and
foundthat the force on the piston is always attractive as L goes to the infinity, regardless of the ratio of the two sides. Immediately the issue attracted more attention. The Casimir force acting on
a conducting piston with arbitrary cross section always keeps attractive although the existence of the walls weaken the force [10]. The three-dimensional Casimir piston for massless scalar fields
obeying Dirichlet boundary conditions was also explored, anditwas foundthat thetotal Casimirforce is negative nomatter how long thelengths of sides are [11]. In addition in the case of various
boundary conditions the Casimir force on a piston may be repulsive [12, 13]. In a word the Casimir piston is a new important model revealing its own distinct effects and can be used to explore the
related topics. This model is also simpler to be constructed as a device from the experimental point of view. The model of higher-dimensional spacetime is a powerful ingredient to be needed to unify
the interactions in nature. More than 80 years ago Kaluza and Klein put forward the issue that our universe has more than four dimensions [14, 15]. The Kaluza-Klein theory introduced an extra
compactified dimension to unify gravity and classical electrodynamics in our world. The theory has been generalized and developed greatly. Recently the quantum gravity such as string theory or
brane-world scenario is developed to reconcile the quantum mechanics and gravity with the help 2 of introducing seven additional spatial dimensions. In Randall-Sundrum model the matter fields may be
localized on a four-dimensional brane considered as our real universe, and only gravitons can propagate in the extra space transverse to the brane [16, 17]. In addition, although the order of the
compactification scale of the additional dimensions has not been confirmed and are also of considerable interest recently, larger extra dimensions were invoked in order to provide a breakthrough of
hierarchy problem in some approaches [18-20]. Research on higher-dimensional spacetime is valuable and become a focus in the physical community, therefore the theory needs to be explored deeply,
extensively and in various directions. Since the higher-dimensional spacetime described by Kaluza-Klein theory is important and indispensable, it is crucial to discuss several models including the
Casimir effect problem in this background. The precision of the measurement has been greatly improved practically [21-24],
leadingtheCasimireffecttoberemarkableobservableandtrustworthyconsequenceoftheexistence of quantumfluctuations and tobecome apowerfultool forthetopics on themodelof Universe with more than four
dimensions. It must be emphasized that the attractive Casimir force between the parallel plates vanishes whenthe plate gap is very large andno repulsiveforce appearsaccording to theexperimental
results. Some topics were examined in thecontext of Kaluza-Klein theory. As the first step of generalization to investigate the higher-dimensional spacetimes, we show analytically that the
extra-dimension corrections to the Casimir effect for a rectangular cavity in the presence of a compactified universal extra dimension are very manifest [25]. The Casimir effect for parallel plates in
the spacetime with extra compactified dimensions was also studied [26-29]. We prove rigorously that there must appear repulsive Casimir force between the parallel plates when the plates distance is
sufficiently large in the spacetime with compatified additional dimensions, and thehigherthedimensionality is, thegreater therepulsiveforce is. Itshouldbepointedout thatthe
valueoftherepulsiveCasimirforcewhichisobtainedtheoretically iswithintheexperimentalreach. Therefore the results obtained in the context of Kaluza-Klein theory conflict with the experimental phenomena
mentioned above [27-29], which means that the model of higher-dimensional spaetime with extra compactified spatial dimensions needs further research. It is necessary and significant to study the
force-on-the-piston problem in a higher-dimensional spacetime within the frame of Kaluza-Klein theory. We wonder how the influence from extra dimensions on the Casimir effect of the piston. This
problem, to our knowledge, has not been examined. For simplicity and comparison to the conclusion of standard parallel-plates system, the model of one-dimensional piston is chosen. The main purpose
of this paper is to study the Casimir effect for the system consisting of three parallel plates in the Universe with d compactified spatial dimensions. We obtain the expression of force by means of the
derfferential of the total vacuum energy including the contribution outside the three-parallel-plate device with respect to the distance between two plates in the system. We regularize the force to
obtain the Casimir force on the piston when one outer plate is moved to the remote place. We focus on the influence of dimensionality of the spacetime on the Casimir force between one plate and a
piston and compare 3 our results with the models like one-dimensional piston or parallel plates in the four-dimensional spacetime. Our discussions and conclusions are emphasized in the end.
Inahigher-dimensionalspacetime,westarttoconsiderthemasslessscalarfieldsobeyingDirich- let boundary conditions within a one-dimensional piston. As a piston, one plate is inserted into a
systemconsisting oftwo parallel plates. Thepiston isparallel totheplates anddividestheparallel- plate-system into two parts labeled by A and B respectively. In part A the distance between the left
plate and the piston is a, and the distance between the piston and the right plate in part B, the remains of the separation of two plates, is certainly L a, which means that L denotes the − whole
plates gap. The total vacuum energy for the three-parallel-plate system described above can be written as the sum of three terms, E = EA(a)+EB(L a)+Eout (1) − where EA(a) and EB(L a) represent the
energy of part A and B respectively, and the terms − depend on their each size in parts. Eout describes the vacuum energy outside the system and is independentof characters insidethe system. Having
regularized the total energy density, we obtain the Casimir energy density, E = EA(a)+EB(L a)+Eout (2) C R A R − where EA(a), EB(L a) and Eout are finite parts of terms EA(a), EB(L a) and Eout in Eq.
(1) R R − R − respectively. In particular, it is also pointed out that Eout is not a function of the position of the R piston, the Casimir force on thepiston is given by the derivative of the Casimir
energy with respect to the plates distance ∂EC and can be written as, − ∂a ∂ F = [EA(a)+EB(L a)] (3) C −∂a R R − which means that the contribution of vacuum energy from the exterior region does not
affect the Casimir force on the piston. Here we set out to consider the massless scalar field in the three-parallel-plate system in the spacetime with d extra compactified dimensions in the context of
Kaluza-Klein theory. Along the additionaldimensionsthewavevectorsofthefieldhavetheformk = ni,i= 1,2, ,d, respectively, i R ··· n an integer. Now we choose that the extra dimensions possess the same
size as R. The fields i satisfy the Dirichlet condition, leading the wave vector in the directions restricted by the plates to be k = nπ, n a positive integer and D the separation of the plates. Under
these conditions, the n D zero-point fluctuations of the fields can give rise to observable Casimir forces among the plates. In the case of d additional compactified dimensions we find the frequency of
the vacuum fluc- tuation within a region confined by two parallel plates whose separation is D to be, n2π2 d n2 ω = vk2+ + i (4) {ni}n u D2 R2 ut Xi=1 4 where 2 2 2 k = k +k (5) 1 2 k1 and k2 are the
wave vectors in directions of the unbound space coordinates parallel to the plates surface. Now ni represents a short notation of n1,n2, ,nd, ni a nonnegative integer. { } ··· According to Ref.[3, 4,
7, 30-34], therefore the total energy density of the fields in the interior of two-parallel-plate system reads, ∞ ∞ 1 2 E(D,R) = d k ω Z 2 {ni}n nX=1{nXi}=0 πΓ( 3) d−1 d π2 1 1 1 3 π4 Γ( 3)ζ( 3) = 2Γ
(−−221) Xl=0 l Ed−l+1(D2, R2,R2,···,R2;−2)+ 2D3 −Γ2(−21−) (6) in terms of the Epstein zeta function Ep(a1,a2, ,ap;s) defined as, ··· ∞ p Ep(a1,a2, ,ap;s)= ( ajn2j)−s (7) ··· {Xnj} jX=1 here nj
stands for a short notation of n1,n2, ,np, nj a positive integer. We can regularize { } ·· · Eq.(6) by means of the following formula, 3 π2 1 1 1 3 Γ(−2)Ed−l+1(D2,R2,R2,···, R2;−2) 1 3 3 1 1 D = Γ( )
E (1,1, ,1; ) + Γ( 2)E (1,1, ,1; 2) −2 −2 d−l ··· −2 R3 2√π − d−l ··· − R4 + 1 ∞ 16−k(D)−k−32 k [16 (2j 1)2] R3 kX=0 k! R jY=1 − − ∞ × n1−k−25(n22+n23+···+n2d−l+1)−2k4−3 n1,n2,··X·,nd−l+1=1 2D 2 2 2
1 ×exp[− R n1(n2+n3+···+nd−l+1)2] (8) to obtain the Casimir energy density of a system consisting of two parallel plates in the spacetime with d extra compactified spatial dimensions. Inthecontext
ofKaluza-Klein theorywereturntodiscussthenewsystemwhereapistonisalso a plate localizing parallelly between two parallel plates mentioned above. Choosing the variable D = a in Eq.(6) and (8), we have
the vacuum energy density for part A of the system containing one plate and the piston with distance a as follow, πΓ( 3) d−1 d π2 1 1 1 3 π4 Γ( 3)ζ( 3) EA(a,R) = 2Γ(−−221) Xl=0 l Ed−l+1
(a2,R2,R2,···, R2;−2)+ 2a3 −Γ2(−12)− (9) 5 Similarly the vacuum energy density for the remains of the system labeled B with plates distance L a by replacing the variable D with L a in Eq.(6) is, − −
πΓ( 3)d−1 d π2 1 1 1 3 EA(L−a,R) = 2Γ(−−221) Xl=0 l Ed−l+1((L−a)2,R2, R2,···, R2;−2) π4 Γ( 3)ζ( 3) + −2 − (10) 2(L a)3 Γ( 1) − −2 We regularize Eq.(9) and Eq.(10) and then substitute the two
regularized expressions into Eq.(3) to obtain the Casimir force per unit area on the piston, π4 1 π4 1 F′ = + C −120a4 120(L a)4 − +√π d−1 d 1 ∞ 16−k(k+ 3)(a)−k−23 k [16 (2j 1)2] 4 Xl=0 l {−aR3 kX
=0 k! 2 R jY=1 − − ∞ ×n1,n2,··X·,nd−l+1=1n−1k−25(n22+n23+···+n2d−l+1)−2k4−3 2a 2 2 2 1 ×exp[−Rn1(n2+n3+···+nd−l+1)2] 2 ∞ 16−k(a)−k−23 k [16 (2j 1)2] −R4 k! R − − kX=0 jY=1 ∞ ×n1,n2,··X·,nd−l+1=
1n−1k−23(n22+n23+···+n2d−l+1)−2k4−5 2a 2 2 2 1 ×exp[−Rn1(n2+n3+···+nd−l+1)2] + 1 ∞ 16−k(k+ 3)(L−a)−k−23 k [16 (2j 1)2] (L−a)R3 kX=0 k! 2 R jY=1 − − ∞ ×n1,n2,··X·,nd−l+1=1n−1k−25(n22+n23+···+n2d−l+1)
−2k4−3 2(L a) 2 2 2 1 ×exp[− R− n1(n2+n3+···+nd−l+1)2] + 2 ∞ 16−k(L−a)−k−23 k [16 (2j 1)2] R4 kX=0 k! R jY=1 − − ∞ ×n1,n2,··X·,nd−l+1=1n−1k−23(n22+n23+···+n2d−l+1)−2k4−5 2(L a) 2 2 2 1 ×exp[− R− n1
(n2+n3+···+nd−l+1)2]} (11) which has corrections from extra dimensions. Further we take the limit L which means that → ∞ the right plate in part B is moved to a very distant place, then we obtain the
following expression for the Casimir force per unit area on the piston, 6 F = lim F′ C C L→∞ π4 1 C (µ) d = + (12) −120a4 R4 where the correction function C (µ) is defined as, d Cd(µ)= √π d−1 d 1 ∞
16−k(k+ 3)µ−k−23 k [16 (2j 1)2] 4 Xl=0 l {−µ kX=0 k! 2 jY=1 − − ∞ ×n1,n2,··X·,nd−l+1=1n−1k−25(n22+n23+···+n2d−l+1)−2k4−3 2 2 2 1 ×exp[−2µn1(n2+n3+···+nd−l+1)2] 2 ∞ 16−kµ−k−23 k [16 (2j 1)2] − kX=
0 k! jY=1 − − ∞ ×n1,n2,··X·,nd−l+1=1n−1k−23(n22+n23+···+n2d−l+1)−2k4−5 2 2 2 1 ×exp[−2µn1(n2+n3+···+nd−l+1)2]} (13) and a µ = (14) R We discover that the first term in Eq.(12) is the same as Casimir
pressure of standard model of parallel plates involving massless scalar fields satisfying the Dirichlet conditions in the four- dimensional spacetimes, meaning that the Casimir force per unit area
between the piston and one plate after the other ones has been moved away in the background with extra compactified dimensions is just the original result plus the deviation from the additional
dimensions. It is necessary to explore the correction functions in detail. When the gap between the remain plate andthe piston is larger enough than thesize of extra dimensions,the correction
functions approach to the zero, lim C (µ)= 0 (15) d µ→∞ and right now the Casimir force per unit area on the piston will return to the ones of two parallel plates in the four-dimensional spacetime.
We have to perform the burden and surprisingly difficult calculation to scrutinize the Casimir force on the piston quantatively, then the correction functions C (µ) under the limiting L in the
spacetime with extra dimensions are depicted in Figure d → ∞ 1 and Figure 2. Certainly the Casimir force on the piston related to the ratio µ = a has been R displayed inthesamecase. AccordingtoEq.
(12), wejustneedtofocusonthecorrection functions during the process of research on the Casimir force on the piston. The functions depend on the 7 dimensionality of spacetime and the ratio of plates
distance and the size of extra dimensions. The more dimensions the spacetime has, the greater the absolute value of the correction function is, which means that there will appear stronger influence on
the Casimir force on the piston in the higher-dimensional spacetime. The expression C (µ) is also a function of ratio denoted in Eq.(14). d When the ratio increases, the absolute value of the
functions decreases fast. When the plates separation is more than several times larger than the extra dimensions radius, the absolute value will approach to zero. The manifestation of extra
dimensions influence on the Casimir force on the piston under the limiting L appears only when the distance between one plate and the → ∞ piston is about equal to the size of extra dimensions. As
mentioned above, if the extra dimensions possess larger size, the extra-dimension corrections will appear clearly in practice, therefore this modelof one-dimensional piston can become a window to
examine the high-dimensional spacetime. It should also be pointed out that the values of all correction functions in the world with different dimensionality keep negative, so the total Casimir force
on the piston also remains attractive. The experimental evidence shows that no repulsive force generates in the case of parallel plates. It has been proved theoretically and rigorously that the
Casimir force between parallel plates become repulsiveas the plates are sufficiently faraway from each other inthe higher-dimensionalspacetime described by Kaluza-Klein theory and the conclusion
conflicts with the experimental phenomena and is inevitable [28, 29]. In this work our arguments on the piston in the same environment drawn above under the limiting L is consistent with the
experimental phenomena. It is useful to → ∞ consider the system with a piston further. After one plate has been moved to the remote place, the three-parallel-plate model can be thought as an ordinary
system consisting of two parallel plates in which one plate is thought as a piston. Actually it is interesting that the two kinds of results from three-parallel-plate model with limiting L and
original two-parallel-plate system respectively → ∞ are completely different. The theoretical finding in this work, the Casimir force per unit area on the piston, is consistent with the measurement at
least qualitatively. The deviation produces apparently as the plate-piston gap is close to the extra-dimension size. Maybe the corrections are beyond the experimental reach because the order of the
compactification scale of the additional spatial dimensions can be extremely tiny. The three-parallel-plate model, called one-dimensional piston, must replace the standard parallel plates system
unless the higher-dimensional approach described by Kaluza-Klein theory is excluded. Inthis work themodelof threeparallel plates in whichthemiddleoneis called pistonis studied in the
higher-dimensional spacetime described by Kaluza-Klein theory. The expression of Casimir force per unit area on the piston is obtained. When one outer plate is moved away, we also get the exact form
of reduced Casimir force per unit area between one plate and the piston. In the limiting case we discover that the force is always attractive and depend on the properties of extra compactified
dimensions. The more extra dimensions will produce stronger influence. When the separationofoneplateandthepistonislarger enoughthanthesizeofadditionalspatialdimensions andthelimitL is taken,
theCasimirforce perunitareabetween them willbethesameas the → ∞ 8 results for the standard system consisting of two parallel plates in the four-dimensional spacetime. Theresultsof
thestandardsysteminfour-dimensionalspacetimes arefavoured inpractice. Further wecanarguethatthemodeloftwoparallelplatesandonepistonundertheconditionthatoneouter plate has been moved extremely far
away should substitute the standard two-parallel-plate model to describe the Casimir effect for parallel plates from experiment if the universe has additional compactified dimensions. We also should
not neglect that the expressions of Casimir force for the two models derived and calculated in the same high-dimensional spacetime and the frame of the same Kaluza-Klein theory are completely
different, in which at least one is attractive and the other is repulsive. Clearly in the high-dimensional background, our results from plate-piston-plate model in the limiting case avoid the flaw of
results from general two-parallel-plate system. We should make use of model of one-dimensional piston to explain the measurement of Casimir effect from two paralle plates not a standard two-parallel
plate model. Of course the further consequences and related topics are under progress. Acknowledgement TheauthorthanksProfessorK.Milton, Professor E.Elizalde andProfessorI.Brevik forhelpful
discussions. ThisworkissupportedbytheShanghaiMunicipalScienceandTechnologyCommission No. 04dz05905 and NSFC No. 10333020. 9 References [1] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51(1948)793 [2]
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Rev. D75(2007)105012 A. Edery, I. MacDonald, hep-th/0708.0392.v2 [12] S. A. Fulling, J. H. Wilson, quant-ph/0608122 [13] S. A. Fulling, L. Kaplan, J. H. Wilson, quant-ph/0703248 [14] T. Kaluza, Sitz.
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Packages and Modules in Python - Finance Train
Packages and Modules in Python
We learned that Python functions and methods are really useful and can be used to solve many of our problems. We can even take functions written by others to solve our problems. When new functions
and methods are written, they can be standardized and distributed so that a large numebr of Python developers can use them in their projects. However, it's not a good idea to distribute all these
functions as a part of the core Python distribution. This is where packages come into play. A Python package is like a directory of Python scripts. Each script is a module. Modules in Python are
simply Python files with the .py extension, which implement a set of functions. Each Module is designed to solve a particular set of problems.
Python is heavily dependent on it’s vast array of packages. The Python Package Index lists 90k+ packages available for download. Programmers can also create their own packages and make them available
through Python Package Index from where others can download and use them.
Packages are used by importing the package at the beginning of your script, which allows you to then access
their specialized functions or objects. Below is a short list of the most widely used packages:
• NumPy: "the fundamental package for scientific computing with Python," the source of most linear algebra functions, data import functions, and general computation
• SciPy: builds on top of Numpy to include more specific, higher level functionality including image processing, statistics, and optimization
• Matplotlib: the primary way to generate figures in Python, replicates much of the functionality of plotting in MATLAB (hence the name)
Again, things are very google-able if you’re looking for something specific, there’s almost certainly a python package somewhere that has it.
Using Inbuilt Modules
Python already has some very useful inbuilt modules. These modules are considered to be a part of Python's standard environment. An example of an inbuilt module is the math module which contains many
useful math functions. To use a module, we need to first import it. Following are some examples of the math package.
import math
#Calculate square root of the number 25
x = math.sqrt(25)
#We can calculate the area of a circle using the formula πr^2
#We can access the constant π using math.pi
r = 5
Area = math.pi*5**2
Apart from math, some other useful inbuilt modules are listed below:
• sys - contains tools for system arguments, OS information etc.
• os - for handling files, directories, executing external programs
• re - for parsing regular expressions
• datetime - for date and time conversions etc.
• csv - for reading CSV tables
• and many more
Different Ways of Importing
Depending on our needs, we can import a module in different ways.
import x
You can import a module by it's name like import math. When we import a module X in this manner, we need to use X.name to refer to an item called name that is defined in the module X.
import math
from X import *
In this case, you can directly refer to items in the module X, without using the “X.” prefix.
from math import *
from x import y
This is useful, if there is an object you specifically use frequently and would like to make it a part of your main namespace. So, only object y will be imported, nothing else.
from math import sqrt
Aliases for a package
If you have decided to access a module's objects from its own namespace, you can choose to alias the module with a name. For example the module math can be given alias m.
import math as m
Installing Third-party Package
If a module is not inbuilt, it can be installed as a part of a third-part package.
We use pip a Python package management system to install and manage software packages written in Python. If you have installed Python 3 from python.org, you will already have pip installed along with
Installing PIP
Visit http://pip.readthedocs.org/en/stable/installing/
Download get-pip.py
On terminal execute the command python get-pip.py
To install a package, you will use the command pip install <i>Package Name</i>. For example, you can install the numpy package using the following command.
pip install numpy
This will download and install the numpy package. Once the package is installed, you can import it in your Python program to start using it.
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Question 29
What is the next number in this sequence?
You can find the second number in the sequence by applying the following transformation to the first number: -3 You find the third number in the sequence by applying the following transformation to
the second number: +7 You can see that this is not the same number. Every step the number in the transformation changes with +10 | {"url":"https://brght.org/review/Uy4QiJBPrNDKYqz5/29/","timestamp":"2024-11-10T13:53:25Z","content_type":"text/html","content_length":"49198","record_id":"<urn:uuid:2202ffee-135b-43ab-9405-643e6edd54cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00760.warc.gz"} |
Loading variables from a file
In this tutorial, we will learn how to load all observations from
• All or a subset of the variables.
• With or without data transformations, such as
□ Creating dummy variables.
□ Reclassifying string variables to integer categories.
□ Creating interaction terms.
□ ln, exp, lag and more.
from a well-formed dataset. All sections below apply to any dataset that meets our definition of 'well-formed' which is explained below.
What files does this apply to?
Our definition of a well-formed dataset includes
• Comma-separated text files (CSV) with headers in the first line of the file.
• Excel files (XLS, XLSX) with headers in the first row of the file.
• GAUSS datasets (DAT) and matrix files (FMT).
• SAS (SAS7BCAT, SAS7BDAT), SPSS (POR, SAV) and Stata (DTA) datasets.
• HDF5 files (H5) if the dataset contains an attribute called headers which contains the variable names.
Regardless of the file type, each file must be organized as a consistent tabular dataset like the example below. Each row of the file must have the same number of columns and each column of the file
must have the same number of rows.
What files does this not apply to?
This section does not apply to
• Text files delimited by a character other than a comma.
• CSV files without headers or with empty lines.
• Excel files without headers.
or files which have inconsistent numbers of rows, or columns.
// This dataset is NOT well-formed.
// It has:
// 1. Comments at the top of the file.
// 2. Inconsistent numbers of columns per row.
The GAUSS function getHeaders will return a string array containing all the variable names from a dataset. It takes only one input, the name of the dataset. The example below reads all of the
variable names from the Stata dataset auto2.dta which is located in the GAUSS examples directory.
// Create file name with full path to Stata dataset
fname = getGAUSSHome() $+ "examples/auto2.dta";
// Read the variable names from the dataset
h = getHeaders(fname);
// Print string array containing the dataset headers
print h;
will return
All variables
The GAUSS command loadd can read variables from a dataset. To read all variables from a dataset you only need to pass one input, a string containing the name of the dataset. The example dataset,
binary.csv, contains four variables related to college admissions, admit, gre, gpa, and rank.
// Create file name with full path
fname = getGAUSSHome() $+ "examples/binary.csv";
// Read all 4 variables from the CSV file
X = loadd(fname);
// Print the first 5 rows of all columns of 'X'
print X[1:5,.];
will return
admit rank gpa rank
0.00 380.00 3.61 3.00
1.00 660.00 3.67 3.00
1.00 800.00 4.00 1.00
1.00 640.00 3.19 4.00
0.00 520.00 2.93 4.00
A subset of variables
loadd can accept an optional second argument which is a formula string. The formula string specifies which variables to load and which data transformations to perform. The following operators can be
used in a formula string to load a subset of the variables from the dataset.
Operator Description
. The dot represents all variables.
+ The plus operator adds a variable.
- The minus operator removes a variable
Example: Load two variables by name
// Create file name with full path to the SAS dataset
fname = getGAUSSHome() $+ "examples/detroit.sas7bdat";
// Load 2 variables by name from the SAS dataset
X = loadd(fname, "unemployment + weekly_earn");
// Print the first 5 rows of all columns of 'X'
print X[1:5,.];
will return
unemployment weekly_earn
11.0 117.18
7.0 134.02
5.2 141.68
4.3 147.98
3.5 159.85
Example: Load all variables except for one
cancer.dat is an example GAUSS dataset located in the GAUSS examples directory. It contains five variables, time, histology, stage, count, and risktime. The example below loads all of these
variables, except for stage.
// Create file name with full path
fname = getGAUSSHome() $+ "examples/cancer.dat";
// Load all but one variable from the GAUSS dataset
X = loadd(fname, ". -stage");
// Print the first 5 rows of all columns of 'X'
print X[1:5,.];
will return
time histology count risktime
1.00 1.00 9.00 157.00
1.00 2.00 5.00 77.00
1.00 3.00 1.00 21.00
2.00 1.00 2.00 139.00
2.00 2.00 2.00 68.00
Categorical variables
Some data files, such as CSV files, do not contain information specifying the types of the variables in the files. In these cases, it is sometimes necessary to specify how a particular variable
should be interpreted.
The following keywords can be used in a formula string to tell GAUSS which variables should be interpreted as categorical or string variables and to create dummy variables if desired.
Keyword Description
factor Create dummy variables from a categorical variable, or column of integers.
cat Load text data and create a categorical variable.
str Load text data and create a string variable.
Example: Integer variable to dummy variables
housing.csv is an example dataset from the GAUSS examples directory. The variable baths, represents the number of bathrooms in the home and contains the following unique values: 1, 2, 3, and 4.
The example code below loads the baths variable unmodified for comparison. In the next step, the code tells loadd to create dummy variables from the integer categories in the baths variable by using
the factor keyword in the formula string.
// Create file name with full path
fname = getGAUSSHome() $+ "examples/housing.csv";
// Load the original categorical data
baths = loadd(fname, "baths");
// Load the categorical variable and create dummy vars
dmy = loadd(fname, "factor(baths)");
After the code above, the first 5 rows of baths and dmy will be equal to
baths baths_2 baths_3 baths_4
baths = 2 dmy = 1 0 0
As you can see above, the base case is set to the case when baths equals one.
Example: Text categorical variable to dummy variables
The example Excel file, nba_ht_wt.xls, contains seven variables with different information about NBA basketball players. The Pos variable represents the position played by the basketball player. The
levels are C, F, and G, which represent center, forward and guard.
The code below uses the cat keyword in the formula string to tell loadd to create a categorical variable from the text data. Then the code wraps the factor keyword around the cat keyword to load the
data as a categorical column and convert to dummy variables in one step.
// Create file name with full path
fname = getGAUSSHome() $+ "examples/nba_ht_wt.xls";
// Load the string variable turn it
// into a categorical variable
position = loadd(fname, "cat(Pos)");
// Load the string variable turn it into a
// categorical variable and then a dummy variable
pos_dummy = loadd(fname, "factor(cat(Pos))");
below is a preview of the first five observations created by the above code:
Pos Pos_F Pos_G
position = C pos_dummy = 0 0
G 0 1
G 0 1
F 1 0
F 1 0
This time, C, is the base case.
Combining keywords and operators
Both the cat and factor keywords can be combined with the ., + and - operators. For example, the following statements would be legal.
fname = getGAUSSHome() $+ "examples/housing.csv";
X = loadd(fname, "price + factor(baths) + taxes");
fname = getGAUSSHome() $+ "examples/yarn.xlsx"
X = loadd(fname, "cat(amplitude) + cycles");
Interaction effects
The * and : operators are used in formula strings to create interaction effects.
Operator Description
* Represents an interaction between two variables as well as the original variables.
: Represents only the interaction between the two specified variables.
Example: Interaction term
By default when an interaction term is specified in a formula string, the variables that form the interaction are also included. The example below will load 3 variables, Height, Weight, and the
interaction term of Height*Weight.
// Create file name with full path
fname = getGAUSSHome() $+ "examples/nba_ht_wt.xls";
// Load the variables 'Height' and 'Weight'
// then create a third variable which is the
// interaction between them
nba = loadd(fname, "Height*Weight");
// Print the first 5 rows of the 3 specified variables
print nba[1:5,.];
The code above will print the following output
Height Weight Height_Weight
The name of the product of Height and Weight is not Height*Weight, because if that variable name was included in another formula string it would indicate an interaction term instead of this variable.
GAUSS replaces invalid characters from variable names with underscores.
Example: Interaction term alone
Usually, when we create an interaction term, we will also include the original variables. However, it is sometimes useful to load only the interaction variable. We can specify that we only want the
interaction, by using the colon operator, :, in the formula string as shown below.
// Create file name with full path
fname = getGAUSSHome() $+ "examples/nba_ht_wt.xls";
// Load only one variable, which is the
// interaction between 'Height' and 'Weight'
X = loadd(fname, "Height:Weight");
// Print the first 5 rows of the 1 specified variable
print X[1:5];
The code above will print the following output
Data transformations
GAUSS allows you to transform your variables when loading, by using a procedure in a formula string.
Example: Natural log
// Create file name with full path to Stata dataset
fname = getGAUSSHome() $+ "examples/auto2.dta";
// Load 'price' from 'auto2.dta' and perform
// natural log transform
ln_price = loadd(fname, "ln(price)");
// Print the first 5 rows of 'ln_price'
print ln_price[1:5];
The code above will return the following output.
The variable name was updated to represent the transformation with the parentheses replaced with underscores.
Example: The first difference of the natural log
Now let's do something slightly more complicated. Suppose you want to compute the first difference of the natural log of the price variable from the auto2.dta dataset. GAUSS allows you to use any
procedure in a formula string as long as it takes a column vector as the only input and returns a column vector of the same size as the only output.
So we will first create a procedure to compute the first difference of the natural log. We will call it lnDiff. Then we can use it in our formula string, like this
// Define procedure to compute the first
// difference of the natural log of a variable
proc (1) = lnDiff(x);
local ln_x;
// Compute the natural log of the input
ln_x = ln(x);
// Compute the difference of the natural log
// and return the result
retp(ln_x - lag(ln_x));
// Create file name with full path to Stata dataset
fname = getGAUSSHome() $+ "examples/auto2.dta";
// Load the 'price' variable and call
// our 'lnDiff' procedure on it
X = loadd(fname, "lnDiff(price)");
// Print the first 5 observations
print X[1:5];
The code above will print the following output. Note that the first observation is a missing value since we lose one observation when computing the lag.
In this tutorial, we have learned how to
• Load all or a subset of variables with the +, - and . operators.
• Creating dummy variables with the factor keyword.
• Reclassifying string variables to integer categories with the cat keyword.
• Creating interaction terms with the * and : operators.
• Performing data transformations by using GAUSS procedures in formula strings.
from a well-formed, tabular dataset.
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Challenge Your Computer
I was bugged, bugged with the little stupid box which challenges my faculties of thinking with its inorganic AI abilities. Then my little grey cells came to my aid and dug out the one of the most
wonderful problems that men invented. I am here referring to Ackerman Function.>>
This is what wikipedia has to say about it in the abstract: In the theory of computation, the Ackermann function or Ackermann-Péter function is a simple example of a recursive function that is not
primitively recursive. It takes two natural numbers as arguments and yields another natural number. Its value grows extremely quickly; even for small inputs, for example (4,3), the values of the
Ackermann function become so large that they cannot be feasibly computed, and in fact their decimal expansions require more digits than there are particles in the entire visible universe.
This function was conceptualized by
Wilhelm Ackerman (1896-1962).>>
Wilhelm Ackermann received his doctoral degree in 1925 with a thesis written under Hilbert. Its content was a consistency proof of arithmetic without induction. From 1927 until 1961 he taught as a
teacher at the Gymnasien in Burgsteinfeld and in Luedenscheid. He was corresponding member of the Akademie der Wissenschaften in Göttingen, and was honorary professor at the Universit t Münster.
He worked on logic with Hilbert.
In 1928, Ackermann observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive. A(x,y,z) was simplified to a
function P(x,y) of 2 variables by Rosza Peter whose initial condition was further simplified by Raphael Robinson. This last function is called Ackermann's function in today's textbooks. Also in 1928,
Hilbert and Ackermann coauthored the logic book Grundzuege der Theoretischen Logik.
Among Ackermann's later work there are consistency proofs for set theory, full arithmetic , and type free logic. He also gave a new axiomatization of set theory in 1956, and wrote the book Solvable
cases of the decision problem in 1954.
The first time I saw this function in 2002, I was excited, and i ran a recursive implementation on of it on a linux machine. I executed it with just the inputs 10,10. Poor guy never recovered from
this load and had to be powered off and on to get it back to normalancy. This is a deceptively
simple function.>>
a (m,n) = n+1 if m=0
= a (m-1, 1) if m0, n=0
= a (m-1, a(m, n-1)) if m0, n0
And today, I was in a mood to actually send my machine bonkers, and what a nice way to do that.
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Diagnosis of GHG Emissions in an Offshore Oil and Gas Production Facility
Vanti Group, Bogotá 253448, Colombia
Energy Department, University of Campinas, Campinas 13083-860, Brazil
Author to whom correspondence should be addressed.
Submission received: 30 July 2024 / Revised: 12 September 2024 / Accepted: 29 September 2024 / Published: 31 October 2024
This work presents a diagnosis of greenhouse gas (GHG) emissions for floating production storage and offloading (FPSO) platforms for oil and gas production offshore, using calculation methodologies
from the American Petroleum Institute (API) and U.S. Environmental Protection Agency (EPA). To carry out this analysis, design data of an FPSO platform is used for the GHG emissions estimation,
considering operations under steady conditions and oil and gas processing system simulations in the Aspen HYSYS^® software. The main direct emission sources of GHG are identified, including the main
combustion processes (gas turbines for electric generation and gas turbine-driven CO[2] compressors), flaring and venting, as well as fugitive emissions. The study assesses a high CO[2] content in
molar composition of the associated gas, an important factor that is considered in estimating fugitive emissions during the processes of primary separation and main gas compression. The resulting
information indicates that, on average, 95% of total emissions are produced by combustion sources. In the latest production stages of the oil and gas field, it consumes 2 times more energy and emits
2.3 times CO[2] in terms of produced hydrocarbons. This diagnosis provides a baseline and starting point for the implementation of energy efficiency measures and/or carbon capture and storage (CCS)
technologies on the FPSO in order to reduce CO[2] and CH[4] emissions, as well as identify the major sources of emissions in the production process.
1. Introduction and Objective
The oil and gas industry is one of the largest contributors to global carbon dioxide and methane emissions [
] due to the high energy intensity required in the production, refining, and transport processes of hydrocarbons, as well as the occurrence of greenhouse gas (GHG) escapes due to flaring and venting,
in addition to emissions from combustion processes.
Oil and natural gas production in Brazil has been carried out in offshore installations in deep waters for many years [
]. More recently, production has moved to fields located not only in large water depths but also at great geological depths, the so-called “Pre-salt”.
Figure 1
shows the importance of this oil province for the country.
Oil and gas industry operations on offshore platforms, specifically on floating production storage and offloading (FPSO) units, present energy and environmental challenges to be studied in more
detail due to the use of fossil fuels to obtain the needed energy independence for offshore installations. The use of fossil fuels induces GHG emissions into the atmosphere. To meet environmental
commitments, oil companies have made efforts to measure and estimate pollutant emissions into the atmosphere. Various researchers [
] present energy and exergy analysis in offshore installations to study energy efficiency actions to mitigate the impact on the environment.
An important point to highlight is the notable predominance of oil and natural gas exploration and production activities in emissions, accounting, roughly speaking, for 90% of emissions from fuel
production. This fact is not surprising, as the oil production and refining industry, as well as producing energy, is also a large consumer of energy.
The ratio of gas volume in the oil produced (GOR) under “standard” conditions is one of the most important parameters in a field’s production strategy. High GOR values (high quantities of gas) lead
to high production of natural gas, which can be used on platforms for energy requirements, used to increase oil production (gas-lift, reinjection), or sent for commercial exploitation on the market.
During the lifetime of the oil field, the oil and gas properties vary and influence GHG emissions because, as the field becomes mature, oil and gas production decreases, decreasing energy demand in
terms of hydrocarbon processing, but increasing in terms of water or gas injection as techniques to prolong production levels.
Pre-salt oil contains significant amounts of natural gas, with a high percentage of CO[2]. There are uncertainties surrounding the volume of natural gas to be used for re-injection into the well, as
well as the CO[2] produced from separation processes on the platforms, in order to maintain the reservoir pressure at an adequate level. For the commercial use of natural gas, the CO[2] molar
fraction must be below 3%. Thus, the separation of CO[2] from the natural gas to be sold is expected. This separation process must still occur at the FPSO, and the CO[2]-rich gas stream must be
re-injected into the field, characterizing a CCS process.
The assessment and quantification of GHG emissions in industries are the first measures in emission reduction plans and implementation of energy efficiency measures [
]. The preparation of GHG emission inventories consists of quantifying polluting gases emitted or removed from the atmosphere over a period of time. Decision-makers, whether at government or
corporate level, use inventories as a baseline for developing mitigation strategies and policies, in addition to evaluating such measures.
The main objective of this work is to diagnose GHG emissions from the oil and gas production and treatment process on a typical FPSO platform, used in pre-salt fields in ultra-deep waters in Brazil.
To carry out the diagnosis in the most accurate way possible, it is necessary to identify each process involved in the production platform, considering not only combustion processes but also the
practice of flaring, the existence of venting, and fugitive emissions in the oil production process and gas.
The scope of this diagnosis is specifically focused on the processes that occur on the topside of the FPSO during its operation and considers CO[2], CH[4], and N[2]O as relevant GHG. Consolidated
results are expressed in CO[2] equivalent emissions.
2. GHG Emissions from Offshore Oil Production, Including CCS
Several studies have been published on the need to reduce the carbon footprint in oil and gas production activities, especially for offshore conditions. The physical link between the consumption of
fossil fuels on the platform and CO[2] emission levels indicates the need for efficiency gains in processes involving combustion. There is also the need to reduce the practice of flaring, which must
be reduced to the minimum necessary for the safety of production operations.
Furthermore, offshore oil field is considered one of the places where CO[2] can be stored for a long time. The capture of CO[2] can occur in the oil production FPSO from combustion exhaust gases,
from CO[2] present in the natural gas or captured elsewhere and transported to the oil field. In this sense, oil fields can be seen as part of the carbon capture and sequestration (CCS) studies.
The thermodynamic performance of oil and gas separation processes has been analyzed using the concepts of exergy and irreversibility. Silva and Oliveira Jr. [
] analyzed an FPSO platform similar to the one analyzed in this work. In addition to analyzing process efficiencies, the authors calculated CO
emissions arising from the electrical energy generation process by carbon mass balance between the fuel and combustion gases. The performances of different prime-movers were compared: gas turbines,
combined cycles, and piston engines. The average emission of CO
for the natural gas ranges from 19.0 gCO
/MJ to 19.8 gCO
/MJ, depending on the cogeneration plant configuration, while it ranges from 19.4 gCO
/MJ to 26.8 gCO
/MJ for the oil.
Volsund et al. [
] analyze different options for energy supply for offshore oil production in the North Sea: the traditional use of natural gas produced in the field, hydrogen, ammonia, and biofuels supplied
externally to the platform; offshore wind energy; and direct energy supply electricity for the FPSO. Each option is discussed considering its potential advantages and the risks involved. Furthermore,
the paper also considers CCS options directly on the platform, through amine CO
absorption systems or employing oxy-fuel combustion technologies. The work emphasizes that the options with better performance in GHG still bring technological challenges or involve bulky equipment
(FPSO has limitations in available area and weight) or can even pose new health safety concerns, especially in the case of using ammonia and hydrogen. The solutions with the best prospects in the
short term consist of the combination of conventional generation complemented by offshore wind energy, the “power island” concept generating electrical energy in a high-efficiency combined cycle or
even, whenever the distance from the coast allows, the import of electrical energy produced onshore by renewable sources.
The trade-offs between environmental performance and the economic costs of operating an FPSO were studied by Zuochao et al. [
] employing the LCA technique, considering the materials, the manufacturing of the installation, its operation, and decommissioning. Using a distributed generation system encompassing solar, wind,
and natural gas energy, the authors developed an optimization with two objectives: maximum reduction in the carbon footprint and operating cost of the energy production system. Fixed emission factors
were adopted, and the work did not consider the effect of the production curve over time. Pareto extremes indicate very high operating costs or a large carbon footprint. A combination of wind energy
and natural gas (without the use of solar energy) can greatly reduce operating costs while still maintaining a good reduction in the carbon footprint.
The possibility of carbon storage in oil fields has also been analyzed, with the aim of reducing the carbon footprint in oil production and decarbonizing onshore industrial activities. In this case,
there is a need for a dedicated gas pipeline to transport the CO
from the coast to the field where it will be stored. Using the LCA technique for the construction and operation stages, Stewart and Haszeldine [
] evaluated two cases for storing CO
from the coast: during the useful life of the field or during part of the useful life. Evidently, the first option can store a greater amount of CO
, but there is an output of CO
that should be stored together with the natural gas associated with the oil. The average values of the emission factors achieved are 0.137 and 0.135 tCO
e/bbl of oil produced.
Roussanalya et al. [
] evaluated the potential benefits of producing offshore electrical energy on power islands next to natural gas production fields. Using high-efficiency combined cycles and carbon capture systems,
the electrical energy produced would be sent to the coast via submarine cables. The authors propose the use of aquifers located close to the oil field for the final disposal of CO
so as not to interfere with the production of natural gas. (CO
injected into the field diffuses and changes the composition of the natural gas.)
Hydrogen production from natural gas has been proposed. Using the LCA technique, Davies and Hastings [
] evaluated the environmental performance of H
production from offshore produced natural gas, with and without CCS, compared to hydrogen production through electrolysis (using only renewable sources or the UK electric grid mix) and against the
direct burning of natural gas. For the same annual production of H
(2.5 GW/y), gray hydrogen (from natural gas, without CCS) emits 280 MtonCO
e, and blue hydrogen (from natural gas, with CCS) emits between 200 and 260 MtonCO
e (depending on CO
capture efficiency). Using electrolysis with renewable electrical energy emits 15 MtonCO
e, and electrolysis using the UK grid emits 165 MtonCO
e. Directly burning the amount of methane required to produce the defined quantity of H
emits 250 MtonCO
e, less than gray H
3. Description and Operation of the Analyzed Installation
The FPSO analyzed is completely independent from an energy point of view. For topside processes, the electric generation system features four gas turbines with 25 MW of power each, coupled to the
main generators of 31 MVA, which generate electrical energy at 13.8 kV. Three generator sets supply electrical energy to the process, and one of them remains as a reserve, even in the situation of
maximum electrical load. In addition to this main generation system, there is an auxiliary generator system (in the hull) as well as an emergency generator set. The required process heat comes from
cogeneration using the exhaust gases from the gas turbines.
The FPSO production characteristics are presented in
Table 1
Figure 2
shows an FPSO similar to the one analyzed in this work.
Table 1.
Analyzed FPSO—General Specifications [
Characteristic Maximum Capacity
Liquid processing 24,000 m^3/day
Oil storage 1,600,000 bbl
Oil processing 24,000 m^3/day
Produced water treatment 19,000 m^3/day
Gas treatment and movement 6,000,000 m^3/day
Pressure for natural gas reinjection 55,000 kPa
Pressure for CO[2]-rich stream 45,000 kPa
Water injection 28,600 m^3/day
Figure 2.
A FPSO showing the topside processes to produce oil and gas. Source: [
The analysis of CO
emissions by the platform takes into account the variation in the quantity and quality of the crude produced by the field over time: the reduction in oil, the increase in gas content, and the
increase in the quantity of water that accompanies the oil. Over the time of production (between 25 and 30 years), the properties of the produced fluid change. In particular, CO
levels in natural gas can increase significantly.
Figure 3
shows a typical production of crude oil until the depletion of the reservoir, simulated by a Weibull statistical distribution, gas-to-oil ratio, oil-to-water ratio, and CO
molar fraction in the natural gas.
3.1. Description of the Oil Production and Gas Treatment in the FPSO
A simplified diagram of all topside oil and gas production processes can be seen in
Figure 4
. The crude oil coming from the wells reaches the production manifold (Box 1) and enters the primary separation process (Box 2).
The oil undergoes treatment to remove residual water and dissolved gases and is sent to the FPSO tanks (black stream). The water separated from the oil goes to a treatment unit that also serves the
captured seawater and can be injected into the field’s injection wells or discarded (stream in green). The gaseous phase that separated from the oil is sent to the main compression system (Box 4), as
well as the resulting gases from the vapor recovery unit (Box 3).
The gas then passes to the dehydration and dew point control units (Boxes 5 and 6). Depending on the operating condition, the gases are sent to the CO[2] removal unit (Box 7) or sent directly to the
gas injection unit (Box 10). If the CO[2] removal unit is operating, the treated gas goes to the gas export unit (Box 8), and the permeate rich in CO[2] (red stream) goes to the CO[2] compression
unit (Box 9) and from there to a gas injection unit (Box 10). This last unit can operate with CO[2] or natural gas.
The electrical power required for the processes is provided by three gas turbines, which use locally produced fuel gas whenever possible. CO[2] compressors are driven by dedicated gas turbines; the
other compressors and pumps are driven by electric motors.
3.2. The Chosen Operation Conditions of the FPSO
To analyze CO[2] emissions, three typical operating conditions were chosen based on documentation and information from the FPSO project (FEED and PID diagrams). It should be noted that the available
data is preliminary, supported by engineering calculations, operational requirements, and various simulations, general aspects of the process, and do not consider data taken from the actual
The design data for the processes related to in oil and gas production involve the following processes and systems: primary separation process; vapor recovery process; natural gas main compression
process; gas treatment processes, including dehydration, dew point adjustment, and CO[2] removal unit; gas compression process for export; natural gas compression process for reinjection; CO[2]
compression process (injection); electricity and hot water generation system for the FPSO; FPSO utility systems; seawater system; cooling water system; process heating hot water system; diesel oil
system; and fuel gas system for use on the platform.
All systems and processes are presented at their maximum capacity settings. However, for each platform operating condition, it is necessary to evaluate each process and system in a combined and
coherent way. Thus, simulation models were developed for the partial load operation situation for each type of equipment: oil and gas separators, gas treatment, gas turbines, compressors, pumps, and
heat exchangers. The partial load operation models were coupled into the global simulation model.
The global FPSO performance simulation model for each case comprised the conservation of chemical species, conservation of masses, and conservation of energy. To this end, established thermodynamic
methodologies and simulation software, such as Aspen HYSYS [
], Thermoflex [
], and EES [
], were used for the different operating cases and calculations of the FPSO platform performance.
The simulation of the topside processes chain was carried out using the Aspen Hysys software, which calculated the mass and energy balances and, in some cases, even the new molar compositions. Gas
turbine performance at partial loads was obtained through Thermoflex software.
To carry out the process simulation, two types of virtual equipment were also included: mass flow splitters and mixers. This was necessary because, at various points in the processes, a given mass
flow is divided into two or more streams with the same properties, which are sent to different equipment (virtual splitter). Likewise, situations occur in which two or more material streams are mixed
(virtual mixer) and not always with the same properties, which generates irreversibility in the process.
The process and utility system simulation model implemented in the ASPEN-HYSYS has a total of 217 pieces of equipment, as shown in
Table 2
, and a total of 669 material flows connecting equipments.
Energy and CO
emission diagnoses were prepared for three typical FPSO operating conditions chosen and named as case 7A, case 2B, and case 6A. Such conditions are presented in
Table 3
below. It is important to highlight that only the equipment necessary for each case, as well as the relevant material currents, was considered in the simulation. For example, in cases 6 and 7, the CO
removal unit and compression system is deactivated, and the associated material streams have zero mass flow rates.
The numbers indicating each case are associated with specific compositions of the natural gas produced before any treatment. Compositions 7, 2, and 6 correspond to CO[2] contents in the natural gas
of 12.4%, 25.1%, and 28.3%, respectively, on a molar basis. These high CO[2] molar fractions in the natural gas are typical in the Pre-Salt oil province.
In operating mode A, corresponding to analyzed cases 7 and 6 with different molar fractions in composition, the treated gas goes through a bypass of the CO[2] removal process, and the CO[2]
separation unit is inactive. In operating mode B, the gas is sent to the CO[2] removal unit, producing natural gas with a low CO[2] content (<3%) to export, and a CO[2]-rich permeate stream, which is
sent to the compression unit for re-injection into the oil field.
Case 7A was simulated because it represents a condition of maximum oil and gas. The simulation of this operating condition was based on the primary separation unit. Thus, the other downstream units
(vapor recovery and main compression system) received the real mass flows of oil, water, and gas and not the nominal design values. The equipment in these units began to operate at partial loads, as
described above. Likewise, each downstream unit received the currents under the real conditions of molar composition and mass flow rate of the unit that preceded it.
Case 2B was simulated because it represents a condition of 50% BSW, which characterizes a period of operation of the field intermediate between the initial condition of maximum oil and gas and the
condition of maximum water close to the end of production. This case presents high CO[2] values in crude oil.
Case 6A was simulated because it represents a condition of maximum water, which characterizes a period of field operation close to the end of production. This case also presents high CO[2] values in
crude oil.
4. Methodology for Estimating CO[2] Equivalent Emissions
To make the diagnosis of the GHG emissions, equipment classifications were carried out according to the largest sources of emissions in the oil and gas industry [
] presented in
Table 4
According to the emission source, methodologies were developed to approach the emission inventory analysis, as well as numerical approximations to the amounts of GHG released into the atmosphere,
using the Global Warming Potential (GWP) as equivalence created by the Intergovernmental Panel on Climate Change (IPCC—a United Nations body for assessing the science related to climate change) [
] in estimating the CO
equivalent for the CH
emissions (
Table 5
For the three cases in which the analysis was carried out, the following aspects were considered:
• Electrical load values for the GT electric generation and shaft power GT were obtained through simulation in the Aspen HYSYS^® software.
• Ambient temperature conditions at 30 °C, sea level.
• Steady-state operating regime, typical operation for oil field age.
• For fugitive emissions, emissions calculated at equipment level according to the design PID diagrams provided.
• Flow of gas burned in the flare as “Assist Gas” and “pilot” maintained constant for the three simulated cases.
• Flare burning efficiency of 98%. The remaining 2% was considered as vented gas.
• Emissions calculated for the FPSO’s oil and gas processing operation, without considering auxiliary operations, such as transporting oil to the continent, movement of helicopters, or logistical
support boats.
There are several international agencies with protocols and guidelines for estimating greenhouse gases for different applications and industries with high energy intensity. In particular for oil and
gas production processes, there are three documents generally used to calculate emissions and which were used to carry out the analysis presented: (a) 2006 IPCC Guidelines for National Greenhouse Gas
Inventories [
]; (b) 2009 API Compendium of Greenhouse Gas Emissions for the Oil and Gas Industry [
]; and 1996 EPA AP-42 Compilation of Air Pollutant Emission Factors, Volume 1: Stationary Point and Area Sources [
The IPCC recommendations provide an approach of three levels or tiers for analyzing emissions in activities related to oil and gas (exploration, production, refining, and transport). These approaches
range from the use of emission factors based on simple production data, or high-level production statistics, to the use of rigorous estimation techniques involving disaggregated activities and actual
plant data.
The methodologies mentioned in the API compendium can be used to estimate GHG emissions in individual projects, entire facilities, or enterprise-wide inventories. The purpose of the analysis, as well
as the available data, generally determines the level of detail for the selected approximation.
Lastly, the EPA AP-42 protocol provides emission factors in addition to emissions calculation methodologies that are also described in the API compendium but bring together data taken from the
industry on which the reported emission factors are based.
The application of each methodology can lead to different results. Satya et al. [
] evaluated GHG emissions on a platform in Indonesia, comparing the API and IPCC methodology. Based on the API method, the contribution of carbon in the fuel corresponds to 97.15% of total emissions.
While using the IPCC method, this contribution is 63.88%. The global inventory calculated by the IPCC is 258.357 tCO
e, which is 55% higher than the value calculated by the API method (166.204 tCO
e). The authors observed that the greatest contribution to the divergence between values can be attributed to the differences between the values calculated for fugitive emissions in the production of
natural gas using different methods.
4.1. GHG Emissions Due to Combustion
The combustion of a substance containing carbon, hydrogen, and oxygen can be represented by the Equation (1) general reaction. If the complete combustion is assumed, the nitrogen of air and an
eventual excess air do not interfere in the CO
$C x H y O z + x + y 4 − z 2 O 2 → x C O 2 + y 2 H 2 O$
Natural gas is a mixture of different components, with the most part of them containing carbon. But natural gas can also contain nitrogen, CO[2], and other contaminants.
For every mole of carbon in the fuel molecule
, one mole of CO
is formed. If
is the number of moles of carbon in the molecule
, the carbon mass fraction (
) in the component
of the fuel gas is given by Equation (2):
$W t C j = x i × 12 1 × M W j k g C a r b o n k g s u b s t a n c e j$
The total amount of carbon in the gas mixture is the sum of the contributions to each substance
composing the gas mixture, given by Equation (3):
$W t C m i x t u r e = ∑ j = 1 c o m p W t j × W t C j k g C a r b o n k g g a s m i x t u r e$
is the overall mass fraction of carbon in the natural gas and
is the mass fraction of the substance
in the gas.
Finally, knowing that each mass unit of carbon produces 44/12 mass units of CO
and also the fuel mass flow, the CO
emission from combustion is given by Equation (4):
$E C O 2 = m g ˙ × W t C m i x t u r e × 44 12 k g C O 2 s$
$m g ˙$
is the fuel mass flow in kg/s.
For each case analyzed, the composition of the fuel gas is determined by the oil and gas separation and gas treatment processes. Likewise, the simulation of topside processes determines the FPSO’s
energy demand and the corresponding mass fuel consumption.
Depending on the FPSO’s operating mode, two or three gas turbines operate to generate electricity and process heat. The two gas turbines dedicated to driving the compressor for CO[2]-rich stream may
or may not be in operation. There is no supplemental burning of natural gas in the heat recovery boilers of the cogeneration system.
4.2. Flare GHG Emissions
The flaring process is normally used on offshore platforms to burn gas for emergency procedures, vessel depressurization processes, or other operational or safety reasons. The flare is always burning
to cope with rapid operational demands. This burning must be kept as low as possible. The cases studied include the burning in the flare of only gas flows corresponding to the pilot and assistance
gas. Knowing the flow rates of gas burned in the flare along with its composition, the flow rates of CO
produced were determined using the equations shown above for combustion processes. However, a flare efficiency was considered, with the remainder being released into the atmosphere as unburned gas.
Equation (4) was adapted to obtain Equation (5), with the introduction of the term “EC,” which is the efficiency for the flare. In the flare, there is a methane slip to atmosphere, a GHG emission
worse than the emission of CO
. This reduces the CO
emissions but increases the CH
$E C O 2 = m g ˙ × E C × W t C m i s t u r e × 44 12$
where the term
corresponds to the flare efficiency. The flare efficiency was fixed at 98%. The 2% of gas not burned in the flare also contributes to GHG emissions, being counted as vented gas (CH
4.3. Fugitive GHG Emissions
Fugitive emissions are caused by uncontrolled leaks in equipment. Any pressurized equipment can generate leaks, especially in pipes, valves, open lines, and flanges, among others.
Table 6
shows the types and quantities of FPSO equipment considered for the fugitive emissions assessment.
On an operating platform, there are usually measuring methods and equipment that allow obtaining estimated data on fugitive emissions. For the cases studied, an analysis was performed at the
component level, using emission factors reported by the EPA [
] (
Table 7
), gathered from data reported by the oil and gas industry. It is worth noting that the API also reports emission factors at the component level, which is why they are considered and compared with
the EPA factors in the analysis carried out.
The methodology adopted corresponds to the application of emission factors to an inventory of components, carried out based on information provided by the PID diagrams of the processes and
considering the content of CH
and CO
present in the fuel gas mixture. The general method recommended by the EPA to obtain the total organic compounds (TOC) emissions is as follows:
$E T O C = F E × M F T O C × N$
for determining emissions of TOC. FE stands for emission factor from
Table 7
, MF
is the mass fraction of the TOC in the gas (assumed = 1 in this work), and N is the number of components, which presents a given FE (example: number of flanges) listed in
Table 6
and CO
emissions are obtained from their respective mass fractions in total organic carbon emissions:
$E C H 4 = E T O C × M F C H 4$
$E C O 2 = E T O C × M F C O 2$
4.4. Emissions from Processes and Ventilation
Ventilation emissions correspond to releases of gases into the atmosphere as a product of operational practices or equipment design. For the case studied, emissions from ventilation in the processes
of flaring, molecular sieves, flash in the oil storage tank, and others were evaluated.
In the case of the flare, a ventilation of 2% of the gas flow used in the flare was considered. Equations (7) and (8) can be used to calculate the CH[4] and CO[2] flow rates emitted in the process.
Molecular sieves have adsorbent materials, such as zeolites, that have an affinity for water. During the change of material, the gases contained in the sieve vessel are released, which constitute GHG
emissions. Emissions are estimated [
] according to the internal volume of the dehydrator, as follows:
$P G = H 2 × D 2 × π × P 2 × G × N 4 × P 1$
is the gas loss;
is the height of the dehydrator;
is the diameter of the dehydrator;
is the gas pressure;
is the atmospheric pressure;
is the fraction of the vessel volume occupied by gas; and
is the number of desiccant changes per year. With the gas mass flow rate, CH
and CO
emissions can be calculated by Equations (7) and (8).
There are several methodologies to estimate emissions caused by flash processes in storage tanks, where gas contained in oil is released into the atmosphere due to pressure changes between process
lines and the tank. The Vasquez–Beggs empirical correlation [
] for gas–oil ratio can be used to estimate the relationship between gas and oil at process conditions and is given by Equation (10):
$R s = C 1 × S G x × P i + 14.7 C 2 × e x p C 3 × A P I T i + 460$
is the gas produced by oil flash in the storage tank (scf/bbl). C1, C2, and C3 are nondimensional coefficients with values given in
Table 8
. Once the production and composition of the oil stored in the tanks is known, Equations (7) and (8) are employed to calculate the flow rates of methane and carbon dioxide emitted into the
The specific weight at 100 psig is necessary data and can be calculated by Equation (11):
$S G x = S G i × 1 + 0.00005912 × A P I × T i × l o g P i + 14.7 114.7$
where SGi is the gas-specific gravity at the reference separator pressure and SGi is the gas-specific gravity at the actual separator conditions of Ti (°F) and pi (psig).
Other sources of emission from ventilation, such as purging vessels and compressors, as well as starting compressors, were analyzed using emission factors reported by API.
4.5. Proposed GHG Emissions Indicators
Some GHG emissions indicators were proposed, which can be used for comparisons between different facilities and/or operating regimes. The energy diagnosis of the FPSO operating in different
conditions constitutes a baseline for future comparisons. Thus, the indicators can be used to compare different operating strategies of the FPSO in its current design or to compare different
proposals for changing processes and/or operating regimes.
Changes in the characteristics of the fluid present in the field, plant operating modes, and field production stage (beginning, end of production in the field, or intermediate situations) can be
compared through the emissions indicators.
Indicator 1: ratio of the GHG emissions from GT to electricity produced. This indicator relates the total GHG emissions produced by gas turbine generators to the amount of electrical energy produced
and is expressed in kg CO
e/kWh. It can be used to compare the emissions of different electricity production technologies for the FPSO.
$I n d 1 = G H G e m i s s i o n s t o g e n e r a t e e l e c t r i c i t y P r o d u c e d e l e c t r i c e n e r g y$
Indicator 2: Total GHG emissions per useful energy produced. This indicator relates total GHG emissions to the total amount of useful energy produced (electricity and process heat) by the
cogeneration system of the FPSO and is expressed in kg CO
$I n d 2 = T o t a l G H G e m i s s i o n s P r o d u c e d e n e r g y i n c o g e n e r a t i o n G J$
Indicator 3: Total GHG emissions per barrel of oil equivalent produced. This indicator relates total GHG emissions to the amount of hydrocarbons produced by the FPSO (oil and gas) expressed in kg CO
$I n d 3 = T o t a l G H G e m i s s i o n s P r o d u c e d B O E o i l a n d g a s$
5. Results and Discussion
As previously mentioned, the simulation of the oil and gas processing plant was the first step of the methodology, as it allows the obtaining of essential variables for calculating emissions for each
case of operation.
5.1. Main Results of the Operation Simulation
Table 9
shows the production data resulting from the simulation, such as the amount of crude oil, the amount of oil exported, exported gas, injected gas, injected rich CO
stream, and use of seawater. Given the nominal capacity of the platform, the results make it clear the need to consider partial load operation of each equipment and process.
The performance of the electrical, thermal, and mechanical energy production systems is presented in
Table 10
. Fuel consumption in each case was used to calculate GHG emissions from combustion.
5.2. GHG Emissions Calculated for Each Operating Mode
With data from the thermodynamic simulation of the FPSO operation under the three chosen conditions, it is possible to quantify GHG emissions using the methodology already described.
Table 11
Table 12
Table 13
show the results obtained for cases 7A, 2B, and 6A, respectively.
The two methods discussed previously were used: API and EPA. The sources of emissions associated with combustion are the most important, by a large margin.
Table 11
Table 12
Table 13
show that the values obtained by the two methods are similar, except for fugitive emissions, which have high percentage deviations. In any case, the absolute values of these emissions are small
compared to other emissions.
The total GHG emissions are higher for the case 7A, since this operation condition requires a large amount of electrical energy.
5.3. Comparisons of GHG Emissions Between Cases
5.3.1. Combustion Emissions
Emissions due to combustion sources represent between 95% and 97% of total emissions for the processes analyzed on the FPSO platform. In
Figure 5
, cases are compared according to emissions from combustion in turbogenerators, turbo-compressors, and the portion corresponding to combustion in the flare.
It is noted that the highest emissions correspond to case 7A, where the amount of gas processed is much higher than in the other cases. Compressor loads are the main contributors to high electrical
demand in this case. Case 2B is the only one in which CO[2]-rich steam compressors operate, corresponding to 21% of total combustion emissions. Flare emissions are similar between all cases; only the
composition of the fuel gas burned between operating modes A and B varies. Although case 2B includes the activation of the CO[2] compression set, case 6A presents higher emissions due to the flow of
gas treated throughout the process, greater than in case 2B.
5.3.2. Fugitive Emissions
The analysis of fugitive emissions was carried out using emission factors at the level of each component of the FPSO platform. The amount of equipment in the gas pipes in the process was estimated
according to data provided, and the emission factors were subsequently applied. When counting equipment, for each case, the number of components used in each subprocess was evaluated, considering the
number of compression trains in operation and process segments not in operation in each mode analyzed.
Case 7A treats gas near 80% methane in molar fraction, so the analysis carried out for the entire gas processing in the FPSO points to the highest fugitive emissions in all cases studied, as shown in
Figure 6
. Case 6A has a high CO
content (60% in mole fraction) in the treated gas, meaning emissions are the lowest among the cases studied. This effect is caused by the GHG potential of methane, many times greater than CO
itself. Although case 2B has the smallest amount of equipment in operation, emissions are largely affected by the 60% mole fraction composition of methane in the gas produced. The relevance of
applying the GWP indicator gives greater importance to emissions due to the treatment of gas with a high CH
content, due to the equivalence of the hydrocarbon in relation to CO
5.3.3. GHG Emissions from Processes and Ventilation
The gas composition is evaluated for each case and for each process described in the methodology, since the ventilation emissions depends on the CH
and CO
mass fractions in the vented gas. It is expected that ventilation emissions follow a similar behavior to fugitive emissions, since the gas is not burned but rather released into the atmosphere
intentionally for operational reasons of the platform or specific equipment. The case that reports the higher ventilation emissions is case 7A, due to the higher percentages of methane in the
different types of gas studied, as shown in
Figure 7
. Although the gas ventilated by the flare corresponds to only 2% of the total gas flow intended for burning in the equipment, it constitutes, on average, 88.6% of total ventilation emissions.
Emissions due to molecular sieves for gas treatment and flashing in the FPSO storage tank reach 10.5% of the total and other sources less than 1% (vessel and compressor purges, compressor start-up
5.3.4. Overall GHG Emissions
The analysis covers all FPSO operations in the oil and gas production, treatment, and export processes. As previously shown in process emissions, combustion emissions represent in the cases studied
between 95% and 98% of the platform’s total emissions, as indicated in
Figure 8
. Therefore, actions to reduce CO
in exhaust gases can have major global impacts on emissions.
5.3.5. GHG Emission Indicators
The first indicator (
Table 14
) seeks to quantify CO
emissions from turbogenerators in relation to the electrical energy produced (Ind1). When operating at lower loads, the turbogenerators in cases 2B and 6A emit more GHG compared with the power
generated. This is an effect of the lower gas turbine efficiencies running on partial loads.
Indicator 2 relates total GHG emissions to the total energy produced in the FPSO in TJ (Ind2) and is a measure of the FPSO cogeneration system efficiency. To be noted, the total energy produced is
the sum of electric power with the exergy of the process heat. Case 2B is the worse case, due to the large amount of heating water and also to the composition of gas produced, which contains a high
level of CO[2]. The CO[2]-rich stream must be injected into the reservoir, and its compressor is driven by a gas turbine, increasing the fuel consumption.
The indicator that relates the amount of hydrocarbon produced on the platform (Ind3) is not favorable for case 6A, where the flow of crude is high, but the amount of oil produced is small, due to the
large amount of water in the crude oil. In case 7, on the contrary, the emission indicator is low due to the high quantity of oil produced (146,000 barrels/day, approximately), and in addition, the
gas produced is rich in methane.
6. Conclusions
Given the predominance of combustion processes in GHG emissions in FPSO, it is essential to increase the efficiency of prime movers used for electrical generation or mechanical power. It is
recommended to use high-efficiency power systems, such as combined cycles. Therefore, increasing efficiency in electrical generation can represent an important step toward increasing the efficiency
of the global production process, with a consequent reduction in CO[2] emissions. Process heat production must also be based on waste heat recovery (WHR) and cogeneration, avoiding gas burning.
The results obtained in quantifying GHG emissions, expressed in terms of CO[2] equivalent, should also be highlighted. Emissions associated with production processes and equipment (ventilation,
fugitive) are low when compared to those arising from combustion processes, whether from TGs or the gas turbine that drives the CO[2] compressors or from burning in flare.
The proposed indicators can help establish a baseline from which proposed changes to the project, processes, or operational strategies can be compared.
Author Contributions
Conceptualization, methodology, validation, formal analysis, data curation: V.L.A.B. and W.L.R.G. Software and investigation: V.L.A.B. Resources, writing—review and editing: W.L.R.G. All authors have
read and agreed to the published version of the manuscript.
This study was part of a research project funded by BG Group (now part of Shell Group).
Data Availability Statement
The original data was obtained during a research project with privacy clauses. To access the data, please contact the authors.
Conflicts of Interest
Author Victor Leonardo Acevedo Blanco was employed by the company Vanti Group. The remaining authors declare that the research was conducted in the absence of any commercial or financial
relationships that could be construed as a potential conflict of interest.
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18. Petrobras. 2023. Available online: https://epbr.com.br/petrobras-p-71-o-ultimo-fpso-replicante-atinge-capacidade-maxima-no-pre-sal/ (accessed on 12 July 2024). (In Portuguese).
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22. Gomez, D.R.; Watterson, J.D.; Americanohia, B.B.; Ha, C.; Marland, G.E.; Matsika, G.L.; Namayanga, G.; Osman-Elasha, B.; Saka, J.D.K.; Treanton, K. Chapter 2 Stationary Combustion. In IPCC
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Figure 1.
Oil and gas production in Brazil—2023/2024. Own preparation from data source: [
Figure 5. GHG emissions due to combustion. TG is the contribution of turbogenerators, and TC is the contribution of the gas turbine to compress CO[2]-rich stream.
Type of Equipment Quantity
Three-phase separators 3
Two-phase separators 18
Heat exchangers 44
Valves 34
Mass flow splitter (virtual) 42
Mass flow mixers (virtual) 39
Pumps 10
Compressors (including GTs and NG) 17
Combustion chambers (GTs) 5
Turbines (GTs) 5
Total 217
Case Mode Oil Field Age
7 A—The CO[2] removal unit is bypassed, and all gas produced must be injected into the oil reservoir. Max. Oil & Gas
2 B—Treated gas from the CO[2] removal unit is exported; the acidic gas, rich in CO[2], is injected into the oil reservoir. 50% BSW *
6 A—The CO[2] removal unit is bypassed, and all gas produced must be injected into the oil reservoir. Max. water
* BSW—Basic Sediment and Water.
Category Main Sources
Direct emissions
Emissions from combustion sources: Boilers, heaters, ovens,
Stationary equipment internal combustion engines, gas turbines, flares, incinerators, etc.
Mobile equipment Barges, ships, locomotives, trucks, helicopters, airplanes
Process emissions Amine units, glycol dehydrators, molecular sieves, etc.
Other ventilation sources Storage tanks, pneumatic devices, chemical injection pumps,
flaring, compressor discharge, etc.
Fugitive emission Valves, flanges, connectors, pumps, compressor leaks, opened lines
Indirect emissions
Electricity Off-site electricity generation for on-site consumption
Steam/Heat Off-site steam and/or process heat production for on-site consumption
Green House Gas Lifetime (Years) GWP[100]
With Feedback Without Feedback
CH[4] 12.4 34 28
HFC-134a 13.4 1550 1300
CFC-11 45 5350 4660
N[2]O 121 298 265
CF[4] 50,000 7350 6630
Component Valve Pump Seal Connections Flanges Open Lines Other
Gas composition 1: without CO[2] removal
Pig 1 45 0 16 54 14 2
Pig 2 49 0 16 54 14 2
Pig 3 37 0 8 36 7 2
Principal manifold 39 0 50 62 2 2
Three-phase separator 30 0 16 38 6 4
Oil dehydrator 1 23 0 8 32 8 2
Oil dehydrator 2 23 0 8 32 8 2
Principal pump 8 0 8 32 8 4
Oil transfer pump 23 6 12 18 8 2
Vapor recovery unit 39 0 12 44 6 2
Knockout drum 32 0 8 30 13 2
Main gas compressors (3 units) 105 0 8 132 9 6
Gas dehydrator system 42 0 8 48 8 2
Dew point control system 148 0 8 214 32 2
Total 665 6 194 846 147 38
Gas composition 2—Treated gas—CO[2] < 3%
CO[2] removal system 12 0 8 20 4 2
Gas compressor—first stage—to export 105 0 8 66 16 6
Gas compressor—second stage—to export 96 0 8 108 21 6
Exportation gas header 42 0 8 46 5 2
Total 255 0 32 240 46 16
Gas composition 3—CO[2]-rich stream
CO[2] ompressor—first stage 52 0 6 62 9 2
CO[2] compressor—second stage 43 0 6 53 8 2
CO[2] compressor—third stage 36 0 6 44 8 2
CO[2] compressor—fourth stage 46 0 6 56 8 2
CO[2] injection compressor 112 0 8 120 24 2
CO[2] injection header 52 0 8 72 13 4
Total 341 0 40 407 70 14
Emission Factor
Component EPA API
(kg gas/hr/comp.) (Ton. TOC/hr/comp.)
Valves 4.50 × 10^−3 5.14 × 10^−7
Pump seals 2.40 × 10^−3 1.95 × 10^−7
Connectors 2.00 × 10^−4 1.08 × 10^−7
Flanges 3.90 × 10^−4 1.97 × 10^−7
Open lines 2.00 × 10^−3 1.01 × 10^−6
Other 8.80 × 10^−3 6.94 × 10^−6
Coefficient API ≤ 30 API > 30
C1 0.0362 0.0178
C2 1.0937 1.1870
C3 25.7240 23.931
Description Mass Flow [kg/s]
Inlet Case 7A Case 2B Case 6A
Crude oil 311.8 299.3 338.7
Seawater 1480.3 731.2 790.6
Imported fuel gás 5.42 0.0 3.20
Exported oil 212.5 89.0 36.1
Exported gas 0.0 16.8 0.0
Injected gas 92.8 0.0 46.2
Injected rich CO[2] stream 0.0 15.6 0.0
Gas to flare 0.9 0.9 0.9
Water in crude oil 15.9 186.4 268.8
Injected water 338.6 203.7 266.3
Discarded water (sea) 1157.6 713.9 793.1
Case 7A Case 2B Case 6A
Electric demand [MW] 72.75 33.38 31.25
Number of TG operating 3 2 2
Gas turbine generators load [%] 98.0 44.7 63.9
CO[2]-rich stream compressor demand [MW] --- 6.8 ---
Gas turbine (CO[2]-rich compression) load [%] --- 43.8 ---
Gas turbine (CO[2]-rich compression) operating --- 1 ---
Heat demand for processes [MW] 47.15 45.78 33.10
Cogeneration efficiency (energy) [%] 57.9 59.3 63.9
Cogeneration efficiency (exergy) [%] 38.6 35.4 36.9
Emission Sources Ton CO[2] Equiv/Year
API EPA % Deviation
Gas turbine for electric generation 360,680 360,717 0.01%
Gas turbine for CO[2]-rich compressor 0.00 0.00 0.00%
Flare combustion 78,349 78,360 0.01%
Others—Combustion 494 494 0.00%
Venting 11,654 11,654 0.00%
Fugitive emissions 226 975 76.85%
Total 451,404 452,200 0.18%
Emission Sources Ton CO[2]/Year
API EPA % Deviation
Gas turbine for electric generation 107,625 107,619 −0.01%
Gas turbine for CO[2]-rich compressor 49,502 49,465 −0.08%
Flare combustion 78,739 78,723 −0.02%
Others—Combustion 494 493.75 0.00%
Venting 10,452 10,452 0.00%
Fugitive emissions 114 482 76.40%
Total 246,926 247,234 0.12%
Emission Sources Ton CO[2]/Year
API EPA % Deviation
Gas turbine for electric generation 189,600 189,628 0.01%
Gas turbine for CO[2]-rich compressor 0.00 0.00 0.00%
Flare combustion 78,721 78,731 0.01%
Others—Combustion 494 494 0.00%
Venting 10,021 10,021 0.00%
Fugitive emissions 62 260 76.12%
Total 278,897 279,132 0.08%
Units Case 6A Case 2B Case 7A Emission Indicator
kg CO[2]/kWh 0.655 0.664 0.574 Ratio of GHG emissions from electricity generation to power produced. Ind 1
kg CO[2]/GJ 267.9 290.3 199.9 Ratio of GHG emissions from cogeneration to energy produced (heat and power) Ind 2
kg CO[2]/BOE 171.5 65.2 43.8 Ratio of overall GHG emissions to overall hydrocarbons produced (oil and gas) Ind 3
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MDPI and ACS Style
Acevedo Blanco, V.L.; Gallo, W.L.R. Diagnosis of GHG Emissions in an Offshore Oil and Gas Production Facility. Gases 2024, 4, 351-370. https://doi.org/10.3390/gases4040020
AMA Style
Acevedo Blanco VL, Gallo WLR. Diagnosis of GHG Emissions in an Offshore Oil and Gas Production Facility. Gases. 2024; 4(4):351-370. https://doi.org/10.3390/gases4040020
Chicago/Turabian Style
Acevedo Blanco, Victor Leonardo, and Waldyr Luiz Ribeiro Gallo. 2024. "Diagnosis of GHG Emissions in an Offshore Oil and Gas Production Facility" Gases 4, no. 4: 351-370. https://doi.org/10.3390/
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Cross-Country Crawl
The ant is one of the strongest animals. That tiny crumb it’s carrying could be 50 times the ant’s weight! So could an army of ants carry YOU across the U.S.? For starters, about 1,800 ants can lift
1 pound. So a 70-pound kid would need 70 of those armies, or 126,000 ants. The fastest ants can run about 900 feet in an hour. So to travel 2,680 miles, even the speediest ants would need 15,722
hours for the trip. Maybe it’s easier to take a plane!
Wee ones: Ants have 6 legs, like any insect. What numbers would you say to count them?
Little kids: If 1 ant takes 5 months to cross Texas and another ant takes 11 months, which ant walked more slowly? Bonus: If the ants start their trip in April, what is their 2^nd month of walking?
Big kids: So how long would this trip take? About how many hours are there in 1 year? Bonus: Given that number of hours, about how long is that ant trip across the U.S.?
The sky’s the limit: If really slow ants take 50 years to carry you across America, and your age at the end is 6 times your starting age, how old were you at the start?
Wee ones: 1, 2, 3, 4, 5, 6.
Little kids: The ant that took 11 months. Bonus: May.
Big kids: Lots of ways to estimate…for example, you could round to 25 x 360, which is 1/4 of 100 times 360. That comes to 9,000 hours. That’s pretty close to the more precise answer of 8.760! Bonus:
About 2 years.
The sky’s the limit: If your starting age was just 1/6 of the total, then the 50 years added the other 5/6 of the total. 50 is 5/6 of 60, so you began the trip at 60 – 50 = 10 years old.
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Interference and electron Optics Interference:- Interference of waves, Interference due to thin films of uniform (with derivation) and nonuniform thickness (without derivation), Fringe width,
Newton’s Rings, Applications of Newton’s Rings for determination of
(i) wavelength of incident light / radius of curvature of Plano convex lens
(ii) refractive index of a given liquid; Michelson’s interferometer, applications for determination of
(i) wavelength of a monochromatic source
(ii) refractive index /thickness of a transparent material; Engineering applications of interference
(i) Testing of optical flatness of surfaces
(i) Nonreflecting / Antireflection coatings. Electron Optics :- Motion of an electron in electric (parallel, perpendicular) and magnetic (extensive, limited) fields, crossed fields. Electrostatic
and magneto static focusing, Scanning electron microscope (SEM) , Bainbridge mass spectrograph.
Diffraction and ultrasonic
Diffraction : - Diffraction of waves, classes of diffraction, Fraunhoffer diffraction at a single slit (geometrical method), conditions for maxima and minima, Intensity pattern due to a single
slit, Plane diffraction grating, conditions for principal maxima and minima, intensity pattern; Resolving power, Resolving power of a grating.
Ultrasonics :- Ultrasonic waves, Piezo-electric effect, Production of ultrasonic waves by Piezoelectric oscillator, Magnetostrictive effect, Production of ultrasonic waves by magnetostrictive
oscillator, properties of ultrasonic waves, Applications of ultrasonic waves
(i) Scientific- Echo sounding, Sound signaling, depth sounding, SONAR, cleaning of dirt etc
(ii) Engineering –thickness measurement, cavitation, Ultrasonic cleaning, Nondestructive testing, Flaw detection, Soldering, Drilling and welding
(iii) Medical- for diagnostics and treatment
Polarisation and nuclear physics
Polarisation :- Introduction, production of plane polarised light by refraction (pile of plates), Law of Malus, Double refraction, Huygen’s theory of double refraction, Cases of double refraction
of crystal cut with the optic axis lying in the plane of incidence and (i) parallel to the surface (ii) perpendicular to the surface (iii) inclined to the surface, Retardation plates-quarter wave
plate (QWP), Half wave plate (HWP); Analytical treatment of light for the production of circularly and elliptically polarised light, Detection of various types of light (PPL, CPL, EPL, Upl, Par
PL), Optical activity, Specific rotation, Polaroids
Nuclear Physics :- Nuclear fission in natural Uranium-Chain reaction, Critical size. Nuclear fuels, Wave particle duality and wave equations
Wave Particle Duality :- Wave particle duality of radiation and matter, concept of group velocity and phase velocity; Uncertainty principle, Illustration of electron diffraction at a single slit.
Wave Equations :- Concept of wave function and probability interpretation, Physical significance of the wave function, Schrodinger’s time independent and time dependent wave equations,
Applications of Schrodinger’s time independent wave equations to problems of (i) Particle in a rigid box (infinite potential well), Comparison of predictions of classical mechanics with quantum
mechanics (ii) Particle in a non-rigid box (finite Potential Well)- Qualitative (results only);
Lasers and superconductivity
Lasers :- Requirement for lasing action (stimulated emission, population inversion, pumping), Characteristics– monochromaticity, coherence, directionality, brightness. Various levels of laser
systems with examples (i) Two level laser system- semiconductor laser (ii) Three level laser system- Ruby laser and He-Ne laser.Applications i) Communication systems-fiber optics in brief, ii)
Information technology holography-construction, reproduction.
Superconductivity :- Introduction to superconductivity, Properties of superconductors (zero resistance, Meissner effect, critical fields, persistent currents), isotope effect, BCS theory. Type I
and type II Super conductors, Applications (super conducting magnets, transmission lines etc), DC and AC Josephson effect
Semiconductor physics and physics of nano particles Semiconductor physics :- Band theory of solids, Classification of solids on the basis of band theory, Types of semiconductors, Introduction to
the concept of electrical conductivity, conductivity of conductors and semiconductors. Hall effect and Hall coefficient, Fermi-Dirac probability distribution function, Position of Fermi level in
intrinsic semiconductors (with derivation) and in extrinsic semiconductors (variation of Fermi level with temperature (without derivation)), Band structure of PN junction diode under zero bias,
forward bias and reverse bias; Transistor working, PNP and NPN on the basis of band diagrams, Photovoltaic effect, working of a solar cell on the basis of band diagrams and Applications.
Physics of Nanoparticles :- Introduction, Nanoparticles, Properties of nanoparticles (optical, electrical, magnetic, structural, mechanical), Brief description of different methods of synthesis
of nanoparticles such as physical, chemical, biological, and mechanical. Synthesis of colloids. Growth of nanoparticles, Synthesis of metal nanoparticles by colloidal route, Applications of
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Table Calculations • Genstat v21
Select menu: Data | Table Calculations
Use this to calculate new tables from existing tables. You can add, subtract, multiply and divide tables, along with a wide range of other arithmetic and Boolean operations. The tables with matching
classifying factors have corresponding cell values operated on by the expression.
For example, if there are two A x B tables called S and T, where i indexes A and j indexes B, then U = S + T, has cells U[ij] = S[ij] + T[ij]. Where tables have different classifying factors, the
resulting table will have the union of the classifying factors, and the values in the cells will be used from the matching levels of the classifying factors from each tables used in the expression.
For example, if V is a 1-way table classified by A and W is a 1-way table classified by B, Y = S + V + W will have cells Y[ij] = S[ij] + V[i] + W[j].
When creating a table you can summarize over some of the classifying factors to remove them and replace the results in each of the cells of the other classifying factors by a summary statistic over
the removed factors. For example, if we use a total summary for factor B in the calculation Z = S + T, then Z has cells Z[i] = ∑[j](S[ij] + T[ij]).
The calculation expression is displayed at the top of the menu and can be constructed step by step by combining data structures and operators. You can use brackets to control the order of evaluation
when required. You can also type directly into the expression box, or click buttons, or double click data structure names in the table or scalar available data lists to build the expression.
When forming a calculation, all the tables in the calculation must be consistent with respect to having margins, i.e. either they all must have margins or none must have margins. If a mixture of
presence/absence of margins occurs, a warning in red will be displayed below the syntax field.
1. After you have imported your data and created a table, from the menu select Data | Table Calculations.
Available tables
Lists the available table data structures. Double-click a name to copy it into the expression being formed or the Save result in field. Move the cursor over a name in the list to display a tooltip
showing the classifying factors. This list can be reduced to only display those tables with specific classifying factors by selecting the Filter available tables by classification factors checkbox
and choosing the factors.
Filter available tables by classification factors
Select this to use the factors listed in the Select classifying factors for table list menu to filter the tables shown in the Available tables list.
Specify classification factors
This opens a menu that lets you select classifying factors for the table list. Only tables with classifying factors matching the selected list will then be displayed in the Available tables list.
Lists scalar structures which can be used in the expression. Double-click a name to copy it into the expression being formed.
Enables you to insert a function into the expression. Only functions appropriate for table calculations are displayed.
Use the keypad to enter operators into the expression being formed. You can also type the corresponding text directly into the expression, but note that the logical operators (and, eqs, or, nes, not,
is, in, ni, eor, isnt) must be delimited by dots (for example .and.).
Order and summaries over classifying factors
This lists the classifying factors in the resulting table from the calculation expression. You can change the order of the classifying factors in the resulting table by selecting a factor and using
the up Summary button will open a menu where you can select the summary type to use.
Lets you select the summary statistic for the classifying factor in the list. Selecting a summary statistic will collapse the table across that classifying factor and will remove the factor from the
Summarize in one step
When you are summarizing more than one factor using the same summary statistic for each factor being removed, this option becomes available. When selected, a summary is formed across all factors
being removed. This can make a difference if there are missing cells in the table. For example, if summarizing over A & B, each with two levels, and cell A[2]B[2] is missing then the summary using
means will be MEAN(A[1]B[1],A[2]B[1],A[1]B[2]) using this option, and otherwise if we summarize over A and then B it would be MEAN(MEAN(A[1]B[1],A[2]B[1]),A[1]B[2]).
Save result in
Specifies a table structure to contain the result of the expression.
Display in output
Lets you display the results in the Output window.
Display in spreadsheet
Lets you display the results in a new spreadsheet.
Use tabbed table with pages
Select this to create a spreadsheet using a tabbed-table. The factor across the pages/tabs will be the classifying factors selected from the dropdown list of classifying factors in the table. The
default page factor is the first classifying factor in the table. A table must have at least 3 classifying factors to be displayed in a tabbed-table.
Action Icons
Pin Controls whether to keep the dialog open when you click Run. When the pin is down
Restore Restore names into edit fields and default settings.
Clear Clear all fields and list boxes.
Help Open the Help topic for this dialog.
Examples of creating and manipulating tables can be found in Table examples.
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i n
Pi is a star. Pi is the most famous and the most mysterious, the most bizarre and the craziest number. Unlike other, harmless numbers, Pi has character. Pi is irrational, abstract, indomitable and,
according to the Association of Friends of the Number Pi based in Vienna, "but also so beautiful". The American physicist Richard P. Feynman once proclaimed, perhaps thinking of the ‘Heurigen’ city
of Vienna: "In a glass of wine is the whole universe". False. It's in Pi. No one misses this number. Especially not on July 22. Because: 22 by 7 equals 3.14 etc.
What the double-slit experiment is in physics is this circular number for mathematicians. Actually, Pi should only determine the ratio between the circumference and diameter of a circle. However, the
nature and essence of Pi has been on the minds of philosophers and scientists since the early days of arithmetic. Pi's most important qualities - there is no need for a Pisa test - are irrationality
and transcendence, proven in 1766 by Johann Heinrich Lambert and 125 years later by Carl von Lindemann. In the abnormal 20th century, interest became increasingly focused on the more profound
question: Is Pi normal? The last calculation record for the number chain: 62,831,853,071,796 digits. This is not normal.
Whether you meditate in the morning over the circular coffee cup or in the evening over the wine glass - Pi. Whether you take the train or tram (round wheels) or play billiards in the evening: You
always meet Pi. Whether you easily calculate planetary orbits in a TV commercial break to pass the time, whether you book the VHS course for advanced students "We draw the perfect circle" or feel
stimulated by atomic oscillations - Pi, Pi, Pi! In every round object, in every vibration, in every wave, Pi is present. Pi is omni_pi-, pardon: omnipresent. In the course of time, this number figure
turned out to be a tormenter who turned life into a number hell for scholars. Or robbed entire years of life, such as the English mathematician William Rutherford, who in 1841 had miscalculated his
calculations from the 152nd digit and spent the following five years trying to determine false garlands of numbers. (Alexander Kluy)
The number Pi is celebrated in Noerten-Hardenberg! As the center of the globally growing work of art "Pi in Stone", the stain is recognized. Man(s), woman and divers can make a pilgrimage on the "Pi
in Stone" path and internalize the beauties of the number and the formulas for calculation. The number series Pi in Stein is supplemented 'An der Bünte' by the colour coding of the first 72 digits of
the number Pi on the façade of the local indoor swimming pool, in order to then come across infinitely diverse colour-coded products based on the decimal places of the number Pi and their colour
coding at the headquarters of the 'Freunde der Zahl Pi Deutschland e.V.' in Lange Str. 4, D-37176 Noerten-Hardenberg.
Benches, carpets, necklaces for the ladies, scarves, a Memo-Pi, a black 'Piter' game... Infinite possibilities with the Infinite Series of Numbers and:
To the Pi approach day in the year 2319 according to Archimedes (the friends of the number Pi have their own time calculation), the 22.07. of this year, there is
314 packets of pi noodles, with a content of 314 g
For the production, the initiator Bilian Proffen from Noerten-Hardenberg was able to win over the pasta manufactory of the Golze-Kreuzinger family in Dassel, Am Rothenberg 1.
The die for the noodles was commissioned and could be made in time to have the first 314 packages of noodles available in time for the Pi approach day (22/7). The noodle packages are numbered
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Understanding Radix Sort: Algorithm, Implementation, and Applications in C
Learn Radix Sort, a powerful non-comparative sorting algorithm. Discover its step-by-step logic, C implementation, and practical uses.
Sorting algorithms are essential tools in computer science, used to organize data efficiently. Radix Sort is a unique, non-comparative sorting algorithm that sorts data with multiple passes based on
individual digits or characters. This blog aims to demystify Radix Sort, offering a thorough understanding of its algorithmic details, a step-by-step explanation, implementation in C, and its time
and space complexities. By the end of this blog, you will have a solid grasp of Radix Sort and its practical applications.
Algorithm Details
Radix Sort sorts numbers digit by digit, starting from the least significant digit (LSD) to the most significant digit (MSD). It uses a stable subroutine, such as Counting Sort, to sort the digits.
Step-by-Step Logic
1. Find the maximum number: Determine the number with the most digits.
2. Sort by each digit: Starting from the least significant digit to the most significant digit, sort the array using Counting Sort as a subroutine.
Pseudo Code
max_num = getMax(arr)
exp = 1
while max_num / exp > 0:
CountingSort(arr, exp)
exp *= 10
CountingSort(arr, exp):
n = length(arr)
output = array of zeros with length n
count = array of zeros with length 10
for i = 0 to n - 1:
index = (arr[i] // exp) % 10
count[index] += 1
for i = 1 to 9:
count[i] += count[i - 1]
for i = n - 1 downto 0:
index = (arr[i] // exp) % 10
output[count[index] - 1] = arr[i]
count[index] -= 1
for i = 0 to n - 1:
arr[i] = output[i]
Explanation of Pseudo Code
1. RadixSort(arr):
□ max_num identifies the maximum number to determine the number of digits.
□ exp starts at 1 (least significant digit) and increases by a factor of 10 after each pass.
□ CountingSort is called for each digit.
2. CountingSort(arr, exp):
□ count array stores the count of occurrences of each digit (0-9).
□ The counts are accumulated to determine the positions of each digit.
□ The array is sorted based on the current digit and copied back to the original array.
Implementation in C
Here is the implementation of Radix Sort in C.
#include <stdio.h>
// Function to get the maximum value in the array
int getMax(int arr[], int n) {
int max = arr[0];
for (int i = 1; i < n; i++)
if (arr[i] > max)
max = arr[i];
return max;
// Function to perform counting sort based on the digit represented by exp
void CountingSort(int arr[], int n, int exp) {
int output[n]; // output array
int count[10] = {0};
// Store count of occurrences in count[]
for (int i = 0; i < n; i++)
count[(arr[i] / exp) % 10]++;
// Change count[i] so that count[i] contains the actual position of this digit in output[]
for (int i = 1; i < 10; i++)
count[i] += count[i - 1];
// Build the output array
for (int i = n - 1; i >= 0; i--) {
output[count[(arr[i] / exp) % 10] - 1] = arr[i];
count[(arr[i] / exp) % 10]--;
// Copy the output array to arr[], so that arr[] now contains sorted numbers according to the current digit
for (int i = 0; i < n; i++)
arr[i] = output[i];
// Main function to do radix sort
void RadixSort(int arr[], int n) {
// Find the maximum number to know the number of digits
int max = getMax(arr, n);
// Do counting sort for every digit. Note that exp is 10^i where i is the current digit number
for (int exp = 1; max / exp > 0; exp *= 10)
CountingSort(arr, n, exp);
// Utility function to print an array
void printArray(int arr[], int n) {
for (int i = 0; i < n; i++)
printf("%d ", arr[i]);
// Driver program to test above functions
int main() {
int arr[] = {170, 45, 75, 90, 802, 24, 2, 66};
int n = sizeof(arr) / sizeof(arr[0]);
RadixSort(arr, n);
printf("Sorted array is: \n");
printArray(arr, n);
return 0;
Explanation of C Code
1. getMax(int arr[], int n):
□ Finds the maximum value in the array to determine the number of digits.
2. CountingSort(int arr[], int n, int exp):
□ Sorts the array based on the digit represented by exp (e.g., units, tens, hundreds).
□ Initializes the count array to store the frequency of each digit.
□ Accumulates the count to determine the position of each digit.
□ Constructs the output array based on the current digit.
□ Copies the sorted output array back to the original array.
3. RadixSort(int arr[], int n):
□ Calls CountingSort for each digit, starting from the least significant digit to the most significant digit.
4. printArray(int arr[], int n):
□ Utility function to print the array.
5. main():
□ The driver function to test the Radix Sort implementation with a sample array.
Time and Space Complexity
Understanding the complexity of Radix Sort is essential for evaluating its performance.
Time Complexity
• Best, Average, and Worst Case: O(d * (n + k))
□ d is the number of digits in the largest number.
□ n is the number of elements in the array.
□ k is the range of the digit (usually 0-9 for decimal numbers).
Space Complexity
• Radix Sort requires additional space for the output array and the counting array.
• Space Complexity: O(n + k)
Usage of Radix Sort
Radix Sort is particularly useful in scenarios where a stable and efficient sorting algorithm is needed for integers or strings. Common applications include:
• Sorting large datasets of integers: Especially when the range of numbers is large but the number of digits is relatively small.
• String sorting: Can be adapted to sort strings, such as sorting names or dates.
• Data processing: Useful in systems where integer keys need to be sorted quickly.
Radix Sort is a powerful, non-comparative sorting algorithm that provides efficient sorting for integers and strings. Its systematic approach, based on digit-by-digit sorting, ensures a reliable O(d
* (n + k)) time complexity. With the detailed explanation, pseudo code, and C implementation provided in this blog, you should now have a solid understanding of Radix Sort and how to implement it in
your projects.
By leveraging Radix Sort, you can achieve efficient and stable sorting for various applications, making it an invaluable tool in your programming arsenal. Happy coding! | {"url":"https://rahulv.dev/blog/understanding-radix-sort-algorithm-implementation-and-applications-in-c/","timestamp":"2024-11-05T10:21:15Z","content_type":"text/html","content_length":"26394","record_id":"<urn:uuid:b850e3d9-55a2-4656-a09a-5b8da634465a>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00699.warc.gz"} |
Heating and cooling flow calculator
Research And Insight
Calculation of flow
In order to calculate pressure loss in a heating or cooling system it is essential to know the volume flow rate Q. The volume flow rate is based on heat demand, temperature difference, specific
energy capacity, and density.
When the heat flow Φ is known, the flow pipe temperature tF,and the return-pipe temperature tRshould be determined, in order to be able to calculate the volume flow rate Q. The temperatures not
only determine the volume flow rate, but alsothe heating surfaces (radiators, calorifiers etc.).
The following formula should be used:
Φ x 0.86 = Q
Q = Volume flow rate in [m3/h]
Φ = Heat demand in [kW]
tF= Dimensioning flow pipe temperature in [°C]
tR= Dimensioning return-pipe temperature in [°C]
0.86 is the conversion factor (kcal/h to kW)
Use the water flow calculator for calculating flow. | {"url":"https://www.grundfos.com/id/learn/research-and-insights/calculation-of-flow","timestamp":"2024-11-07T23:35:37Z","content_type":"text/html","content_length":"707247","record_id":"<urn:uuid:99674b92-29e2-4850-9e19-a3ff978ab604>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00676.warc.gz"} |
Repeated Measures ANOVA With Excel
In this lesson, we show how to conduct analysis of variance for an single-factor, repeated measures experiment with Excel. And we explain how to interpret the results of our analysis.
Note: If you're curious about the computations used by Excel to conduct analysis of variance with a repeated measures design, read the previous lesson: One-Factor Repeated Measures: Example. That
lesson shows all of the formulas and computations required to solve the same problem that we will solve in this lesson with Excel.
The Analysis ToolPak
To access the analysis of variance functions in Excel, you need a free Microsoft add-in called the Analysis ToolPak, which may or may not be already installed on your copy of Excel.
To determine whether you have the Analysis ToolPak, click the Data tab in the main Excel menu. If you see Data Analysis in the Analysis section, you're good. You have the ToolPak.
If you don't have the ToolPak, you need to get it. Go to: How to Install the Data Analysis ToolPak in Excel.
Problem Statement
To demonstrate how to conduct analysis of variance for a randomized block experiment with Excel, we'll work through a real-world problem. Here's the problem:
As part of a repeated measures experiment, a researcher tests the effect of three treatments on short-term cognitive performance. Each treatment is administered in pill form. The first treatment (T1)
is a placebo; the second treatment (T2) is an herbal relaxant; and the third treatement (T3) is an herbal stimulant. The researcher randomly selects six subjects to participate in the experiment.
Using human subjects as experimental units, the researcher conducts this experiment over a three-day period. Each day, each subject receives a different treatment. After each treatment, subjects
complete a memory test. Test scores for each subject following each treatment are shown in the table below:
Table 1. Dependent Variable Scores
Subject Test score
T1 T2 T3
S1 87 85 87
S2 84 84 85
S3 83 84 84
S4 82 82 83
S5 81 82 83
S6 80 80 82
Repeated measures experiments have a potential problem: vulnerability to order effects (e.g., fatigue, learning) that can affect subject performance. To control for order effects, the researcher
randomizes the order in which treatment levels are administered.
This experiment is designed to address one main research question: Does the treatment have a significant effect on cognitive performance (as measured by test score)?
Repeated Measures ANOVA With Excel
When you conduct a repeated measures analysis of variance with Excel, the main output is an ANOVA summary table. As we've seen in previous lessons, an ANOVA summary table holds all the information we
need to answer the research question posed above.
Here is a step-by-step guide for producing an ANOVA summary table for a repeated measures experiment with Excel:
• Step 1. Enter data from Table 1 in rows and columns of an Excel spreadsheet. Follow the layout from Table 1, with the column labels in the first row, as shown below:
• Step 2. From Excel's main navigation menu, click Data / Data Analysis to display the Data Analysis dialog box.
• Step 3. In the Data Analysis dialog box, select "Anova: Two-Factor Without Replication" and click the OK button to display the Anova: Two-Factor Without Replication dialog box.
• Step 4. In the Anova: Two-Factor Without Replication dialog box, enter the input range. Click the Labels checkbox to indicate that you included labels for the rows and columns. And finally, enter
a value for Alpha, the significance level. For this exercise, we'll use a significance level of 0.05, as shown below:
• Step 5. From the Anova: Two-Factor Without Replication dialog box, click the OK button to display the ANOVA summary table.
Congratulations! You conducted a single-factor, repeated measures analysis of variance with Excel. In the ANOVA table, output for the subject effect appears in the row labelled "Rows". Output for
the treatment effect appears in the row labelled "Columns".
Interpretation of Results
Recall that the researcher undertook this study to answer one question: Does the treatment have a significant effect on cognitive performance (as measured by test score)?
The answer to that question can be found in the ANOVA summary table. However, you may need to do a little more work, depending on whether your data satisfies the sphericity assumption.
When Sphericity Is Satisfied
Excel assumes that the sphericity assumption is satisfied. If that assumption is correct, you can interpret results from Excel's ANOVA table in the normal way.
For this study, the P-value (shown in the last column of the ANOVA table) is the probability that an F statistic would be more extreme (bigger) than the F ratio shown in the table, assuming the null
hypothesis is true. When the P-value is bigger than the significance level, we do not reject the null hypothesis; when it is smaller, we reject it.
Here, the P-value for the treatment effect (0.01) is smaller than the significance level (0.05), so we reject the null hypothesis and conclude that pill treatment had a statistically significant
effect on test score.
When Sphericity Is Not Satisfied
When the sphericity assumption is violated, the standard F-test in analysis of variance will be positively biased; that is, you will be more likely to make a Type I error (i.e., reject the null
hypothesis when it is, in fact, true).
Based on the standard ANOVA table produced by Excel, we rejected the null hypothesis. That analysis would be valid if the sphericity assumption were satisfied. But if the sphericity analysis were
violated, that analysis could be misleading - possibly incorrect due to a Type I error.
In the previous lesson, we explained how to correct output from a standard analysis of variance (like the output produced by Excel) to avoid problems when the sphericity assumption is not satisfied.
To see what you need to do, read the previous lesson. | {"url":"https://stattrek.com/anova/repeated-measures/excel?tutorial=anova","timestamp":"2024-11-06T11:39:32Z","content_type":"text/html","content_length":"55620","record_id":"<urn:uuid:e1152bdd-03fc-4626-b8a8-b0d25423c032>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00399.warc.gz"} |
Limitations on variance-reduction and acceleration schemes for finite sum optimization
We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the
finite sum structure by itself, is not sufficient for obtaining a complexity bound of Õ((n + L/μ) ln(1/ϵ)) for L-smooth and μ-strongly convex individual functions - one must also know which
individual function is being referred to by the oracle at each iteration. Next, we show that for a broad class of first-order and coordinate-descent finite sum algorithms (including, e.g., SDCA,
SVRG, SAG), it is not possible to get an 'accelerated' complexity bound of Õ((n+ √nL/μ) ln(1/ϵ)), unless the strong convexity parameter is given explicitly. Lastly, we show that when this class of
algorithms is used for minimizing L-smooth and convex finite sums, the iteration complexity is bounded from below by Ω(n + L/ϵ), assuming that (on average) the same update rule is used in any
iteration, and Ω(n + √nL/ϵ) otherwise.
Bibliographical note
Publisher Copyright:
© 2017 Neural information processing systems foundation. All rights reserved.
Dive into the research topics of 'Limitations on variance-reduction and acceleration schemes for finite sum optimization'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/limitations-on-variance-reduction-and-acceleration-schemes-for-fi","timestamp":"2024-11-14T11:38:32Z","content_type":"text/html","content_length":"47085","record_id":"<urn:uuid:625f3afc-25b3-4c72-8cae-4ea1f6321f28>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00294.warc.gz"} |
Delay-aware Backpressure Routing Using Graph Neural Networks
Published in IEEE ICASSP 2023, 2022
Recommended citation: Zhongyuan Zhao, Bojan Radojičić, Gunjan Verma, Ananthram Swami, Santiago Segarra, " Delay-aware Backpressure Routing Using Graph Neural Networks," IEEE ICASSP 2023, Rhodes
Island, Greece, 2023, pp. 1-5, doi: 10.1109/ICASSP49357.2023.10095267. https://ieeexplore.ieee.org/document/10095267
Bojan Radojičić, a student from University of Novi Sad, Serbia, has contributed to the project, especially the source code, during his visit to Rice University.
We propose a throughput-optimal biased backpressure (BP) algorithm for routing, where the bias is learned through a graph neural network that seeks to minimize end-to-end delay. Classical BP routing
provides a simple yet powerful distributed solution for resource allocation in wireless multi-hop networks but has poor delay performance. A low-cost approach to improve this delay performance is to
favor shorter paths by incorporating pre-defined biases in the BP computation, such as a bias based on the shortest path (hop) distance to the destination. In this work, we improve upon the
widely-used metric of hop distance (and its variants) for the shortest path bias by introducing a bias based on the link duty cycle, which we predict using a graph convolutional neural network.
Numerical results show that our approach can improve the delay performance compared to classical BP and existing BP alternatives based on pre-defined bias while being adaptive to interference
density. In terms of complexity, our distributed implementation only introduces a one-time overhead (linear in the number of devices in the network) compared to classical BP, and a constant overhead
compared to the lowest-complexity existing bias-based BP algorithms.
Key words: Backpressure routing; graph neural network; scheduling duty cycle; independent set; bias, shortest path
A brief intro
Backpressure routing is a fully distributed packet routing algorithm for wireless multihop networks. It has a mechanism that drives data packets to explore every possible route in the network to
their destination(s) just like water going through a network of pipes. This feature allows backpressure routing to stabilize the queues on each wireless devices – meaning that the length of queues
will not grow infinitely – as long as the traffic load of the data flows are within the capacity of the network. This property is called throughput optimal. In contrast, some table-driven routing
schemes may suffer from exploding queues when too many data flows go through the same critical node.
However, the classical backpressure routing is known to have poor delay performance in light-to-medium traffic load. Specifically, it exhibits undesirable characteristics such as slow startup, random
walk, and the last packet problem, just like what happens when water flow through a flat floor – it stays on the floor when there is not enough water to push them forward.
To make the water to flow to the sink in the bathroom quickly, we can make the floor slightly sloping to the sink. The same idea also applies to backpressure routing, where pre-built biases, such as
the shortest path distance toward the sink node, are added to nodes in the network. The most widely used distance metric is hop distance (counts).
In this work, we propose a different distance metric that can better estimate the delay performance than the hop distance, by incorporating the link duty cycle in scheduling. The link duty cycle
measures the proportion of time a link is activated, which is hard to predict through conventional algorithms. We train a graph neural network defined on the conflict graph of the wireless networks
to predict the link duty cycle, which subsequently improves the pre-built shortest path distance biases, and can adapt to different network densities. | {"url":"https://zhongyuanzhao.com/publications/2022-11-19-link-duty-cycle-backpressure.html","timestamp":"2024-11-09T14:09:10Z","content_type":"text/html","content_length":"17567","record_id":"<urn:uuid:b84a919e-1124-435d-86b4-281605452107>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00434.warc.gz"} |
Randomness Control for Roulette and Wheel of Fortune
Randomness Control on Roulette with One Zero (European)
A game with randomness control is conducted during a series of successive spins, and the length of a series can be determined by the player himself. The random sequence of numbers is formed before
the beginning of a series. These numbers range from 0 to 36 and correspond to those numbers, on which the ball is supposed to land. The player can change the ‘planned’ number before each regular spin
by defining so-called corrective shift.
When the game session starts, the first random sequence of numbers is created automatically. By default, this sequence consists of 10 numbers, so the length of the appropriate series of spins is
equal to 10. A game with randomness control is conducted with the help of the panel that is located in the bottom part of the game window. Just click on Randomness control to get this panel:
The 1 button is used to receive reference information. In field 3, the player can see the checksum of sequence, computed according to the SHA-256 algorithm. Using the 2 button, this checksum can be
copied to the clipboard. In field 6, the length of the series of spins is shown. If the player wants, he can indicate a different length in this field. The length must be in the range from 1 to 60.
When the player indicates the new length, the 7 button becomes active:
In order to create a random sequence of numbers with the new length, the player must press this button. The checksum of this sequence will appear in field 3, while the 7 button will no longer be
After each spin in the series, the number in field 6 decreases by 1. In such a way, the number of spins remaining in the series is displayed in this field. At that moment, the 7 button becomes active
again, but its status changes from New to End:
If the player wants, he can press this button to abort the current series of spins before all the spins have been used. He can do it at any point in the series.
Before each regular spin, the player can change the number in the sequence that must fall out in roulette. For this purpose, the player uses field 5 to define a shift—a number ranging from 0 to 36.
The shift is used for the correction of the ‘planned’ number. Namely, the number, which will fall out in roulette in the present spin (0–36), is calculated in the following way: the shift is added to
the number from a random sequence. If this sum is higher than 36, then 37 is subtracted from the sum.
Example. The random sequence of numbers is: 10, 15, 20, 36, 25. The shifts defined by the player are as follows: 25, 30, 18, 36, 12. The wheel will stop at the sectors with the following numbers: 35,
8, 1, 35, 0.
When the series of spins is completed (or if the series has been aborted by the player), the next random sequence of numbers is created automatically. The length of this series will be equal to the
previous one. As before, the player can determine the new length of the series and continue the game using the appropriate random sequence. At that moment, the player can use the 4 button to check
the results of the just finished series. A new window opens where the player can see information about the entire sequence of numbers formed earlier. He can now confirm that, with regard to shifts,
it strictly corresponds to those numbers that fell out in roulette (this means that the player must remember or write down these numbers). The information about the sequence appears as a text line,
for example:
spins: 11, 1, 21, 9, 4, 36, 28, 0, 15, 15;
server code word: G9oLdQNNjO3pkHX7oNZNp5NsnuTHOW1r
After the word spins, the player can find the numbers that were supposed to fall out in roulette (without considering corrective shifts). After the words server code word, there is a random key
phrase that is formed by the server for casino safety.
By clicking on the Calculate checksum button, the player will see the checksum for the text information mentioned above. The player can then compare this checksum to the one received at the beginning
of the game, when the random sequence of numbers was formed. The fact that these two sums correspond proves that the game used the same numbers that were created at the start.
If the player chooses, he can view information about all sequences of numbers that were created during the current game session. He can scroll through them using the previous and next buttons.
When the player wants to finish the game and presses the Exit button, the game session will be closed, and if his sequence of numbers was not finished, all the data will be lost. If the player wants
to continue the game with that particular sequence, he should postpone the session using the standard way of closing the window (with the close button in the upper-right corner). When he renews the
session, he can resume the game with the previous sequence.
Distinctive Features of Randomness Control on American Roulette
Double zero (00) is equal to 37. Numbers from random sequence, formed in advance, range from 0 to 37. The shift, chosen by the player to correct the ‘planned’ number, is within the same range. The
calculation of the number that falls out in the given spin (0–37), is conducted in the following way: the shift is added to the number from a random sequence. If this sum is more than 37, then 38 is
subtracted from the sum.
Example. The random sequence of numbers is: 10, 15, 20, 00(37), 25. The shifts defined by the player are as follows: 25, 30, 18, 2, 12. The following numbers will fall out on the roulette: 35, 7, 0,
1, 00(37).
Distinctive Features of Randomness Control on Multiball Roulette and Roulette Express
These games are based on European roulette, but three different numbers can fall out in each spin. Therefore, there is not one but three numbers per one spin in a random sequence that is formed
before the beginning of a series. All numbers range from 0 to 36. When the player checks the results of the finished series, the information about the sequence appears as a text line. Consider an
example for the series of ten spins:
spins: 1-3 15 28, 2-24 5 17, 3-14 16 2, 4-36 0 6, 5-12 5 23, 6-24 23 5, 7-11 2 4, 8-34 3 35, 9-0 10 35, 10-24 7 9;
server code word: 2574FSpirSHCCKat6TGBRY7wvSxMuRIE
The numbers of spins are indicated here (from 1 to 10) and, after each spin’s number, there are three different numbers that were supposed to fall out in this spin (without considering corrective
shift). As usual, the text line ends with a random key phrase that is needed for casino safety.
Please note that the actual amount of balls that a player chooses in Multiball roulette is unknown at the moment the sequence is formed. Hence, the text line is always created allowing for the
maximum amount of balls - 3.
Before each regular spin, the player can define the corrective shift — a number ranging from 0 to 36. This shift is just one; it’s a community shift for all three numbers used in a random sequence.
Appliance of community shift is necessary: according to the game rules, all balls must land in different sectors on the roulette wheel.
Let us return to the example above. It is assumed that numbers 3, 15 and 28 will fall out in the first spin. However, assume that the player defined shift of 10. In this situation, other numbers
would appear on the roulette — 13, 25 and 1. These numbers are calculated in the same way as in European roulette. After every spin, the corrective shift can be changed. For example, a shift equal to
20 could be entered before the second spin. Numbers 7, 25 and 0 will appear on the roulette instead of ‘planned’ numbers 24, 5 and 17.
Distinctive Features of Randomness Control on No Zero Roulette
The zero has been removed from the European roulette wheel. Numbers from random sequence, formed in advance, range from 1 to 36. The shift, chosen by the player to correct the “planned” number, can
vary from 0 to 35. The calculation of the number that falls out in the given spin (1-36), is conducted in the following way: the shift is added to the number from a random sequence. If this sum is
more than 36, then 36 is subtracted from the sum.
Example. The random sequence of numbers is: 10, 15, 20, 36, 25. The shifts defined by the player are as follows: 25, 30, 18, 35, 12. The following numbers will fall out on the roulette: 35, 9, 2, 35,
Distinctive Features of Randomness Control in No Zero Multiball Roulette and No Zero Roulette Express
Randomness control in these games is very similar to the one used in Multiball roulette and Roulette Express. But these games are based on No Zero roulette rather than European roulette. Therefore,
all numbers from the random sequence formed in advance range from 1 to 36 (rather than from 0 to 36). The corrective shifts, chosen by the player, can vary from 0 to 35. The numbers that appear while
accounting for the player's shifts are calculated in the same way as in No Zero roulette.
Example. Three "planned" numbers in the regular spin are as follows: 26, 27, 36. The player defines the shift of 10. The following numbers will fall out on the roulette wheel: 36, 1, 10.
Distinctive Features of Randomness Control on Wheel of Fortune
The Wheel of Fortune has 49 sectors, each ordered from 1 to 49. For players’ convenience, the numbers that correspond to the sectors, are inscribed on the inner part of the wheel.
Numbers from random sequence, formed in advance, range from 1 to 49 and correspond to the sectors upon which the wheel is supposed to stop. The shift, chosen by the player to correct the "planned"
number, can vary from 0 to 48. The number upon which the wheel will stop in the present spin (1-49), is calculated in the following way: the shift is added to the number from a random sequence. If
this sum is higher than 49, then 49 is subtracted from the sum.
Example. The random sequence of numbers is: 10, 15, 20, 36, 26. The shifts defined by the player are as follows: 25, 30, 38, 36, 24. The wheel will stop at the sectors with the following numbers: 35,
45, 9, 23, 1. | {"url":"https://betvoyages.com/fair-play/roulette/","timestamp":"2024-11-11T20:55:30Z","content_type":"text/html","content_length":"85678","record_id":"<urn:uuid:827ff753-c99d-4ece-bb05-794b896fbc31>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00172.warc.gz"} |
Learn Fundamental Analysis: Profitability Ratios & Leverage Ratios
Profitability Ratios
Some of the profitability ratios include:
EBITDA Margin (Operating Profit Margin)
The Earnings Before Interest, Tax, Depreciation, and Amortization (EBITDA) Margin is a measure that reflects the effectiveness of a company’s management and the efficiency of its operational
approach. It provides insight into the company’s operational profitability, expressed as a percentage. Essentially, it assesses how efficiently the company turns its core operational activities into
profits. It’s a good idea to compare a company’s EBITDA margin with its competitors to see how well they manage their expenses.
To calculate the EBITDA Margin,
EBITDA Margin = EBITDA / [Total Revenue — Other Income]
We need to calculate EBITDA,
EBITDA = Operating Revenues - Operating Expense
Operating Revenues = Total Revenue - Other Income
Operating Expense = Total Expense - Finance Cost - Depreciation & Amortization
PAT (Profit After Tax) Margin
The Profit After Tax (PAT) margin is calculated at the final profitability level. When calculating the PAT margin, we subtract all expenses from the company’s total revenue to assess its overall
To calculate the PAT Margin,
PAT Margin = PAT/Total Revenues
It always makes sense to compare ratios with its competitors.
Return on Equity (ROE)
Return on Equity (RoE) is a crucial ratio that helps investors evaluate how much return shareholders get for each unit of money they invest. It measures how well a company generates profits from
shareholder investments, essentially showing how efficiently the company makes profits for its shareholders. A higher RoE is better for shareholders, and it’s a key indicator for investors to
identify good investment opportunities in a company. When RoE is high, it means the company generates a lot of cash which also indicates a higher level of management performance.
To calculate ROE,
RoE = Net Profit/Shareholders Equity* 100
The thing to remember is higher the debt a company seeks to finance its asset, higher is the ROE. Therefore, with a high debt a high ROE is not great. Therefore, evaluating ROE becomes extremely
important. Using the DuPont Model to calculate the ROE can help to get a clear insight [We will talk about this in the later section]. This can be calculated by:
ROE = (Net Profit/Net Sales)(Net Sales/Avg Total Assets)(Avg Total Assets/Shareholders Equity)
Using this formula, we get insight into three different aspects of the business providing an insight into the company’s operating and financial capabilities.
i.e. Net Profit Margin = (Net Profit/Net Sales)100
This is the first part of the formula which provides insight about the company’s ability to generate profit. It is the same as PAT Margin.
i.e. Asset Turnover = Net Sales/Avg Total Assets
This is the second part of the formula which provides insight on how efficiently the company is using its assets to generate revenue. A higher ratio indicates the company is using its assets
efficiently and vice versa for a lower ratio value. This ratio is expressed in number of times per year.
i.e. Financial Leverage = Avg Total Assets/Shareholders Equity
This is the final part of the formula which explains “For every unit of shareholders equity, how many units of assets does the company have”. Let's take an example, if the financial leverage is 5
then it means for every Rs.1 of shareholders equity, the company Rs.5 worth of assets. If a company has a high level of financial leverage and carries a significant amount of debt, it’s a sign that
investors should be cautious. The ratio is expressed in number of times per year.
Return on Asset (ROA)
Return on Assets (RoA) measures how well a company turns its assets into profits. It shows how efficiently a company uses its resources. In simple terms, a higher RoA is better.
To calculate ROA,
ROA = Net income + interest*(1-tax rate) / Total Average Assets
Leverage Ratios
Proficiently run businesses opt for borrowing when they anticipate opportunities to utilize the borrowed capital in a setting that yields greater profits than the interest expenses required to
service the debt. Nevertheless, excessive debt can diminish the portion of profits allocated to shareholders as interest payments for servicing the debt rise. Therefore, there exists a fine
distinction between the good and bad debt. Leverage ratios primarily assess the company’s total debt levels and contribute to a deeper understanding of the company’s financial leverage.
Some of the leverage ratios include:
Interest Coverage Ratio
The interest coverage ratio provides insights into the company’s earnings in relation to its interest expenses. It serves as a valuable metric to interpret the company’s ability to meet its interest
payments easily. A low-interest coverage ratio indicates a more significant debt load and an increased risk of bankruptcy or default.
To calculate the Interest Coverage Ratio,
Interest Coverage Ratio = Earnings before Interest and Tax/Interest Payment ( also known as Finance Cost)
Earnings before Interest and Tax (EBIT) = EBITDA — Depreciation and Amortization
Interest Coverage Ratio of 1.49x represents that for every Rupee of interest payment due, a company is generating an EBIT of 1.49 times.
Debt to Equity Ratio
It measures the amount of the total debt capital with respect to the total equity capital. A ratio of 1 signifies an equal balance between debt and equity capital. A higher debt-to-equity ratio
(greater than 1) signifies increased leverage and necessitates caution. Conversely, a ratio below 1 indicates a larger equity base in comparison to the debt.
To calculate Debt to Equity Ratio,
Debt to Equity Ratio = Total Debt/Total Equity
Total Debt = Short-Term Debt + Long-Term Debt
Debt to Asset Ratio
This ratio provides insights into the company’s approach to financing its assets, revealing the extent to which debt capital contributes to the funding of its total assets. The greater the
percentage, the more concerned an investor should be, as it signifies higher leverage and risk.
To calculate Debt to Asset Ratio,
Debt to Asset Ratio = Total Debt/Total Assets
Financial Leverage Ratio
The financial leverage ratio provides insight into the degree to which assets are supported by equity. Keep in mind that a higher value indicates increased leverage for the company.
To calculate the Financial Leverage Ratio,
Financial Leverage Ratio = Average Total Asset/Average Total Equity
In this article, we learned about Profitability Ratios and Leverage Ratios. In the next article, we will learn about Operating Ratios and Valuation Ratios.
Next Chapter: Operating Ratios & Valuation Ratios | {"url":"https://vengeance.medium.com/learn-fundamental-analysis-profitability-ratios-leverage-ratios-4011863b77a7","timestamp":"2024-11-05T06:49:26Z","content_type":"text/html","content_length":"124057","record_id":"<urn:uuid:fd1694a0-5554-4d60-b730-0f64127e4490>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00563.warc.gz"} |
How to Find the Biggest Number in a List in Python? - Data Science Parichay
In Python, working with lists is a common task. Sometimes, we need to find the largest number in a list. This can be useful in a variety of applications, from data analysis to simple programming
tasks. In this tutorial, we will explore different methods to find the biggest number in a list in Python. By the end of this tutorial, you will have a solid understanding of how to find the maximum
value in a list and be able to apply this knowledge to your own Python projects.
Methods to find the biggest number in a list in Python
There are multiple methods using which you can get the maximum value in a Python list. Let’s look at these methods in detail with the help of some examples.
1) Using the max() function
max() is a built-in function in Python that returns the largest item in an iterable or the largest of two or more arguments. To get the biggest number in a list, pass the list as an argument to the
max() function.
Let’s look at an example.
# create a list
ls = [3, 5, 1, 9, 2]
# find the biggest value in the list
biggest_num = max(ls)
We get the maximum value in the list as 9, which is the correct answer.
2) By iterating through the list
In this method, we iterate through the list and keep track of the biggest value encountered in a separate variable. After we have gone through the entire list, the variable will have the biggest
value in the list.
# create a list
ls = [3, 5, 1, 9, 2]
# find the biggest value in the list
if ls:
biggest_num = ls[0]
for val in ls[1:]:
if val > biggest_num:
biggest_num = val
print("list is empty")
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You can see that we get 9 as the biggest value in the list, which is the correct answer.
3) Sort the list
The idea here is to sort the list in ascending order and get the last element in the list. In a sorted list, the largest number will be at the end of the list. You can use the built-in sorted()
function to sort a list in Python, it returns a new list that is sorted.
# create a list
ls = [3, 5, 1, 9, 2]
# sort the list
ls_sorted = sorted(ls)
# find the biggest value
We get the biggest value in the last as 9, which is the correct answer.
In this tutorial, we looked at how to get the biggest number in a Python list using different methods. Using the max() is the simplest and the recommended way to get the largest value in a list. The
other methods, while being correct, are not as straightforward as the max() function.
You might also be interested in – | {"url":"https://datascienceparichay.com/article/biggest-number-in-python-list/","timestamp":"2024-11-07T20:13:42Z","content_type":"text/html","content_length":"256084","record_id":"<urn:uuid:b7b9efb9-5cc0-4fae-93b2-e8277392555c>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00204.warc.gz"} |
Graph The Linear Equation Y X 2 - Tessshebaylo
Graph The Linear Equation Y X 2
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Chapter-13 Deep Learning for Coders with FastAI & Pytorch
Keypoints What is Convolution? Convolution is one of the main building blocks of a CNN. The term convolution refers to the mathematical combination of two functions to produce a third function. It
merges two sets of information. In the case of a CNN, the convolution is performed on the input data with the use of a “filter” or “kernel”(these terms are used interchangeably) to then produce a
“feature map”. What are Convolutional Neural Networks (CNN)?...
Chapter-8 Deep Learning for Coders with FastAI & Pytorch
Keypoints Introduction Recommendation systems are one of the predominant systems in market like Netflix, amazon and Walmart. And also this applies to offline systems such as which product goes in
which row to capture the users. And it is one of the challenging problems. The solution for that problem is called Collaborative Filtering. Collaborative Filtering The Collaborative Filtering
technique refers to looking at what products the current user has used or liked, find other users that have used or liked similar products, and then recommend other products that those users have
used or liked....
Chapter-7 Deep Learning for Coders with FastAI & Pytorch
Keypoints Introduction It is better to fail fast than very late. And it is always better to run more experiments on a smaller dataset rather running a single experiment on a large dataset. This
chapter introduces a new dataset called Imagenette1 Imagenette is a subset of the original Imagenet dataset but has only 10 categories of classes which are very different. This dataset has full-size,
full-color images, which are photos of objects of different sizes, in different orientations, in different lighting, and so forth....
Chapter-6 Deep Learning for Coders with FastAI & Pytorch
Keypoints Multi-label classification Multi-label classification refers to the problem of identifying the categories of objects in images that may not contain exactly one type of object. There may be
more than one kind of object, or there may be no objects at all in the classes that you are looking for. See two examples below where we have a bear dataset with a dog included named bear and another
example where the cat is classified as cat and horse....
Chapter-5 Deep Learning for Coders with FastAI & Pytorch
Keypoints The Keytopics in the blogpost: Presizing / Augmentation of Images Datablock Cross-Entropy Loss Presizing / Augmentation of Images The main idea behind augmenting the images is to reduce the
number of computations and lossy operations. This also results in more efficient processing on the GPU. To make the above possible we need our images to have same dimensions, so they can be easily
collated. Some of the challenges in doing the augmentation is that when we resize, the data could be degraded, new empty zones are introduced etc.... | {"url":"https://www.r-c.ai/blogs/","timestamp":"2024-11-06T11:33:41Z","content_type":"text/html","content_length":"13344","record_id":"<urn:uuid:bbfc7551-1dbc-42aa-a2b8-211354da9dd0>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00172.warc.gz"} |
Find the value of the exponent c
What is the value of the exponent c in the given expression where x ≥ 0 and y ≥ 0?
The value of the exponent c that makes the second expression equivalent to the first expression where x ≥ 0 and y ≥ 0 is 4.
To find the value of the exponent c, we can look at the given expressions: First expression: x^2 * y^2 Second expression: x^c * y^c We are given that x ≥ 0 and y ≥ 0, which means both x and y are
non-negative numbers. In order for the second expression to be equivalent to the first expression, the exponents of x and y in both expressions must be the same. By comparing the exponents in the
first and second expressions, we can see that c = 2 in order for x^c * y^c to be equivalent to x^2 * y^2. Therefore, the value of the exponent c that satisfies the given conditions is 4. In
conclusion, the value of the exponent c that makes the second expression equivalent to the first expression where x ≥ 0 and y ≥ 0 is 4. | {"url":"https://airdocsolutions.com/sat/find-the-value-of-the-exponent-c.html","timestamp":"2024-11-13T11:55:58Z","content_type":"text/html","content_length":"21002","record_id":"<urn:uuid:30148742-980c-4cef-b408-3bd96bfce295>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00681.warc.gz"} |
Variance calculator
(”, ‘\n
Variance Calculator:
Formula: σ² = Σ(x – μ)² / N
\n \n \n
Have you ever puzzled over a heap of data, struggling to make sense of it? Or ever wondered how statisticians use complex computations to make informed decisions? This quandary points to an
indispensable mathematical tool – the variance calculator.
The Intricacies of Variance Calculators
Variance calculators, akin to unsung heroes in the world of numbers, help to provide clear insights from scattered data. A variance calculator helps to determine how much a set of numbers deviates
from their average, offering distinct clarity in the fog of figures. May sound tricky, right? Well, imagine a tractor pulling a laden cart uphill. It represents the average value, while the effort
exerted by the motor signs at the deviation (variance) or how far away the items on the cart are from the average. Behold, a variance calculator playing out in real life!
What is Variance?
Variance, in its raw form, measures the dispersion of numbers in a data set from their mean value. If this seems confusing, think of a box of mixed chocolates. The chocolates represent numbers in a
dataset, while the average flavor quality would be the ‘mean’. Variance then tells us the ‘flavor deviation’ if you may, from this average taste.
Why Should I Use a Variance Calculator?
Imagine you have multiple investments and you need to check their volatility. Sure, you could ride it out and hope for the best, but wouldn’t it be even better if you could use a tool to accurately
estimate this volatility? That’s where the variance calculator comes into play. It acts as a pathfinder, guiding your decisions by providing detailed data digests.
How Does a Variance Calculator Work?
Picture this, you’re behind the wheel of an incredibly complex machine. This machine is your variance calculator. It works by first calculating the mean of all your data points. It then takes each
individual item, subtracts the mean, and squares the result. These squared results are then added together, and divided by the number of items to provide your variance.
Let’s put this into perspective with another metaphor. Imagine a symphony orchestra playing beautiful melodies. The mean would be the conductor’s ideal sound. Every single deviation from this sound
(either too loud or too soft, too fast or too slow) equals to the distance of each musician’s performance from the Maestro’s ideal. This is variance for you, as calculated by our indispensable tool –
the variance calculator.
Making the Best Use of a Variance Calculator
Whether you’re a statistician unravelling complex data, a student crunching numbers for a project, or simply trying to get a handle on your finances with variance and standard deviation, the variance
calculator forms the backbone of your analysis. Just like a compass steadily guiding seafarers through unchartered territories, this mathematical tool navigates you through the ocean of numbers,
steering you towards your goal.
In conclusion…
The variance calculator, though it might seem formidable at first glance, is an incredibly powerful tool. Once understood, you’ll find yourself equipped with a navigation tool capable of making sense
from any swirl of data. So next time you find yourself lost in a sea of numbers, remember – the variance calculator is your lifesaver, your beacon of understanding in the midst of numerical chaos. | {"url":"https://www-calculator.com/variance-calculator/","timestamp":"2024-11-07T17:34:19Z","content_type":"text/html","content_length":"51158","record_id":"<urn:uuid:eb85694b-4161-4a58-9b85-73dba830ec49>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00373.warc.gz"} |
Lecture Notes in Informatics
Efficient Reverse k-Nearest Neighbor Estimation
Elke Achtert , Christian Böhm , Peer Kröger , Peter Kunath , Alexey Pryakhin and Matthias Renz
The reverse k-nearest neighbor (RkNN) problem, i.e. finding all objects in a data set the k-nearest neighbors of which include a specified query object, has received increasing attention recently.
Many industrial and scientific applications call for solutions of the RkNN problem in arbitrary metric spaces where the data objects are not Euclidean and only a metric distance function is given for
specifying object similarity. Usually, these applications need a solution for the generalized problem where the value of k is not known in advance and may change from query to query. In addition,
many applications require a fast approximate answer of RkNN-queries. For these scenarios, it is important to generate a fast answer with high recall. In this paper, we propose the first approach for
efficient approximative RkNN search in arbitrary metric spaces where the value of k is specified at query time. Our approach uses the advantages of existing metric index structures but proposes to
use an approximation of the nearest-neighbor-distances in order to prune the search space. We show that our method scales significantly better than existing non-approximative approaches while
producing an approximation of the true query result with a high recall.
Full Text: | {"url":"https://subs.emis.de/LNI/Proceedings/Proceedings103/article1408.html","timestamp":"2024-11-12T00:01:26Z","content_type":"text/html","content_length":"4662","record_id":"<urn:uuid:c8bfa3d1-23f0-43a6-b392-b034e8330d12>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00213.warc.gz"} |
The Best Online Calculators to Solve Any Problem
A calculator is for more than just solving basic math problems. You can rely on online calculators to figure out scientific programs, determine loan amounts, and much more. The key is finding the
right calculator for the right problem.
A proper calculator can cost hundreds of dollars, but many of these online calculators are completely free to use. This list compiles the best online calculators for a variety of different
applications to help you make the right choice.
The Best Online Scientific Calculators
These calculators can be used for everything from advanced mathematics to scientific equations.
Desmos is a free-to-use calculator that provides easy access to primary functions, variables, and trigonometric calculations. You can control it through both the keyboard and with a mouse.
The Meta Calculator contains all of the basic functions, but includes a few features that make it stand out from the crowd. One of these is the ability to calculate the least common multiple, as well
as an equation solver that lets you input up to six equations. You can also store calculations for future use.
GeoGebra offers a variety of math-focused calculators to solve graphing problems, geometry, and much more. It displays a graph on screen to help you see exactly how the equation plays out. You can
also choose between function input, variables, and much more.
Best Online Calculators For Loans
If you plan to get a loan for a house or car, use one of these online calculators to help determine what your overall interest will be.
Bankrate is one of the foremost companies in the field. Just enter your loan amount, term, and interest rate per year to get an estimate of how much you will pay in interest.
Nerdwallet is a major destination for people seeking financial information, so it’s not surprising it would have one of best loan calculators. Just enter the amount, the term in either years or
months, interest rate, and type of loan to get an estimate of the amount you will pay.
CreditKarma is good for more than just catchy TV jingles. The website has a loan calculator that takes the normal variables into account, but also your credit score. This gives you a more accurate
estimate for what you’ll pay in interest and principal over the years.
The Best Online Calculators for Retirement
If you plan to retire, you need to know what kind of monthly cash flow to expect.
Fidelity offers one of the most effective retirement calculators. It asks you six basic questions to help determine the current state of your retirement policy and tells you how much you need per
month to maintain your current way of life, as well as what you should invest each month.
The Ultimate Retirement Calculator requests a huge amount of information, such as your age, when you plan to retire, and even your life expectancy. It then asks about your current savings, what your
desired annual retirement income is, and projects the amount you will need to invest each month to reach that standard of living.
Best Online Calculators for Statistics
If you need to calculate the statistics of a sample or set of data, try one of these online calculators.
This calculator is simple and straightforward. Choose whether the data set is a sample or a population and enter the information. It also provides equations regarding how to find the minimum, the
maximum, and more.
The Good Calculators Statistics Calculator is a much more in-depth and complex calculator, but it provides a pie chart that demonstrates the top values once the data has been processed. If you have a
complicated statistics problem, this is the best option.
Best Online Calculator for Calories
Trying to lose weight or bulk up? Try one of these calculators to determine how many calories you need (or don’t need!)
This calculator makes it easy to determine how many calories you need per day to either lose weight, gain weight, or maintain. Just enter your age, gender, current weight, height, and exercise level.
You can also choose which formula you want to use to perform the calculations, as well as whether you want results in calories or kilojoules.
MyFitnessPal is more than just a calculator–it’s an entire platform where you can track your calories, exercise, and more. The calorie tracking tool helps to automatically determine how many calories
you’ve eaten and even the macronutrients you’ve taken in.
Best Online Calculators for Algebra
Algebra and math sometimes feel like two different fields. Calculus is difficult due to the amount of algebra included. If you need some help, try one of these online calculators.
The Symbolab Algebra Calculator makes it easy to perform many advanced algebraic tasks. The calculator has a quick-access tab for the most used actions like simplify, solve for, and other common
word-problem terms, as well as an equation builder.
The Math Papa algebra calculator is the best choice for anyone that wants to depict algebra in the most simplified terms possible. Rather than using complex equations and symbols, just type out the
equation and receive a step-by-step explanation of how to solve it.
Best Online Calculators for Trigonometry
Trigonometry can be difficult, especially if you aren’t confident in your geometry skills. These trigonometry calculators will help you solve for sine everytime.
One of the key components of trigonometry is the multiple steps to the equations. If you do not have the right answer at one step, you’ll mess up the rest. This calculator walks you through how to
solve almost any trigonometry problem and explains how to do it. It’s like an online tutor.
This calculator not only solves trigonometry problems, but it also explains the concepts and provides videos you can watch to help you better understand the math. It’s an easy to use calculator,
too–just click the operations and enter the numbers.
When faced with math, it’s a good idea to know how to do it without a calculator, but you can certainly use a computer to help you double-check your answers.
source https://www.online-tech-tips.com/cool-websites/the-best-online-calculators-to-solve-any-problem/ | {"url":"https://appresima.com/the-best-online-calculators-to-solve-any-problem/","timestamp":"2024-11-11T13:41:45Z","content_type":"text/html","content_length":"137355","record_id":"<urn:uuid:eecf5264-d66d-409f-baa9-c3754e8ba684>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00382.warc.gz"} |
Interactive Q-Q Plots in R using Plotly | R-bloggersInteractive Q-Q Plots in R using Plotly
Interactive Q-Q Plots in R using Plotly
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In a recent blog post, I introduced the new R package, manhattanly, which creates interactive manhattan plots using the plotly.js engine.
In this post, I describe how to create interactive Q-Q plots using the manhattanly package. Q-Q plots tell us about the distributional assumptions of the observed test statistics and are common
visualisation tools in statistical analyses.
Visit the package website for full details and example usage.
Quick Start
The following three lines of code will produce the Q-Q plot below
qqly(HapMap, snp = "SNP", gene = "GENE")
Notice that we have added two annotations (the SNP and nearest GENE), that are revealed when hovering the mouse over a point. This feature of interactive Q-Q plots adds a great deal of information to
the plot without cluttering it with text.
The Data
Inspired by the heatmaply package by Tal Galili, we split the tasks into data pre-processing and plot rendering. Therefore, we can use the manhattanly::qqr function to get the data used to produce a
Q-Q plot. This allows flexibility in the rendering of the plot, since any graphics package, such as plot in base R can make used to create the plot.
The plot data is derived using the manhattanly::qqr function:
qqrObject <- qqr(HapMap)
## List of 6 ## $ data :'data.frame': 14412 obs. of 3 variables: ## ..$ P : num [1:14412] 6.75e-10 3.41e-09 3.95e-09 4.71e-09 5.02e-09 ... ## ..$ OBSERVED: num [1:14412] 9.17 8.47 8.4 8.33 8.3 ... ##
..$ EXPECTED: num [1:14412] 4.46 3.98 3.76 3.61 3.51 ... ## $ pName : chr "P" ## $ snpName : logi NA ## $ geneName : logi NA ## $ annotation1Name: logi NA ## $ annotation2Name: logi NA ## - attr(*,
"class")= chr "qqr"
## P OBSERVED EXPECTED ## 4346 6.75010e-10 9.170690 4.459754 ## 4347 3.41101e-09 8.467117 3.982633 ## 4344 3.95101e-09 8.403292 3.760784 ## 4338 4.70701e-09 8.327255 3.614656 ## 4342 5.02201e-09
8.299122 3.505512 ## 4341 6.22801e-09 8.205651 3.418362
This qqrObject which is of class qqr can also be passed to the manhattanly::qqly function to produce the inteactive Q-Q plot above:
Related Work
This work is based on the qqman package by Stephen Turner. It produces similar manhattan and Q-Q plots as the qqman::manhattan and qqman::qq functions; the main difference here is being able to
interact with the plot, including extra annotation information and seamless integration with HTML. | {"url":"https://www.r-bloggers.com/2016/06/interactive-q-q-plots-in-r-using-plotly/","timestamp":"2024-11-08T22:00:36Z","content_type":"text/html","content_length":"89352","record_id":"<urn:uuid:bb252a68-bdfa-4b64-8d02-a7dcc1fa80d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00645.warc.gz"} |
Get Reliable Maths Pre-calculus Textbook Solutions
Get Pre calculus Solution Manuals from CFS
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LSTM vs GRU: Experimental Comparison
A Recurrent Neural Network is a type of Artificial Neural Network that contains shared neuron layers between its inputs through time. This allows us to model temporal data such as video sequences,
weather patterns or stock prices. There are many ways to design a recurrent cell, which controls the flow of information from one time-step to another. A recurrent cell can be designed to provide a
functioning memory for the neural network. Two of the most popular recurrent cell designs are the Long Short-Term Memory cell (LSTM) and the Gated Recurrent Unit cell (GRU).
Read the rest of the article at Mindboard’s Medium channel.
Advantage function in Deep Reinforcement learning
Deep reinforcement learning involves building a deep learning model which enables function approximation between the input features and future discounted rewards values also called Q values. We have
seen how we can effectively get these q values and create a map consisting of input features and corresponding set of q values in this article.
This map of input features and all possible q values at a given state enables the Reinforcement learning agent get an overall picture of environment which further helps the agent in choosing the
optimal path.
Read the rest of the article at Mindboard’s Medium channel.
Time-Series Prediction Using A Simple RNN
For deeper networks, the obsession with image classification tasks seems to have also caused tutorials to appear on the more complex convolutional neural networks. This is great if you’re into that
sort of thing, however, if someone is more interested in data with timeframes then recurrent neural networks (RNNs) come in handy.
Read the rest of the article at Mindboard’s Medium channel
A time series contains a sequence of data points observed at specific intervals over time. A time series prediction uses a model to predict future values based on previously observed values. The
natural temporal order of time series data makes analysis of time series different from cross-sectional or spatial data analyses, neither of which depends on a time component.
Time series predictions can be useful in a variety of settings, from processing signal data streaming from a sensor at an industrial site to monitoring trends in a financial market or maintaining
inventory in a commercial setting. In all these scenarios, recent data can be used to inform predictions about future goal values.
Read the rest of the article at Mindboard’s Medium channel.
Input Window Size for Deep Recurrent Reinforcement Learning
Deep Recurrent Reinforcement Learning makes use of a Recurrent Neural Network (RNN), such as Long Short-Term Memory (LSTM) or Gated Recurrent Unit (GRU) based networks, for learning a value function
that maps environment states to action values. Recurrent Neural Networks are useful for modeling time-series data since the network maintains a memory, learning to retain useful information from
inputs of prior model inferences. Every time the model is called, the memory is updated in correspondence with the current inputs.
Read the rest of the article at Mindboard’s Medium channel.
Scaling Reward Values for Improved Deep Reinforcement Learning
Deep Reinforcement Learning involves using a neural network as a universal function approximator to learn a value function that maps state-action pairs to their expected future reward given a
particular reward function. This can be done many different ways. For example, a Monte Carlo based algorithm will observe total rewards following state-action pairs from a complete episode to make
build training data for the neural network. Alternatively, a Temporal Difference approach would use incremental rewards from single time-steps and bootstrap off of predicted future rewards from the
latest version of the value function model. However, no matter what approach is taken, it is important that the neural network is being efficiently fitted to the data in order to optimize the
learning algorithm. There are many factors that determine a neural networks ability to fit to training data. In this post we will examine how scaling our outputs can affect our rate of convergence.
Read the rest of the article at Mindboard’s Medium channel.
Crowd Density Estimation
In the light of problems caused due to poor crowd management, such as crowd crushes and blockages, there is an increasing need for computational models which can analyze highly dense crowds using
video feeds from surveillance cameras. Crowd counting is a crucial component of such an automated crowd analysis system. This involves estimating the number of people in the crowd, as well as the
distribution of the crowd density over the entire area of the gathering. Identifying regions with crowd density above the safety limit can help in issuing prior warnings and can prevent potential
crowd crushes. Estimating the crowd count also helps in quantifying the significance of the event and better handling of logistics and infrastructure for the gathering.
Read the rest of the article at Mindboard’s Medium channel.
Training Recurrent Neural Networks on Long Sequence
Deep Recurrent Neural Networks (RNN) are a type of Artificial Neural Network that takes the networks previous hidden state as part of its input, effectively allowing the network to have a memory.
This makes RNNs useful for modeling sequential or time-series data such as stock prices. However, training RNNs on sequences greater than a few hundred time steps can be difficult. In this post, we
will explore three tools that can allow for more efficient training of RNN models with long sequences: Optimizers, Gradient Clipping, and Batch Sequence Length.
Read the rest of the article at Mindboard’s Medium channel. | {"url":"https://masala.one/2019/page/2/","timestamp":"2024-11-09T06:07:34Z","content_type":"text/html","content_length":"71127","record_id":"<urn:uuid:e1f77147-005c-4423-9a07-5aece5b97289>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00308.warc.gz"} |
Competition Levels among Students in Class
Calculation the Competition Levels among Students in Class and the Relation to the Teacher or Lecture Fairness in Teaching
Measuring the level of competition among students or students taking place in the classroom and its relation to the level of fairness of teachers or lecturers teaching on this calculator is made by
using the Harmony in Gradation formula.
This method is very important because during this time the majority of teachers and lecturers who teach in the class never introspect whether he was fair to his students in the class.
In many cases, students who do not succeed in a school are due to receiving unfair actions from their teachers and/or schoolmates, and also because of the unfair school policies. This calculator can
be used to measure the extent to which students in a school have received fair treatment by considering the student’s test score. If it is found that the level of fairness in the education process is
not good, then the teachers are advised to examine the unfairness that may happen to some of their students.
The result of the calculation using the Harmony in Gradation Index states that: (1) index > = 0.75 indicates that the competition among students in the class is Balanced which means that the class
has run normally, where the students ‘or students’ ability to absorb the lesson is relatively even, and the teacher or lecturers have taught fairly; (2) 0.75 index = 0.5 indicates the existence of
competition among students in the class is less balanced, which means the class has run less normal, where the ability of students to absorb the lesson is relatively uneven and/or teachers or
lecturers have taught with a bit unfair; (3) index 0.5 indicates the existence of competition among students in the class is very unbalanced, which means the class has been running abnormally, where
the ability of students or students to absorb the lesson is very uneven and/or teachers or lecturers have taught unfair.
Calculator Procedure:
1. Fill in N = number of students in the class, then enter
2. Fill in the test scores of each student in each available field, then click CALCULATE.
Number of Students in the Classroom (N):
Index of Perfect Competition: | {"url":"https://www.haryadi.org/competition-levels-among-students-in-class/","timestamp":"2024-11-05T00:48:58Z","content_type":"text/html","content_length":"303862","record_id":"<urn:uuid:07be7eaf-efba-48b5-8935-d929fab10e19>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00889.warc.gz"} |
Gill (imperial) to Liter Converter
Enter Gill (imperial)
⇅ Switch toLiter to Gill (imperial) Converter
How to use this Gill (imperial) to Liter Converter 🤔
Follow these steps to convert given volume from the units of Gill (imperial) to the units of Liter.
1. Enter the input Gill (imperial) value in the text field.
2. The calculator converts the given Gill (imperial) into Liter in realtime ⌚ using the conversion formula, and displays under the Liter label. You do not need to click any button. If the input
changes, Liter value is re-calculated, just like that.
3. You may copy the resulting Liter value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Gill (imperial) to Liter?
The formula to convert given volume from Gill (imperial) to Liter is:
Volume[(Liter)] = Volume[(Gill (imperial))] × 0.1420653125
Substitute the given value of volume in gill (imperial), i.e., Volume[(Gill (imperial))] in the above formula and simplify the right-hand side value. The resulting value is the volume in liter, i.e.,
Calculation will be done after you enter a valid input.
Consider that a recipe calls for 2 gills (imperial) of milk.
Convert this volume from gills (imperial) to Liter.
The volume in gill (imperial) is:
Volume[(Gill (imperial))] = 2
The formula to convert volume from gill (imperial) to liter is:
Volume[(Liter)] = Volume[(Gill (imperial))] × 0.1420653125
Substitute given weight Volume[(Gill (imperial))] = 2 in the above formula.
Volume[(Liter)] = 2 × 0.1420653125
Volume[(Liter)] = 0.2841
Final Answer:
Therefore, 2 gi (imp) is equal to 0.2841 L.
The volume is 0.2841 L, in liter.
Consider that a pub serves a drink in 1 gill (imperial) portions.
Convert this serving size from gills (imperial) to Liter.
The volume in gill (imperial) is:
Volume[(Gill (imperial))] = 1
The formula to convert volume from gill (imperial) to liter is:
Volume[(Liter)] = Volume[(Gill (imperial))] × 0.1420653125
Substitute given weight Volume[(Gill (imperial))] = 1 in the above formula.
Volume[(Liter)] = 1 × 0.1420653125
Volume[(Liter)] = 0.1421
Final Answer:
Therefore, 1 gi (imp) is equal to 0.1421 L.
The volume is 0.1421 L, in liter.
Gill (imperial) to Liter Conversion Table
The following table gives some of the most used conversions from Gill (imperial) to Liter.
Gill (imperial) (gi (imp)) Liter (L)
0.01 gi (imp) 0.00142065313 L
0.1 gi (imp) 0.01420653125 L
1 gi (imp) 0.1421 L
2 gi (imp) 0.2841 L
3 gi (imp) 0.4262 L
4 gi (imp) 0.5683 L
5 gi (imp) 0.7103 L
6 gi (imp) 0.8524 L
7 gi (imp) 0.9945 L
8 gi (imp) 1.1365 L
9 gi (imp) 1.2786 L
10 gi (imp) 1.4207 L
20 gi (imp) 2.8413 L
50 gi (imp) 7.1033 L
100 gi (imp) 14.2065 L
1000 gi (imp) 142.0653 L
Gill (imperial)
The Imperial gill is a unit of measurement used to quantify liquid volumes, particularly in the UK and countries using the Imperial system. It is defined as 5 fluid ounces or approximately 142.065
milliliters. Historically, the gill was used for measuring smaller quantities of liquids, such as beverages and medicinal preparations. Today, while its use has declined, it is still recognized in
some contexts and historical documents, providing a measure for small liquid volumes consistent with the Imperial system.
The liter is a unit of measurement used to quantify liquid volumes and is part of the metric system. It is defined as the volume of a cube with sides each measuring 10 centimeters, equivalent to
1,000 cubic centimeters or 1 cubic decimeter. The liter has been widely adopted globally for its simplicity and ease of use in measuring liquids and gases. Historically, the liter was introduced to
provide a standard metric unit for consistent measurements across various scientific, industrial, and everyday applications. Today, it is commonly used in cooking, scientific research, and trade to
ensure accurate and standardized volume measurements.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Gill (imperial) to Liter in Volume?
The formula to convert Gill (imperial) to Liter in Volume is:
Gill (imperial) * 0.1420653125
2. Is this tool free or paid?
This Volume conversion tool, which converts Gill (imperial) to Liter, is completely free to use.
3. How do I convert Volume from Gill (imperial) to Liter?
To convert Volume from Gill (imperial) to Liter, you can use the following formula:
Gill (imperial) * 0.1420653125
For example, if you have a value in Gill (imperial), you substitute that value in place of Gill (imperial) in the above formula, and solve the mathematical expression to get the equivalent value in | {"url":"https://convertonline.org/unit/?convert=gill_imperial-liter","timestamp":"2024-11-09T19:55:12Z","content_type":"text/html","content_length":"93280","record_id":"<urn:uuid:47a6ff28-03a2-40bd-98bd-72368f1509fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00510.warc.gz"} |
R programming for beginners (GV900) – Can.Do.So
In this lesson, we will learn the standard deviation and the standard error.
First, load the packages we will use in this lesson.
The standard deviation is a measure of how spread out the values are from the mean.
Why we use n()-1 instead of n() in the variance formula? When we calculate the variance, we use the mean of the sample, which itself is estimated from the sample. For example, if we have a sample of
10 numbers, the mean of the 10 numbers is decided by the 10 numbers. But if we know the mean of the 10 numbers, you are free to choose the 10 numbers. Actually, not 10 numbers. When you’ve chosen 9
numbers, you will find that you cannot freely choose the 10th number if you wish get the pre-decided mean. Therefore, we only have 9 degrees of freedom in this case.
The degrees of freedom is the number of independent observations in a sample minus the number of population parameters that must be estimated from sample data.
The standard error is the standard deviation of the sampling distribution of a statistic.
# 100 samples # Don't worry if you don't understand this code. We will learn the for loop function in the future lessons. age_mean <- numeric(100) # Loop to generate 100 age_mean values for (i in
1:100) { age_mean[i] <- BEPS %>% select(age) %>% slice_sample(n = 100, replace = TRUE) %>% summarise(age_mean = mean(age)) %>% pull(age_mean) } # Display the first few age_mean values age_means <-
data.frame(sample = 1: 100, age_mean)
We can see that the se calculated by \(sd \over \sqrt(n)\) is very close to the sd of the 100 age_mean values we calculated before.
We can easily notice that the standard error of the mean is dependent on the sample size (\(n\)). The larger the sample size, the smaller the standard error of the mean.
However, if we increase the sample times but keep the sample size unchanged, the standard error of the mean will not change significantly.
In this lesson, we learned the standard deviation and the standard error.
The standard error of the mean is dependent on the sample size (\(n\)). The larger the sample size, the smaller the standard error of the mean.
In the following lessons, we will use sampling distribution and the standard error to calculate the confidence interval of the mean, and do hypothesis testing. | {"url":"https://www.candoso.org/docs/r/gv900/011","timestamp":"2024-11-05T21:46:47Z","content_type":"text/html","content_length":"58670","record_id":"<urn:uuid:f9ce56cd-c38b-4fcd-ba6e-30dae93429ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00295.warc.gz"} |
TI-Basic Developer
The max( Command
Returns the maximum of two elements or of a list.
• for two numbers: max(x,y)
• for a list: max(list)
• comparing a number to each element of a list: max(x,list) or max(list,x)
• pairwise comparing two lists: max(list1,list2)
1. MATH to access the math menu.
2. RIGHT to access the NUM submenu.
3. 7 to select max(, or use arrows.
Alternatively, press:
1. 2nd LIST to access the list menu.
2. LEFT to access the MATH submenu.
3. 2 to select max(, or use arrows.
max(X,Y) returns the largest of the two numbers X and Y. max(list) returns the largest element of list. max(list1,list2) returns the pairwise maxima of the two lists. max(list1,X) (equivalently, max
(X,list1)) returns a list whose elements are the larger of X or the corresponding element of the original list.
{4 3}
{2 3}
Unlike comparison operators such as < and >, max( can also compare complex numbers. To do this, both arguments must be complex — either complex numbers or complex lists: max(2,i) will throw an error
even though max(2+0i,i) won't. In the case of complex numbers, the number with the largest absolute value will be returned. When the two numbers have the same absolute value, the first one will be
returned: max(i,-i) returns i and max(-i,i) returns -i.
Advanced Uses
max( can be used in Boolean comparisons to see if at least one of a list is 1 (true) — useful because commands like If or While only deal with numbers, and not lists, but comparisons like L₁=L₂
return a list of values. In general, the behavior you want varies, and you will use the min( function or the max( function accordingly.
Using max( will give you a lenient test — if any one element of the list is 1 (true), then the max( of the list is true — this is equivalent to putting an or in between every element. For example,
this tests if K is equal to any of 24, 25, 26, or 34 (the getKey arrow key values):
:If max(K={24,25,26,34
:Disp "ARROW KEY
To get the element of a real list in Ans with the greatest absolute value, use imag(max(iAns)) or max(abs(Ans)).
max( can be also used along with min( to constrain a value between a lower and upper number:
:max(-1,min(1,100)) // returns 1 because 1 is between -1 & 100
:max(-1,min(1,0)) // returns 0 because 1 is not between -1 & 0
where the bounds for which the number 1 must fall between are first argument of max( and the second argument of min( in the above code.
Error Conditions
• ERR:DATA TYPE is thrown when comparing a real and a complex number. This can be avoided by adding +0i to the real number (or i^4 right after it, for those who are familiar with complex numbers)
• ERR:DIM MISMATCH is thrown, when using max( with two lists, if they have different dimensions.
Related Commands | {"url":"http://tibasicdev.wikidot.com/max","timestamp":"2024-11-11T09:50:42Z","content_type":"application/xhtml+xml","content_length":"31734","record_id":"<urn:uuid:9424f607-13d7-4476-9dbb-093eab673a03>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00620.warc.gz"} |
The group generated by the following permutations :
Type one permutation per line. Help
Related tool:
, database of (abstract) groups of order up to 255.
The most recent version This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: calculator of permutation groups based on GAP: symmetric, alternating, transitive, primitive, etc. interactive exercises, online calculators and plotters, mathematical recreation and
• Keywords: interactive mathematics, interactive math, server side interactivity, algebra, group_theory, permutation | {"url":"https://wims.divingeek.com/wims/wims.cgi?lang=en&+module=tool%2Falgebra%2Fpermgroup.en","timestamp":"2024-11-07T12:10:20Z","content_type":"text/html","content_length":"7595","record_id":"<urn:uuid:adc1ad7f-a353-4efa-b2f4-18d4921aaf4c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00597.warc.gz"} |
🌹Make Math Fun with Fact Fluency for ELLs - Fun to Teach
Hello teachers ~
Today we are going to be diving into all things related to math fact fluency. Fact fluency is a skill that students especially English Language Learners need to develop in order to be successful with
math. This post will unpack what fact fluency is, any misconceptions, activities to teach it, and how to make it work for your ELLs. Let’s dive in! 😁
⭐What is math fact fluency?
💧What is 6 x 1?
💧What is 5 x 2?
💧What is 4 x 3?
If you are able to recall the answer to these questions in 2 seconds or less, you may have developed something called math fact fluency.
In simple terms, math fact fluency occurs when a student instantly recalls the answer to a basic math problem. As students repeatedly practice simple addition, subtraction, multiplication, and
division equations, they commit the answers to their long-term memory. When students are able to recall these answers instantly, they have achieved math fact fluency.
However, math fact fluency is also more than that. In order to possess this skill, students need to feel comfortable with numbers and enjoy working with them. Completing multiplication drills is
monotonous and boring for students, but participating in fun activities can help them build their math fact fluency in meaningful ways.
Math fact fluency vs. Math fact automaticity
The internet is filled with terms related to math facts. Sometimes it’s difficult to know what the difference between them is! Two common terms that can easily be interchanged are math fact fluency
and math fact automaticity. Let’s discuss the difference below.
👆Math Fact Fluency – the ability to solve math problems quickly and fluently. Fluency in math refers to being able to manipulate numbers to find an answer quickly. For example, if you posed the
question, “how could I solve the problem 3 x 4?” students with fact fluency may know the answer is 12, but their basis in visualizing and manipulating numbers is what has caused them to possess the
ability to solve this problem so quickly.
✌Math Fact Automaticity – being able to automatically provide the answer to a question without thinking.
While automaticity is a part of math fact fluency, we want to instill a desire to love math and number manipulation, not just spewing out an answer like a robot.
⭐Lesson Plan
Looking for a way to teach multiplication fact fluency to your students? Take a look at the lesson plan below!
Math fact fluency can only be built once students have foundational skills. To teach students multiplication fact fluency, you must first teach students how to multiply. Great ways to do this include
visual activities like arrays, repeated addition, or creating equal groups with manipulatives. These activities will get students’ minds turning with the understanding of how multiplication works.
This is vital for students to understand before they move into fact fluency.
Wondering how to build a background for ELL students? Vocabulary is the cornerstone of learning for language learners. Pull a few keywords from the topic and teach them to students before beginning
your math unit. This ensures that students know and understand the language needed to interact with the topic. As students learn, play games with students to reinforce the vocabulary. Need help
getting started? My math vocabulary bundle in English and Spanish includes keywords and pictures for several topics.
If your students already have a strong foundation of how multiplying works, you can move them into fact fluency. To practice fact fluency, we must give students repetitive, engaging opportunities to
practice solving equations.
There are several activities that I LOVE for teaching multiplication fact fluency. Let’s explore one of them together.
For math games and activities, I especially enjoy using flashcards and game boards. We use both for the multiplying game from my Multiplication Fact Fluency Games Bundle that we are going to explore
in this lesson.
First, each student takes a game board, like the one below. (I have several printable game boards in my resource.)
Next, students take turns drawing an equation from the pile of facedown equation cards in the center of the table. Example equation cards are below:
Once students draw their equation cards, they read the equation aloud and try to determine the answer. As they are solving, their group mates try to solve the equation as well to be sure that the
student who is answering the question gives the correct answer. Once the student solves the problem, he or she looks to see if the answer is on their game board. If it is, they cover the answer with
the equation card itself. If it isn’t, they put the equation card in the discard pile. Play passes to the next player. The game ends when a player has filled his or her entire board with equation
Students love playing this game because they have the opportunity to compete against their group mates to fill up their board first! What student doesn’t love trying to be the first to win? This game
helps build math fact fluency because students are practicing math in a meaningful way. Students are engaged as they repeatedly attempt to solve math equations with their group mates.
Have ESL students in your class? Because fact fluency is primarily number-based, the skill itself requires little language. As long as students know the basic vocabulary surrounding the topic,
students at all language levels, newcomers included, can participate in these math games with ease. This is also a great unit to have newcomer students practice number vocabulary in English during
small group time.
Get this game and many others in my multiplication bundle. This resource also contains timed tests and other multiplication worksheets and activities. Covering numbers 0-12, this unit is all you need
to keep your students engaged during math small group, whole group, or as extra homework practice within your multiplication unit!
Your students are right around the corner from achieving math fact fluency. Let this resource help them practice in a meaningful way!
Click the pictures below to take a peek at my resources. 👀
Happy Teaching! 💜 | {"url":"https://funtoteach.com/2022/05/%F0%9F%8C%B9make-math-fun-with-fact-fluency-for-ells-2/","timestamp":"2024-11-13T08:03:52Z","content_type":"text/html","content_length":"118059","record_id":"<urn:uuid:18e7d6de-871c-4606-876b-a78fec8ba3d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00186.warc.gz"} |
The Stacks project
Remark 29.51.3. An alternative to Lemma 29.51.1 is the statement that a quasi-finite morphism is finite over a dense open of the target. This will be shown in More on Morphisms, Lemma 37.45.2.
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Modeling and Coefficient Identification of Cortical Bone Milling Forces of Ball-End Milling Cutter for Orthopaedic Robot
School of Mechanical and Electronic Engineering, Shandong University of Science and Technology, 266590 Qingdao, China
Journal of Industrial Intelligence
Volume 1, Issue 4, 2023
Pages 229-240
Received: 11-11-2023,
Revised: 12-15-2023,
Accepted: 12-24-2023,
Available online: 12-30-2023
View Full Article|Download PDF
When cutting the hard cortical bone layer, orthopedic robots are prone to cutting chatter and thermal damage due to force and heat. Accurately establishing a model of cortical bone milling force and
assessing the milling force in suppressing cortical bone cutting chatter, reducing cutting thermal damage, and optimizing process parameters is of great significance. This study aims to deeply
explore the issues of modeling and coefficient identification of the milling force model of the orthopedic robot ball-end milling cutter for cortical bone, and to establish a theoretical model
related to the milling state for analyzing the stability of robot milling chatter. The milling force model of the orthopedic robot ball-end milling cutter was constructed using the micro-element
method, and a milling coefficient identification model was established based on the average milling force model. The coefficients were identified using the least squares method, and the cortical bone
milling force model for the orthopedic robot ball-end milling cutter was established and experimentally verified. The experimental results show that the milling force curve calculated is basically
consistent with the actual measured curve in terms of values and trend, verifying the accuracy of the established milling force model, and providing a theoretical basis for the study of robot
cortical bone milling chatter.
Keywords: Ball-end milling cutter, Micro-element method, Cortical bone, Milling force model, Coefficient identification
Cite this:
APA Style
IEEE Style
BibTex Style
MLA Style
Chicago Style
Tian, H. Q. & Ma, H. Q. (2023). Modeling and Coefficient Identification of Cortical Bone Milling Forces of Ball-End Milling Cutter for Orthopaedic Robot. J. Ind Intell., 1(4), 229-240. https://
©2023 by the author(s). Published by Acadlore Publishing Services Limited, Hong Kong. This article is available for free download and can be reused and cited, provided that the original published
version is credited, under the
CC BY 4.0 license
Figure 1. Geometric model of ball-end milling cutter. (a) Ball-end milling cutter coordinate system; (b) $XOY$ plane projection
Table 1. Slot milling experimental data of cortical bone | {"url":"https://www.acadlore.com/article/JII/2023_1_4/jii010404","timestamp":"2024-11-06T09:40:21Z","content_type":"text/html","content_length":"307739","record_id":"<urn:uuid:0d16c9ef-0229-41fb-bf2d-57566d61bc42>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00310.warc.gz"} |
Average speed
Average speed [#permalink]
Sarada drove the same route to work each morning, Monday through Friday, in a particular week. On Monday and Tuesday she averaged 20 miles per hour, and on her three remaining work days she averaged
30 miles per hour.
Quantity A : Sarada’s average speed for all five morning commutes
Quantity B : 26 miles per hour
For the above Q, the answer is B. Could anyone explain how? | {"url":"https://gre.myprepclub.com/forum/average-speed-34696.html","timestamp":"2024-11-10T11:39:12Z","content_type":"application/xhtml+xml","content_length":"178638","record_id":"<urn:uuid:fccf7b36-871e-4737-af64-aeaa812e872a>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00196.warc.gz"} |
Find The Prime Factors Of The Numbers Worksheet
Find The Prime Factors Of The Numbers Worksheet serve as foundational devices in the world of mathematics, supplying an organized yet functional platform for students to explore and grasp numerical
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Find The Prime Factors Of The Numbers Worksheet
Find The Prime Factors Of The Numbers Worksheet -
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Prime Factorization Worksheet Page
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Finding Factors Of A Number Worksheet For Kids
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Free Worksheets For Prime Factorization Find Factors Of A Number
Free Worksheets For Prime Factorization Find Factors Of A Number Gcf And Lcm Worksheets
Check more of Find The Prime Factors Of The Numbers Worksheet below
How To Find The Prime Factors Using Factor Tree A Plus Topper Https www aplustopper
Prime Factorization Worksheet Pdf
Prime Factorization Using Repeated Division solutions Examples Videos
Find Factors Worksheet
Prime Factorization Worksheet Pdf
Prime Factorization Worksheet Of Numbers Up To 1 000 Grade 6 Math Worksheets Samples
Free Worksheets For Prime Factorization Find Factors Of A Number
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Factoring Worksheets K5 Learning
Find the factors of numbers between 4 and 100 Prime factors prime factor trees Divisibility rules Multiples of whole numbers Grade 5 factoring worksheets Factoring numbers to prime factors up to 100
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Find the factors of numbers between 4 and 100 Prime factors prime factor trees Divisibility rules Multiples of whole numbers Grade 5 factoring worksheets Factoring numbers to prime factors up to 100
500 List all the factors of numbers up to 100 Prime factor trees More divisibility rules Greatest common factor GCF of 2 numbers 1
Prime Factorization Worksheet Pdf
Prime Factorization Worksheet Pdf
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Prime Factorization Worksheet Page | {"url":"https://szukarka.net/find-the-prime-factors-of-the-numbers-worksheet","timestamp":"2024-11-03T15:58:26Z","content_type":"text/html","content_length":"26171","record_id":"<urn:uuid:d701c4af-8152-4e17-841d-37411d41eab9>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00828.warc.gz"} |
What is 209 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
If you’re wondering what 209 Celsius is in Fahrenheit, then you’ve come to the right place. 209 degrees Celsius is equal to 408.2 degrees Fahrenheit.
Understanding temperature conversions can be useful for a variety of reasons. Whether you’re a student studying science or simply trying to understand the weather forecast in different units, knowing
how to convert between Celsius and Fahrenheit is an important skill.
Celsius and Fahrenheit are two different units of temperature measurement. While Celsius is commonly used in most countries around the world, Fahrenheit is primarily used in the United States. The
conversion formula for Celsius to Fahrenheit is:
(Fahrenheit) = (Celsius x 9/5) + 32
Using this formula, we can convert 209 degrees Celsius to Fahrenheit:
F = (209 x 9/5) + 32
F = 377.2 + 32
F = 408.2
So, 209 degrees Celsius is equivalent to 408.2 degrees Fahrenheit. This means that if the temperature is 209 degrees Celsius, it would also be 408.2 degrees Fahrenheit.
Knowing this conversion can be helpful in a variety of situations. For example, if you’re traveling to a country that uses Fahrenheit and you’re used to Celsius, it can be useful to have a general
idea of what the temperature will feel like in Fahrenheit. Additionally, if you’re studying science or working in a field that requires temperature measurements, understanding how to convert between
different units can be beneficial.
When it comes to understanding temperature conversions, there are a few key points to keep in mind. First, it’s important to remember that Celsius and Fahrenheit are two different scales, and their
zero points are based on different reference points. In Celsius, the freezing and boiling points of water are defined as 0 and 100 degrees, respectively. In Fahrenheit, these points are defined as 32
and 212 degrees.
Another important concept to understand is the relative nature of temperature measurements. While 209 degrees Celsius may seem very hot, the equivalent temperature in Fahrenheit, 408.2 degrees, might
seem extreme to someone who is more accustomed to Celsius. This is because the two scales measure temperature relative to different reference points.
In conclusion, 209 degrees Celsius is equivalent to 408.2 degrees Fahrenheit. Understanding how to convert between Celsius and Fahrenheit can be useful in a variety of situations, from traveling to
studying science. By keeping in mind the conversion formula and the relative nature of temperature measurements, you can easily make the conversion between these two common units of temperature | {"url":"https://converttemperatureintocelsius.info/what-is-209celsius-in-fahrenheit/","timestamp":"2024-11-05T21:55:18Z","content_type":"text/html","content_length":"73244","record_id":"<urn:uuid:a98c5c67-c810-4def-8d80-ed73f7e24348>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00505.warc.gz"} |
Carbon-14 dating can be used to determine the age of objects up to
Carbon-14 dating can be used to determine the age of objects up to
Cosmic rays hitting the radiocarbon dating has been on fossil 5, pressure, how radiometric dating is left, and since 1947, scientists dig out an object. Given sample from a radiometric dating is
produced in trying to determine the absolute dating can calculate the sign in radiometric dating. Measuring things used scientific dating be a new home the radiocarbon dating is a known as.
Accordingly, most commonly accepted radiocarbon dating method would be dated by measuring carbon-14 decays. Beyond, 568 years old object, pressure, and fossils approximate age of not 2. Researchers
can use carbon-14 is now, the date objects. Relative dating - is used scientific dating is replenished in read here that were created in ancient fossil.
By dating has allowed key tool archaeologists could be accelerated mass spectrometer used to 80, on. Here is used an artifact, carbon-14 decreases. Luckily, 14c can be applied to. Various methods by
using recorded. I recently shared a method of 5730 years after one half-life of carbon dating is one. When determining how have a half-life decay to tell you how the age dating is one. Using
distilled water or cloth, we can. Scientists use a technique is known as redporn fossil 5, 000 years. Thus the number of an organic compounds and exploits the sample that the world february 9, used
to calibrate carbon-14 isotope will not only works.
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fossils younger man. Discovery of earth's oldest inhabitants. https://cottonperu.com/blogs-about-dating-over-50/ carbon dating carefully applied that commonly used on. Biology radioactive isotope
used to about 5, we can work out the mark. Scientists do not used the radioisotope carbon 14 it can calculate how can. Unfortunately, but it's not perfect. Scientists use carbon-based objects ranging
from the known ages of 14c, the radioisotope under 40, 700 years there are carbon-12 and. Relative dating is naturally in decays. As radiocarbon dating is, carbon-13 and year. Q: background
information _____ carbon dating. Most of which an important part of the decay into our best estimate the relative dating always been able to organic remains between.
Technique of relative dating can be used to determine the actual age of a fossil
The bottom layers have a method of laboratory techniques are very powerful method of a fossil is older. Geologists use radiometric dating is used same age constraints step 1. Comparing it is the law
of. Other techniques to decay in a fossil? As many intrusions to determine the fossils and form other fossils are two methods used by analyzing fossils to. Explain why both are of the closer the.
Archaeologists close in years old, that layer is all of igneous rocks can use fossils. They may not result in the richer the.
Radiocarbon dating can be used to determine the age of brainly
However, stratigraphy, is the grand. Who are a middle-aged man looking for those who've tried and contrast with relative dating uses radioactive element is to determine the amount of. In the
environment ceases and thuqeibah. Radio-Carbon dating is continually replaced, the half-life and absolute age can be used to determine the age iii occupation in. For dating can only rarely applied,
and the ages of fossilsoption a variety of relative dating. Note that radiometric dating method: t ln nf/no / -0. Can just switch to meet eligible single man looking for more details. Students will
focus on the age and history of different isotopes of objects that. Carbon date of determining the amount of objects containing organic material like rocks can use radioactive carbon-14 content. When
it can just switch to normal browsing. How far is alive, can radiation dating or range in the amount of carbon dating involves determining the ratios of old. Carbon dating can be grouped, you had a
sample is carbon-14 dating also know the lens?
Can the carbon dating technique be used to determine the age of a diamond explain why or why not
Specifically, miller's team has its codiscoverer, they have been on many articles on the graphite in other dating. Libbys method is not be used by measuring their age of. Very quickly radiocarbon
capture from an age on samples. Scaling method of the age dating is not limited to. Results are used as do not a time that an interplay. Radiometric dating methods of contradiction that are
infection-control measures used to nitrogen of. Reference to determine the age. Click this does not used chronometric technique.
Radiocarbon dating can be used to determine the age of what type of materials
Measuring their distributions in sufficient quantity can be. Learn more than about 100, wood and. Any time can the year we use of. Recent past by the surrounding rocks: bomb radiocarbon dating, 2018
radiocarbon date of sample. We use carbon-based materials that can be. Using a huge difference between the age of a falsely young radiocarbon dating, 000 years. Igneous rock types: the age of age of
carbon-14 in earth's oldest metamorphic rocks. Calculate the age of carbon dating be found experimentally in the limitation in prehistory. Isotopes can be used in the three principal techniques to
determine the type of. Here are used for geologic materials. Most suitable types of two major types. Use of species that are three principal techniques – typological dating of. | {"url":"https://jkl331.jp/carbon-14-dating-can-be-used-to-determine-the-age-of-objects-up-to/","timestamp":"2024-11-11T09:51:58Z","content_type":"text/html","content_length":"51497","record_id":"<urn:uuid:79ac8818-059d-4d3e-8b3d-01871360789a>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00049.warc.gz"} |
How to Use the Excel Functions ISNUMBER, ISTEXT and ISNONTEXT
IS functions are really fun to work with. They’ll return Boolean values (TRUE or FALSE) but can prove super useful when used the right way.
The ISNUMBER and ISTEXT function belong to the same IS family of Excel.
What are these functions used for, and how do you use them? Let’s learn all this and more in the guide below 😀
Also, if you want to practice these functions in real time, download our sample workbook here.
How to use ISNUMBER in Excel
The Excel ISNUMBER function lies in the category of information functions. Its syntax is pretty simple and easy to use.
ISNUMBER checks a cell to see if the given value is a number. The value could be a simple number or the result of a formula.
It returns TRUE or FALSE and is often used in combination with other formulas. Let’s see the syntax below.
The ISNUMBER syntax only uses a single argument – value.
The value argument is the value we want to check. It can be a formula, cell reference, number, or anything else 🔟
To see how the ISNUMBER function works, read on.
We will apply the formula to the following data set.
Here’s the result the ISNUMBER function returns.
As evident, column A contains the values to be checked. Column B contains the results of the function and column C shows the formula used.
Column B returns TRUE for all numerical values and FALSE for all text strings.
Note that the ISNUMBER function returns FALSE for the number #854. That’s because it is not a number. The value is left aligned by default, indicating that it is not a number but stored as text
ISNUMBER and IF formula example
Let’s see an example of the ISNUMBER and IF function 🤓
We have the following example data.
We want to find the salary of an employee in row 6 in the given data set.
But we want to keep the data private. To do that, we have used the IF function that will return the salary amount only if we use ID No.
If anyone tries to find the salary using names, the function will return an empty cell value.
The ISNUMBER function becomes important here as it checks if the given cell value is a number. If the selected cell is not a number, it will return a blank cell.
Let’s see this in action now, shall we? 👀
1. Select a cell.
2. Apply the formula as:
3. Add the ISNUMBER formula and cell value to check.
4. Enter the value_if_true and value_if_false arguments as:
=IF(ISNUMBER(A6), C6, ” “)
And tada! Excel returns the expected result.
That’s because cell A6 contains a number, and we get the score corresponding to that ID No.
If we had used a cell reference containing a text string, for instance, cell B6, the function would return a blank cell. See the image below for reference.
As expected, we get an empty cell because cell B6 contains a text string.
Pretty simple, no? 😉
ISNUMBER SEARCH formula example
The SEARCH function looks up a cell for a specific substring. It returns the location of the value, but when compared with ISNUMBER, it returns the answer as TRUE or FALSE.
Let’s see an example of using the ISNUMBER and SEARCH functions together.
We have the following sample data.
Say we want to see if Roll No. 20 exists in cell B5. To do that:
1. Select a cell.
2. Enter the formula as:
3. Add the SEARCH function.
4. Enter the Roll No. and row you want to look for its location.
=ISNUMBER(SEARCH(20, B5)
5. Press Enter.
Excel returns the result as TRUE.
Note that if we had used the SEARCH function alone, it would have returned a value “1”. But since we used it in combination with the ISNUMBER function, it returned TRUE.
That’s because the value is a number and exists in a specified location. If any of these conditions had not been fulfilled, Excel would have returned FALSE.
How to use ISTEXT in Excel
The ISTEXT function is similar to ISNUMBER. They have the same syntax arguments and perform the same kind of operation.
The only difference is that the ISTEXT function checks a cell for a text value, as evident from the name. It returns TRUE for text strings and FALSE for numeric values.
Its syntax is as follows:
where value is the value to be checked.
We will use the previously used example data for the ISTEXT function.
Upon applying the ISTEXT function, we get the results:
The ISTEXT function returns TRUE for text values and FALSE for numbers.
Note that ISTEXT returns TRUE for the value #854. This proves that it is a text value as seen in the earlier ISNNUMBER example.
ISTEXT and IF formula example
Let’s see an example of the ISTEXT function with IF.
We will use the following example data.
We want to find the salary of Peter, but we don’t want to use his ID No. 😕
To do that, we will use the ISTEXT function, which will only show the salary when we enter his name. If anyone tries to check the salary using his ID No., it will return a blank cell.
Let’s see how to do it below.
1. Select a cell.
2. Enter the formula as:
3. Enter the ISTEXT function.
4. Add the arguments for ISTEXT and IF functions.
=IF(ISTEXT(B7, C7, ” “)
5. Press Enter.
Excel returns the result as:
If the cell reference used in ISTEXT function contained a number, the result would have been something like this:
As visible, the ISTEXT function returns a blank cell because cell A7 is not a text string.
How to use ISNONTEXT
The ISNONTEXT and ISNUMBER functions are identical. Both have the same purpose and the same syntax.
The ISNONTEXT function returns TRUE when a cell contains a numerical value. If it is a text string, it will return FALSE – same as in the case of ISNUMBER.
Syntax of the ISNONTEXT function is:
Where value refers to the value to be checked.
It’s that simple to use.
A couple of tries are all you need to master this function 🧐
That’s it – Now what?
In this article, we learned how to use the ISNUMBER and ISTEXT functions. We also saw how to use the ISNONTEXT function and its uses in the practical world.
These functions are easy to learn and use. But identifying the utility of ISNUMBER and ISTEXT can be slightly difficult on the surface.
The power of these functions is unleashed when they are combined with other functions. You’ve already seen a quick glance through our examples. A little practice can help you master these functions
in no time.
Some common functions you can combine the ISNUMBER and ISTEXT functions with include VLOOKUP, IF, SUMIF and others.
You can learn them in my 30-minute free email course that teaches these functions and more. It’s delivered right to your inbox only at the cost of your email address. So join now! 🤗
Frequently asked questions | {"url":"https://spreadsheeto.com/isnumber-istext-functions/","timestamp":"2024-11-14T07:51:42Z","content_type":"text/html","content_length":"317534","record_id":"<urn:uuid:33744f6d-8330-4e73-8681-b4e15cea582f>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00193.warc.gz"} |
Joanna Niezen
Lecturer & Faculty Teaching Fellow
Department of Mathematics
Faculty of Science
• Ph.D. Discrete Mathematics · University of Victoria · 2020
Joanna enjoys helping students achieve their mathematical goals, whatever they may be. She makes a conscious effort to pique the curiosity of her students and build excitement around course material.
Joanna is passionate about equity and diversity in mathematics and works hard to be inclusive to all students. She earned a doctorate in Discrete Mathematics at the University of Victoria, studying
the existence of designs with special structures. Joanna’s interest in combinatorics extends to abstract algebra, linear algebra, number theory, and all sorts of math puzzles. She is also interested
in the psychology of development and learning. | {"url":"http://www.sfu.ca/math/people/faculty/jniezen.html","timestamp":"2024-11-06T04:44:40Z","content_type":"text/html","content_length":"52317","record_id":"<urn:uuid:740ae69f-c5ac-4f19-8405-46aa446f560a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00807.warc.gz"} |
subroutine sbbcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, B22D, B22E, WORK, LWORK, INFO)
subroutine sgghd3 (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
subroutine sgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
subroutine sggqrf (N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
subroutine sggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
subroutine sggsvp3 (JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
subroutine sgsvj0 (JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO)
SGSVJ0 pre-processor for the routine sgesvj.
subroutine sgsvj1 (JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO)
SGSVJ1 pre-processor for the routine sgesvj, applies Jacobi rotations targeting only particular pivots.
subroutine shsein (SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO)
subroutine shseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO)
subroutine sla_lin_berr (N, NZ, NRHS, RES, AYB, BERR)
SLA_LIN_BERR computes a component-wise relative backward error.
subroutine sla_wwaddw (N, X, Y, W)
SLA_WWADDW adds a vector into a doubled-single vector.
subroutine slals0 (ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO)
SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
subroutine slalsa (ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)
SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine slalsd (UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, IWORK, INFO)
SLALSD uses the singular value decomposition of A to solve the least squares problem.
real function slansf (NORM, TRANSR, UPLO, N, A, WORK)
subroutine slarscl2 (M, N, D, X, LDX)
SLARSCL2 performs reciprocal diagonal scaling on a vector.
subroutine slarz (SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
SLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
subroutine slarzb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
SLARZB applies a block reflector or its transpose to a general matrix.
subroutine slarzt (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
SLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine slascl2 (M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a vector.
subroutine slatrz (M, N, L, A, LDA, TAU, WORK)
SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
subroutine sopgtr (UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
subroutine sopmtr (SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
subroutine sorbdb (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)
subroutine sorbdb1 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
subroutine sorbdb2 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
subroutine sorbdb3 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
subroutine sorbdb4 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO)
subroutine sorbdb5 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
subroutine sorbdb6 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
recursive subroutine sorcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK,
IWORK, INFO)
subroutine sorcsd2by1 (JOBU1, JOBU2, JOBV1T, M, P, Q, X11, LDX11, X21, LDX21, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, WORK, LWORK, IWORK, INFO)
subroutine sorg2l (M, N, K, A, LDA, TAU, WORK, INFO)
SORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine sorg2r (M, N, K, A, LDA, TAU, WORK, INFO)
SORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine sorghr (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
subroutine sorgl2 (M, N, K, A, LDA, TAU, WORK, INFO)
subroutine sorglq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
subroutine sorgql (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
subroutine sorgqr (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
subroutine sorgr2 (M, N, K, A, LDA, TAU, WORK, INFO)
SORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).
subroutine sorgrq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
subroutine sorgtr (UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
subroutine sorm2l (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine sorm2r (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine sormbr (VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine sormhr (SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine sorml2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
subroutine sormlq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine sormql (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine sormqr (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine sormr2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).
subroutine sormr3 (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, INFO)
SORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).
subroutine sormrq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine sormrz (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine sormtr (SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
subroutine spbcon (UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, IWORK, INFO)
subroutine spbequ (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
subroutine spbrfs (UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
subroutine spbstf (UPLO, N, KD, AB, LDAB, INFO)
subroutine spbtf2 (UPLO, N, KD, AB, LDAB, INFO)
SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
subroutine spbtrf (UPLO, N, KD, AB, LDAB, INFO)
subroutine spbtrs (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
subroutine spftrf (TRANSR, UPLO, N, A, INFO)
subroutine spftri (TRANSR, UPLO, N, A, INFO)
subroutine spftrs (TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
subroutine sppcon (UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO)
subroutine sppequ (UPLO, N, AP, S, SCOND, AMAX, INFO)
subroutine spprfs (UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
subroutine spptrf (UPLO, N, AP, INFO)
subroutine spptri (UPLO, N, AP, INFO)
subroutine spptrs (UPLO, N, NRHS, AP, B, LDB, INFO)
subroutine spstf2 (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine spstrf (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine ssbgst (VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, INFO)
subroutine ssbtrd (VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
subroutine ssfrk (TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C)
SSFRK performs a symmetric rank-k operation for matrix in RFP format.
subroutine sspcon (UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
subroutine sspgst (ITYPE, UPLO, N, AP, BP, INFO)
subroutine ssprfs (UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
subroutine ssptrd (UPLO, N, AP, D, E, TAU, INFO)
subroutine ssptrf (UPLO, N, AP, IPIV, INFO)
subroutine ssptri (UPLO, N, AP, IPIV, WORK, INFO)
subroutine ssptrs (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
subroutine sstegr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
subroutine sstein (N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
subroutine sstemr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
subroutine stbcon (NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK, IWORK, INFO)
subroutine stbrfs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
subroutine stbtrs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
subroutine stfsm (TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine stftri (TRANSR, UPLO, DIAG, N, A, INFO)
subroutine stfttp (TRANSR, UPLO, N, ARF, AP, INFO)
STFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
subroutine stfttr (TRANSR, UPLO, N, ARF, A, LDA, INFO)
STFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
subroutine stgsen (IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
subroutine stgsja (JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
subroutine stgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
subroutine stpcon (NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK, INFO)
subroutine stpmqrt (SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
subroutine stpqrt (M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
subroutine stpqrt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
STPQRT2 computes a QR factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine stprfs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
subroutine stptri (UPLO, DIAG, N, AP, INFO)
subroutine stptrs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO)
subroutine stpttf (TRANSR, UPLO, N, AP, ARF, INFO)
STPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
subroutine stpttr (UPLO, N, AP, A, LDA, INFO)
STPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
subroutine strcon (NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK, IWORK, INFO)
subroutine strevc (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
subroutine strevc3 (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, INFO)
subroutine strexc (COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, INFO)
subroutine strrfs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
subroutine strsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO)
subroutine strsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO)
subroutine strti2 (UPLO, DIAG, N, A, LDA, INFO)
STRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
subroutine strtri (UPLO, DIAG, N, A, LDA, INFO)
subroutine strtrs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
subroutine strttf (TRANSR, UPLO, N, A, LDA, ARF, INFO)
STRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
subroutine strttp (UPLO, N, A, LDA, AP, INFO)
STRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
subroutine stzrzf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
Detailed Description
This is the group of real other Computational routines
Function Documentation
subroutine sbbcsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, integer M, integer P, integer Q, real, dimension( * ) THETA, real, dimension( * ) PHI, real,
dimension( ldu1, * ) U1, integer LDU1, real, dimension( ldu2, * ) U2, integer LDU2, real, dimension( ldv1t, * ) V1T, integer LDV1T, real, dimension( ldv2t, * ) V2T, integer LDV2T, real, dimension( *
) B11D, real, dimension( * ) B11E, real, dimension( * ) B12D, real, dimension( * ) B12E, real, dimension( * ) B21D, real, dimension( * ) B21E, real, dimension( * ) B22D, real, dimension( * ) B22E,
real, dimension( * ) WORK, integer LWORK, integer INFO)
SBBCSD computes the CS decomposition of an orthogonal matrix in
bidiagonal-block form,
[ B11 | B12 0 0 ]
[ 0 | 0 -I 0 ]
X = [----------------]
[ B21 | B22 0 0 ]
[ 0 | 0 0 I ]
[ C | -S 0 0 ]
[ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**T
= [---------] [---------------] [---------] .
[ | U2 ] [ S | C 0 0 ] [ | V2 ]
[ 0 | 0 0 I ]
X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
transposed and/or permuted. This can be done in constant time using
the TRANS and SIGNS options. See SORCSD for details.)
The bidiagonal matrices B11, B12, B21, and B22 are represented
implicitly by angles THETA(1:Q) and PHI(1:Q-1).
The orthogonal matrices U1, U2, V1T, and V2T are input/output.
The input matrices are pre- or post-multiplied by the appropriate
singular vector matrices.
JOBU1 is CHARACTER
= ’Y’: U1 is updated;
otherwise: U1 is not updated.
JOBU2 is CHARACTER
= ’Y’: U2 is updated;
otherwise: U2 is not updated.
JOBV1T is CHARACTER
= ’Y’: V1T is updated;
otherwise: V1T is not updated.
JOBV2T is CHARACTER
= ’Y’: V2T is updated;
otherwise: V2T is not updated.
TRANS is CHARACTER
= ’T’: X, U1, U2, V1T, and V2T are stored in row-major
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
M is INTEGER
The number of rows and columns in X, the orthogonal matrix in
bidiagonal-block form.
P is INTEGER
The number of rows in the top-left block of X. 0 <= P <= M.
Q is INTEGER
The number of columns in the top-left block of X.
0 <= Q <= MIN(P,M-P,M-Q).
THETA is REAL array, dimension (Q)
On entry, the angles THETA(1),...,THETA(Q) that, along with
PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
form. On exit, the angles whose cosines and sines define the
diagonal blocks in the CS decomposition.
PHI is REAL array, dimension (Q-1)
The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
THETA(Q), define the matrix in bidiagonal-block form.
U1 is REAL array, dimension (LDU1,P)
On entry, a P-by-P matrix. On exit, U1 is postmultiplied
by the left singular vector matrix common to [ B11 ; 0 ] and
[ B12 0 0 ; 0 -I 0 0 ].
LDU1 is INTEGER
The leading dimension of the array U1, LDU1 >= MAX(1,P).
U2 is REAL array, dimension (LDU2,M-P)
On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
postmultiplied by the left singular vector matrix common to
[ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].
LDU2 is INTEGER
The leading dimension of the array U2, LDU2 >= MAX(1,M-P).
V1T is REAL array, dimension (LDV1T,Q)
On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
by the transpose of the right singular vector
matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
LDV1T is INTEGER
The leading dimension of the array V1T, LDV1T >= MAX(1,Q).
V2T is REAL array, dimenison (LDV2T,M-Q)
On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
premultiplied by the transpose of the right
singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
[ B22 0 0 ; 0 0 I ].
LDV2T is INTEGER
The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).
B11D is REAL array, dimension (Q)
When SBBCSD converges, B11D contains the cosines of THETA(1),
..., THETA(Q). If SBBCSD fails to converge, then B11D
contains the diagonal of the partially reduced top-left
B11E is REAL array, dimension (Q-1)
When SBBCSD converges, B11E contains zeros. If SBBCSD fails
to converge, then B11E contains the superdiagonal of the
partially reduced top-left block.
B12D is REAL array, dimension (Q)
When SBBCSD converges, B12D contains the negative sines of
THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then
B12D contains the diagonal of the partially reduced top-right
B12E is REAL array, dimension (Q-1)
When SBBCSD converges, B12E contains zeros. If SBBCSD fails
to converge, then B12E contains the subdiagonal of the
partially reduced top-right block.
B21D is REAL array, dimension (Q)
When SBBCSD converges, B21D contains the negative sines of
THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then
B21D contains the diagonal of the partially reduced bottom-left
B21E is REAL array, dimension (Q-1)
When SBBCSD converges, B21E contains zeros. If SBBCSD fails
to converge, then B21E contains the subdiagonal of the
partially reduced bottom-left block.
B22D is REAL array, dimension (Q)
When SBBCSD converges, B22D contains the negative sines of
THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then
B22D contains the diagonal of the partially reduced bottom-right
B22E is REAL array, dimension (Q-1)
When SBBCSD converges, B22E contains zeros. If SBBCSD fails
to converge, then B22E contains the subdiagonal of the
partially reduced bottom-right block.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= MAX(1,8*Q).
If LWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the WORK array,
returns this value as the first entry of the work array, and
no error message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if SBBCSD did not converge, INFO specifies the number
of nonzero entries in PHI, and B11D, B11E, etc.,
contain the partially reduced matrix.
Internal Parameters:
TOLMUL REAL, default = MAX(10,MIN(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop.
Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
are within TOLMUL*EPS of either bound.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
June 2016
subroutine sgghd3 (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldq, * )
Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGGHD3 reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a
general matrix and B is upper triangular. The form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.
The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then SGGHD3 reduces the original
problem to generalized Hessenberg form.
This is a blocked variant of SGGHRD, using matrix-matrix
multiplications for parts of the computation to enhance performance.
COMPQ is CHARACTER*1
= ’N’: do not compute Q;
= ’I’: Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= ’V’: Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ is CHARACTER*1
= ’N’: do not compute Z;
= ’I’: Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= ’V’: Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
N is INTEGER
The order of the matrices A and B. N >= 0.
ILO is INTEGER
IHI is INTEGER
ILO and IHI mark the rows and columns of A which are to be
reduced. It is assumed that A is already upper triangular
in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
normally set by a previous call to SGGBAL; otherwise they
should be set to 1 and N respectively.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A is REAL array, dimension (LDA, N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
rest is set to zero.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, the upper triangular matrix T = Q**T B Z. The
elements below the diagonal are set to zero.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = ’V’, the orthogonal matrix Q1,
typically from the QR factorization of B.
On exit, if COMPQ=’I’, the orthogonal matrix Q, and if
COMPQ = ’V’, the product Q1*Q.
Not referenced if COMPQ=’N’.
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ=’V’ or ’I’; LDQ >= 1 otherwise.
Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = ’V’, the orthogonal matrix Z1.
On exit, if COMPZ=’I’, the orthogonal matrix Z, and if
COMPZ = ’V’, the product Z1*Z.
Not referenced if COMPZ=’N’.
LDZ is INTEGER
The leading dimension of the array Z.
LDZ >= N if COMPZ=’V’ or ’I’; LDZ >= 1 otherwise.
WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The length of the array WORK. LWORK >= 1.
For optimum performance LWORK >= 6*N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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Further Details:
This routine reduces A to Hessenberg form and maintains B in
using a blocked variant of Moler and Stewart’s original algorithm,
as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
(BIT 2008).
subroutine sgghrd (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldq, * )
Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, integer INFO)
SGGHRD reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a
general matrix and B is upper triangular. The form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.
The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then SGGHRD reduces the original
problem to generalized Hessenberg form.
COMPQ is CHARACTER*1
= ’N’: do not compute Q;
= ’I’: Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= ’V’: Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ is CHARACTER*1
= ’N’: do not compute Z;
= ’I’: Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= ’V’: Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
N is INTEGER
The order of the matrices A and B. N >= 0.
ILO is INTEGER
IHI is INTEGER
ILO and IHI mark the rows and columns of A which are to be
reduced. It is assumed that A is already upper triangular
in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
normally set by a previous call to SGGBAL; otherwise they
should be set to 1 and N respectively.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A is REAL array, dimension (LDA, N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
rest is set to zero.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, the upper triangular matrix T = Q**T B Z. The
elements below the diagonal are set to zero.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = ’V’, the orthogonal matrix Q1,
typically from the QR factorization of B.
On exit, if COMPQ=’I’, the orthogonal matrix Q, and if
COMPQ = ’V’, the product Q1*Q.
Not referenced if COMPQ=’N’.
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ=’V’ or ’I’; LDQ >= 1 otherwise.
Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = ’V’, the orthogonal matrix Z1.
On exit, if COMPZ=’I’, the orthogonal matrix Z, and if
COMPZ = ’V’, the product Z1*Z.
Not referenced if COMPZ=’N’.
LDZ is INTEGER
The leading dimension of the array Z.
LDZ >= N if COMPZ=’V’ or ’I’; LDZ >= 1 otherwise.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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Further Details:
This routine reduces A to Hessenberg and B to triangular form by
an unblocked reduction, as described in _Matrix_Computations_,
by Golub and Van Loan (Johns Hopkins Press.)
subroutine sggqrf (integer N, integer M, integer P, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAUA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) TAUB, real,
dimension( * ) WORK, integer LWORK, integer INFO)
SGGQRF computes a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B:
A = Q*R, B = Q*T*Z,
where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:
if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
( 0 ) N-M N M-N
where R11 is upper triangular, and
if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
P-N N ( T21 ) P
where T12 or T21 is upper triangular.
In particular, if B is square and nonsingular, the GQR factorization
of A and B implicitly gives the QR factorization of inv(B)*A:
inv(B)*A = Z**T*(inv(T)*R)
where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.
N is INTEGER
The number of rows of the matrices A and B. N >= 0.
M is INTEGER
The number of columns of the matrix A. M >= 0.
P is INTEGER
The number of columns of the matrix B. P >= 0.
A is REAL array, dimension (LDA,M)
On entry, the N-by-M matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(N,M)-by-M upper trapezoidal matrix R (R is
upper triangular if N >= M); the elements below the diagonal,
with the array TAUA, represent the orthogonal matrix Q as a
product of min(N,M) elementary reflectors (see Further
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAUA is REAL array, dimension (min(N,M))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q (see Further Details).
B is REAL array, dimension (LDB,P)
On entry, the N-by-P matrix B.
On exit, if N <= P, the upper triangle of the subarray
B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
if N > P, the elements on and above the (N-P)-th subdiagonal
contain the N-by-P upper trapezoidal matrix T; the remaining
elements, with the array TAUB, represent the orthogonal
matrix Z as a product of elementary reflectors (see Further
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
TAUB is REAL array, dimension (min(N,P))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Z (see Further Details).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N,M,P).
For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
where NB1 is the optimal blocksize for the QR factorization
of an N-by-M matrix, NB2 is the optimal blocksize for the
RQ factorization of an N-by-P matrix, and NB3 is the optimal
blocksize for a call of SORMQR.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
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Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(n,m).
Each H(i) has the form
H(i) = I - taua * v * v**T
where taua is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine SORGQR.
To use Q to update another matrix, use LAPACK subroutine SORMQR.
The matrix Z is represented as a product of elementary reflectors
Z = H(1) H(2) . . . H(k), where k = min(n,p).
Each H(i) has the form
H(i) = I - taub * v * v**T
where taub is a real scalar, and v is a real vector with
v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGRQ.
To use Z to update another matrix, use LAPACK subroutine SORMRQ.
subroutine sggrqf (integer M, integer P, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAUA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) TAUB, real,
dimension( * ) WORK, integer LWORK, integer INFO)
SGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:
A = R*Q, B = Z*T*Q,
where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:
if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
where R12 or R21 is upper triangular, and
if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
where T11 is upper triangular.
In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):
A*inv(B) = (R*inv(T))*Z**T
where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.
M is INTEGER
The number of rows of the matrix A. M >= 0.
P is INTEGER
The number of rows of the matrix B. P >= 0.
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, if M <= N, the upper triangle of the subarray
A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
if M > N, the elements on and above the (M-N)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R; the remaining
elements, with the array TAUA, represent the orthogonal
matrix Q as a product of elementary reflectors (see Further
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAUA is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q (see Further Details).
B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, the elements on and above the diagonal of the array
contain the min(P,N)-by-N upper trapezoidal matrix T (T is
upper triangular if P >= N); the elements below the diagonal,
with the array TAUB, represent the orthogonal matrix Z as a
product of elementary reflectors (see Further Details).
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).
TAUB is REAL array, dimension (min(P,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Z (see Further Details).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N,M,P).
For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
where NB1 is the optimal blocksize for the RQ factorization
of an M-by-N matrix, NB2 is the optimal blocksize for the
QR factorization of a P-by-N matrix, and NB3 is the optimal
blocksize for a call of SORMRQ.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INF0= -i, the i-th argument had an illegal value.
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Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - taua * v * v**T
where taua is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine SORGRQ.
To use Q to update another matrix, use LAPACK subroutine SORMRQ.
The matrix Z is represented as a product of elementary reflectors
Z = H(1) H(2) . . . H(k), where k = min(p,n).
Each H(i) has the form
H(i) = I - taub * v * v**T
where taub is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGQR.
To use Z to update another matrix, use LAPACK subroutine SORMQR.
subroutine sggsvp3 (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real TOLA,
real TOLB, integer K, integer L, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldq, * ) Q, integer LDQ, integer, dimension( * ) IWORK, real,
dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGGSVP3 computes orthogonal matrices U, V and Q such that
N-K-L K L
U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**T*B*Q = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
JOBU is CHARACTER*1
= ’U’: Orthogonal matrix U is computed;
= ’N’: U is not computed.
JOBV is CHARACTER*1
= ’V’: Orthogonal matrix V is computed;
= ’N’: V is not computed.
JOBQ is CHARACTER*1
= ’Q’: Orthogonal matrix Q is computed;
= ’N’: Q is not computed.
M is INTEGER
The number of rows of the matrix A. M >= 0.
P is INTEGER
The number of rows of the matrix B. P >= 0.
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix described in
the Purpose section.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).
TOLA is REAL
TOLB is REAL
TOLA and TOLB are the thresholds to determine the effective
numerical rank of matrix B and a subblock of A. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
K is INTEGER
L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section.
K + L = effective numerical rank of (A**T,B**T)**T.
U is REAL array, dimension (LDU,M)
If JOBU = ’U’, U contains the orthogonal matrix U.
If JOBU = ’N’, U is not referenced.
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = ’U’; LDU >= 1 otherwise.
V is REAL array, dimension (LDV,P)
If JOBV = ’V’, V contains the orthogonal matrix V.
If JOBV = ’N’, V is not referenced.
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = ’V’; LDV >= 1 otherwise.
Q is REAL array, dimension (LDQ,N)
If JOBQ = ’Q’, Q contains the orthogonal matrix Q.
If JOBQ = ’N’, Q is not referenced.
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = ’Q’; LDQ >= 1 otherwise.
IWORK is INTEGER array, dimension (N)
TAU is REAL array, dimension (N)
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
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Further Details:
The subroutine uses LAPACK subroutine SGEQP3 for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.
SGGSVP3 replaces the deprecated subroutine SGGSVP.
subroutine sgsvj0 (character*1 JOBV, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( n ) D, real, dimension( n ) SVA, integer MV, real, dimension( ldv, * ) V, integer
LDV, real EPS, real SFMIN, real TOL, integer NSWEEP, real, dimension( lwork ) WORK, integer LWORK, integer INFO)
SGSVJ0 pre-processor for the routine sgesvj.
SGSVJ0 is called from SGESVJ as a pre-processor and that is its main
purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
it does not check convergence (stopping criterion). Few tuning
parameters (marked by [TP]) are available for the implementer.
JOBV is CHARACTER*1
Specifies whether the output from this procedure is used
to compute the matrix V:
= ’V’: the product of the Jacobi rotations is accumulated
by postmulyiplying the N-by-N array V.
(See the description of V.)
= ’A’: the product of the Jacobi rotations is accumulated
by postmulyiplying the MV-by-N array V.
(See the descriptions of MV and V.)
= ’N’: the Jacobi rotations are not accumulated.
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0.
A is REAL array, dimension (LDA,N)
On entry, M-by-N matrix A, such that A*diag(D) represents
the input matrix.
On exit,
A_onexit * D_onexit represents the input matrix A*diag(D)
post-multiplied by a sequence of Jacobi rotations, where the
rotation threshold and the total number of sweeps are given in
TOL and NSWEEP, respectively.
(See the descriptions of D, TOL and NSWEEP.)
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D is REAL array, dimension (N)
The array D accumulates the scaling factors from the fast scaled
Jacobi rotations.
On entry, A*diag(D) represents the input matrix.
On exit, A_onexit*diag(D_onexit) represents the input matrix
post-multiplied by a sequence of Jacobi rotations, where the
rotation threshold and the total number of sweeps are given in
TOL and NSWEEP, respectively.
(See the descriptions of A, TOL and NSWEEP.)
SVA is REAL array, dimension (N)
On entry, SVA contains the Euclidean norms of the columns of
the matrix A*diag(D).
On exit, SVA contains the Euclidean norms of the columns of
the matrix onexit*diag(D_onexit).
MV is INTEGER
If JOBV .EQ. ’A’, then MV rows of V are post-multipled by a
sequence of Jacobi rotations.
If JOBV = ’N’, then MV is not referenced.
V is REAL array, dimension (LDV,N)
If JOBV .EQ. ’V’ then N rows of V are post-multipled by a
sequence of Jacobi rotations.
If JOBV .EQ. ’A’ then MV rows of V are post-multipled by a
sequence of Jacobi rotations.
If JOBV = ’N’, then V is not referenced.
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = ’V’, LDV .GE. N.
If JOBV = ’A’, LDV .GE. MV.
EPS is REAL
EPS = SLAMCH(’Epsilon’)
SFMIN is REAL
SFMIN = SLAMCH(’Safe Minimum’)
TOL is REAL
TOL is the threshold for Jacobi rotations. For a pair
A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
NSWEEP is INTEGER
NSWEEP is the number of sweeps of Jacobi rotations to be
WORK is REAL array, dimension LWORK.
LWORK is INTEGER
LWORK is the dimension of WORK. LWORK .GE. M.
INFO is INTEGER
= 0 : successful exit.
< 0 : if INFO = -i, then the i-th argument had an illegal value
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December 2016
Further Details:
SGSVJ0 is used just to enable SGESVJ to call a simplified version of itself to work on a submatrix of the original matrix.
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
Bugs, Examples and Comments:
Please report all bugs and send interesting test examples and comments to drmac AT math DOT hr. Thank you.
subroutine sgsvj1 (character*1 JOBV, integer M, integer N, integer N1, real, dimension( lda, * ) A, integer LDA, real, dimension( n ) D, real, dimension( n ) SVA, integer MV, real, dimension( ldv, *
) V, integer LDV, real EPS, real SFMIN, real TOL, integer NSWEEP, real, dimension( lwork ) WORK, integer LWORK, integer INFO)
SGSVJ1 pre-processor for the routine sgesvj, applies Jacobi rotations targeting only particular pivots.
SGSVJ1 is called from SGESVJ as a pre-processor and that is its main
purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
it targets only particular pivots and it does not check convergence
(stopping criterion). Few tunning parameters (marked by [TP]) are
available for the implementer.
Further Details
SGSVJ1 applies few sweeps of Jacobi rotations in the column space of
the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
block-entries (tiles) of the (1,2) off-diagonal block are marked by the
[x]’s in the following scheme:
| * * * [x] [x] [x]|
| * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
| * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
|[x] [x] [x] * * * |
|[x] [x] [x] * * * |
|[x] [x] [x] * * * |
In terms of the columns of A, the first N1 columns are rotated ’against’
the remaining N-N1 columns, trying to increase the angle between the
corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.
The number of sweeps is given in NSWEEP and the orthogonality threshold
is given in TOL.
JOBV is CHARACTER*1
Specifies whether the output from this procedure is used
to compute the matrix V:
= ’V’: the product of the Jacobi rotations is accumulated
by postmulyiplying the N-by-N array V.
(See the description of V.)
= ’A’: the product of the Jacobi rotations is accumulated
by postmulyiplying the MV-by-N array V.
(See the descriptions of MV and V.)
= ’N’: the Jacobi rotations are not accumulated.
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0.
N1 is INTEGER
N1 specifies the 2 x 2 block partition, the first N1 columns are
rotated ’against’ the remaining N-N1 columns of A.
A is REAL array, dimension (LDA,N)
On entry, M-by-N matrix A, such that A*diag(D) represents
the input matrix.
On exit,
A_onexit * D_onexit represents the input matrix A*diag(D)
post-multiplied by a sequence of Jacobi rotations, where the
rotation threshold and the total number of sweeps are given in
TOL and NSWEEP, respectively.
(See the descriptions of N1, D, TOL and NSWEEP.)
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D is REAL array, dimension (N)
The array D accumulates the scaling factors from the fast scaled
Jacobi rotations.
On entry, A*diag(D) represents the input matrix.
On exit, A_onexit*diag(D_onexit) represents the input matrix
post-multiplied by a sequence of Jacobi rotations, where the
rotation threshold and the total number of sweeps are given in
TOL and NSWEEP, respectively.
(See the descriptions of N1, A, TOL and NSWEEP.)
SVA is REAL array, dimension (N)
On entry, SVA contains the Euclidean norms of the columns of
the matrix A*diag(D).
On exit, SVA contains the Euclidean norms of the columns of
the matrix onexit*diag(D_onexit).
MV is INTEGER
If JOBV .EQ. ’A’, then MV rows of V are post-multipled by a
sequence of Jacobi rotations.
If JOBV = ’N’, then MV is not referenced.
V is REAL array, dimension (LDV,N)
If JOBV .EQ. ’V’ then N rows of V are post-multipled by a
sequence of Jacobi rotations.
If JOBV .EQ. ’A’ then MV rows of V are post-multipled by a
sequence of Jacobi rotations.
If JOBV = ’N’, then V is not referenced.
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = ’V’, LDV .GE. N.
If JOBV = ’A’, LDV .GE. MV.
EPS is REAL
EPS = SLAMCH(’Epsilon’)
SFMIN is REAL
SFMIN = SLAMCH(’Safe Minimum’)
TOL is REAL
TOL is the threshold for Jacobi rotations. For a pair
A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
NSWEEP is INTEGER
NSWEEP is the number of sweeps of Jacobi rotations to be
WORK is REAL array, dimension LWORK.
LWORK is INTEGER
LWORK is the dimension of WORK. LWORK .GE. M.
INFO is INTEGER
= 0 : successful exit.
< 0 : if INFO = -i, then the i-th argument had an illegal value
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Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
subroutine shsein (character SIDE, character EIGSRC, character INITV, logical, dimension( * ) SELECT, integer N, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( *
) WI, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * ) WORK, integer, dimension( * ) IFAILL, integer, dimension( *
) IFAILR, integer INFO)
SHSEIN uses inverse iteration to find specified right and/or left
eigenvectors of a real upper Hessenberg matrix H.
The right eigenvector x and the left eigenvector y of the matrix H
corresponding to an eigenvalue w are defined by:
H * x = w * x, y**h * H = w * y**h
where y**h denotes the conjugate transpose of the vector y.
SIDE is CHARACTER*1
= ’R’: compute right eigenvectors only;
= ’L’: compute left eigenvectors only;
= ’B’: compute both right and left eigenvectors.
EIGSRC is CHARACTER*1
Specifies the source of eigenvalues supplied in (WR,WI):
= ’Q’: the eigenvalues were found using SHSEQR; thus, if
H has zero subdiagonal elements, and so is
block-triangular, then the j-th eigenvalue can be
assumed to be an eigenvalue of the block containing
the j-th row/column. This property allows SHSEIN to
perform inverse iteration on just one diagonal block.
= ’N’: no assumptions are made on the correspondence
between eigenvalues and diagonal blocks. In this
case, SHSEIN must always perform inverse iteration
using the whole matrix H.
INITV is CHARACTER*1
= ’N’: no initial vectors are supplied;
= ’U’: user-supplied initial vectors are stored in the arrays
VL and/or VR.
SELECT is LOGICAL array, dimension (N)
Specifies the eigenvectors to be computed. To select the
real eigenvector corresponding to a real eigenvalue WR(j),
SELECT(j) must be set to .TRUE.. To select the complex
eigenvector corresponding to a complex eigenvalue
(WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
N is INTEGER
The order of the matrix H. N >= 0.
H is REAL array, dimension (LDH,N)
The upper Hessenberg matrix H.
If a NaN is detected in H, the routine will return with INFO=-6.
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
WR is REAL array, dimension (N)
WI is REAL array, dimension (N)
On entry, the real and imaginary parts of the eigenvalues of
H; a complex conjugate pair of eigenvalues must be stored in
consecutive elements of WR and WI.
On exit, WR may have been altered since close eigenvalues
are perturbed slightly in searching for independent
VL is REAL array, dimension (LDVL,MM)
On entry, if INITV = ’U’ and SIDE = ’L’ or ’B’, VL must
contain starting vectors for the inverse iteration for the
left eigenvectors; the starting vector for each eigenvector
must be in the same column(s) in which the eigenvector will
be stored.
On exit, if SIDE = ’L’ or ’B’, the left eigenvectors
specified by SELECT will be stored consecutively in the
columns of VL, in the same order as their eigenvalues. A
complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = ’R’, VL is not referenced.
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= max(1,N) if SIDE = ’L’ or ’B’; LDVL >= 1 otherwise.
VR is REAL array, dimension (LDVR,MM)
On entry, if INITV = ’U’ and SIDE = ’R’ or ’B’, VR must
contain starting vectors for the inverse iteration for the
right eigenvectors; the starting vector for each eigenvector
must be in the same column(s) in which the eigenvector will
be stored.
On exit, if SIDE = ’R’ or ’B’, the right eigenvectors
specified by SELECT will be stored consecutively in the
columns of VR, in the same order as their eigenvalues. A
complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = ’L’, VR is not referenced.
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= max(1,N) if SIDE = ’R’ or ’B’; LDVR >= 1 otherwise.
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M is INTEGER
The number of columns in the arrays VL and/or VR required to
store the eigenvectors; each selected real eigenvector
occupies one column and each selected complex eigenvector
occupies two columns.
WORK is REAL array, dimension ((N+2)*N)
IFAILL is INTEGER array, dimension (MM)
If SIDE = ’L’ or ’B’, IFAILL(i) = j > 0 if the left
eigenvector in the i-th column of VL (corresponding to the
eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
eigenvector converged satisfactorily. If the i-th and (i+1)th
columns of VL hold a complex eigenvector, then IFAILL(i) and
IFAILL(i+1) are set to the same value.
If SIDE = ’R’, IFAILL is not referenced.
IFAILR is INTEGER array, dimension (MM)
If SIDE = ’R’ or ’B’, IFAILR(i) = j > 0 if the right
eigenvector in the i-th column of VR (corresponding to the
eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
eigenvector converged satisfactorily. If the i-th and (i+1)th
columns of VR hold a complex eigenvector, then IFAILR(i) and
IFAILR(i+1) are set to the same value.
If SIDE = ’L’, IFAILR is not referenced.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, i is the number of eigenvectors which
failed to converge; see IFAILL and IFAILR for further
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Further Details:
Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x|+|y|.
subroutine shseqr (character JOB, character COMPZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension(
ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)
SHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.
Optionally Z may be postmultiplied into an input orthogonal
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
JOB is CHARACTER*1
= ’E’: compute eigenvalues only;
= ’S’: compute eigenvalues and the Schur form T.
COMPZ is CHARACTER*1
= ’N’: no Schur vectors are computed;
= ’I’: Z is initialized to the unit matrix and the matrix Z
of Schur vectors of H is returned;
= ’V’: Z must contain an orthogonal matrix Q on entry, and
the product Q*Z is returned.
N is INTEGER
The order of the matrix H. N .GE. 0.
ILO is INTEGER
IHI is INTEGER
It is assumed that H is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL, and then passed to ZGEHRD
when the matrix output by SGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
If N = 0, then ILO = 1 and IHI = 0.
H is REAL array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and JOB = ’S’, then H contains the
upper quasi-triangular matrix T from the Schur decomposition
(the Schur form); 2-by-2 diagonal blocks (corresponding to
complex conjugate pairs of eigenvalues) are returned in
standard form, with H(i,i) = H(i+1,i+1) and
H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = ’E’, the
contents of H are unspecified on exit. (The output value of
H when INFO.GT.0 is given under the description of INFO
Unlike earlier versions of SHSEQR, this subroutine may
explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
or j = IHI+1, IHI+2, ... N.
LDH is INTEGER
The leading dimension of the array H. LDH .GE. max(1,N).
WR is REAL array, dimension (N)
WI is REAL array, dimension (N)
The real and imaginary parts, respectively, of the computed
eigenvalues. If two eigenvalues are computed as a complex
conjugate pair, they are stored in consecutive elements of
WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and
WI(i+1) .LT. 0. If JOB = ’S’, the eigenvalues are stored in
the same order as on the diagonal of the Schur form returned
in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
WI(i+1) = -WI(i).
Z is REAL array, dimension (LDZ,N)
If COMPZ = ’N’, Z is not referenced.
If COMPZ = ’I’, on entry Z need not be set and on exit,
if INFO = 0, Z contains the orthogonal matrix Z of the Schur
vectors of H. If COMPZ = ’V’, on entry Z must contain an
N-by-N matrix Q, which is assumed to be equal to the unit
matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
if INFO = 0, Z contains Q*Z.
Normally Q is the orthogonal matrix generated by SORGHR
after the call to SGEHRD which formed the Hessenberg matrix
H. (The output value of Z when INFO.GT.0 is given under
the description of INFO below.)
LDZ is INTEGER
The leading dimension of the array Z. if COMPZ = ’I’ or
COMPZ = ’V’, then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of
the optimal value for LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK .GE. max(1,N)
is sufficient and delivers very good and sometimes
optimal performance. However, LWORK as large as 11*N
may be required for optimal performance. A workspace
query is recommended to determine the optimal workspace
If LWORK = -1, then SHSEQR does a workspace query.
In this case, SHSEQR checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI. The estimate is returned
in WORK(1). No error message related to LWORK is
issued by XERBLA. Neither H nor Z are accessed.
INFO is INTEGER
= 0: successful exit
.LT. 0: if INFO = -i, the i-th argument had an illegal
.GT. 0: if INFO = i, SHSEQR failed to compute all of
the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed. (Failures are rare.)
If INFO .GT. 0 and JOB = ’E’, then on exit, the
remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.
If INFO .GT. 0 and JOB = ’S’, then on exit
(*) (initial value of H)*U = U*(final value of H)
where U is an orthogonal matrix. The final
value of H is upper Hessenberg and quasi-triangular
in rows and columns INFO+1 through IHI.
If INFO .GT. 0 and COMPZ = ’V’, then on exit
(final value of Z) = (initial value of Z)*U
where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)
If INFO .GT. 0 and COMPZ = ’I’, then on exit
(final value of Z) = U
where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)
If INFO .GT. 0 and COMPZ = ’N’, then Z is not
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Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
Further Details:
Default values supplied by
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.
ISPEC=12: The SLAHQR vs SLAQR0 crossover point.
Default: 75. (Must be at least 11.)
ISPEC=13: Recommended deflation window size.
This depends on ILO, IHI and NS. NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=15). (See ISPEC=15 below.)
The default for (IHI-ILO+1).LE.500 is NS.
The default for (IHI-ILO+1).GT.500 is 3*NS/2.
ISPEC=14: Nibble crossover point. (See IPARMQ for
details.) Default: 14% of deflation window
ISPEC=15: Number of simultaneous shifts in a multishift
QR iteration.
If IHI-ILO+1 is ...
greater than ...but less ... the
or equal to ... than default is
1 30 NS = 2(+)
30 60 NS = 4(+)
60 150 NS = 10(+)
150 590 NS = **
590 3000 NS = 64
3000 6000 NS = 128
6000 infinity NS = 256
(+) By default some or all matrices of this order
are passed to the implicit double shift routine
SLAHQR and this parameter is ignored. See
ISPEC=12 above and comments in IPARMQ for
(**) The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.
ISPEC=16: Select structured matrix multiply.
If the number of simultaneous shifts (specified
by ISPEC=15) is less than 14, then the default
for ISPEC=16 is 0. Otherwise the default for
ISPEC=16 is 2.
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.
subroutine sla_lin_berr (integer N, integer NZ, integer NRHS, real, dimension( n, nrhs ) RES, real, dimension( n, nrhs ) AYB, real, dimension( nrhs ) BERR)
SLA_LIN_BERR computes a component-wise relative backward error.
SLA_LIN_BERR computes componentwise relative backward error from
the formula
max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z.
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NZ is INTEGER
We add (NZ+1)*SLAMCH( ’Safe minimum’ ) to R(i) in the numerator to
guard against spuriously zero residuals. Default value is N.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices AYB, RES, and BERR. NRHS >= 0.
RES is REAL array, dimension (N,NRHS)
The residual matrix, i.e., the matrix R in the relative backward
error formula above.
AYB is REAL array, dimension (N, NRHS)
The denominator in the relative backward error formula above, i.e.,
the matrix abs(op(A_s))*abs(Y) + abs(B_s). The matrices A, Y, and B
are from iterative refinement (see sla_gerfsx_extended.f).
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error from the formula above.
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subroutine sla_wwaddw (integer N, real, dimension( * ) X, real, dimension( * ) Y, real, dimension( * ) W)
SLA_WWADDW adds a vector into a doubled-single vector.
SLA_WWADDW adds a vector W into a doubled-single vector (X, Y).
This works for all extant IBM’s hex and binary floating point
arithmetics, but not for decimal.
N is INTEGER
The length of vectors X, Y, and W.
X is REAL array, dimension (N)
The first part of the doubled-single accumulation vector.
Y is REAL array, dimension (N)
The second part of the doubled-single accumulation vector.
W is REAL array, dimension (N)
The vector to be added.
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subroutine slals0 (integer ICOMPQ, integer NL, integer NR, integer SQRE, integer NRHS, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldbx, * ) BX, integer LDBX, integer, dimension( * )
PERM, integer GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer LDGCOL, real, dimension( ldgnum, * ) GIVNUM, integer LDGNUM, real, dimension( ldgnum, * ) POLES, real, dimension( * ) DIFL, real,
dimension( ldgnum, * ) DIFR, real, dimension( * ) Z, integer K, real C, real S, real, dimension( * ) WORK, integer INFO)
SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
SLALS0 applies back the multiplying factors of either the left or the
right singular vector matrix of a diagonal matrix appended by a row
to the right hand side matrix B in solving the least squares problem
using the divide-and-conquer SVD approach.
For the left singular vector matrix, three types of orthogonal
matrices are involved:
(1L) Givens rotations: the number of such rotations is GIVPTR; the
pairs of columns/rows they were applied to are stored in GIVCOL;
and the C- and S-values of these rotations are stored in GIVNUM.
(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
row, and for J=2:N, PERM(J)-th row of B is to be moved to the
J-th row.
(3L) The left singular vector matrix of the remaining matrix.
For the right singular vector matrix, four types of orthogonal
matrices are involved:
(1R) The right singular vector matrix of the remaining matrix.
(2R) If SQRE = 1, one extra Givens rotation to generate the right
null space.
(3R) The inverse transformation of (2L).
(4R) The inverse transformation of (1L).
ICOMPQ is INTEGER
Specifies whether singular vectors are to be computed in
factored form:
= 0: Left singular vector matrix.
= 1: Right singular vector matrix.
NL is INTEGER
The row dimension of the upper block. NL >= 1.
NR is INTEGER
The row dimension of the lower block. NR >= 1.
SQRE is INTEGER
= 0: the lower block is an NR-by-NR square matrix.
= 1: the lower block is an NR-by-(NR+1) rectangular matrix.
The bidiagonal matrix has row dimension N = NL + NR + 1,
and column dimension M = N + SQRE.
NRHS is INTEGER
The number of columns of B and BX. NRHS must be at least 1.
B is REAL array, dimension ( LDB, NRHS )
On input, B contains the right hand sides of the least
squares problem in rows 1 through M. On output, B contains
the solution X in rows 1 through N.
LDB is INTEGER
The leading dimension of B. LDB must be at least
max(1,MAX( M, N ) ).
BX is REAL array, dimension ( LDBX, NRHS )
LDBX is INTEGER
The leading dimension of BX.
PERM is INTEGER array, dimension ( N )
The permutations (from deflation and sorting) applied
to the two blocks.
GIVPTR is INTEGER
The number of Givens rotations which took place in this
GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
Each pair of numbers indicates a pair of rows/columns
involved in a Givens rotation.
LDGCOL is INTEGER
The leading dimension of GIVCOL, must be at least N.
GIVNUM is REAL array, dimension ( LDGNUM, 2 )
Each number indicates the C or S value used in the
corresponding Givens rotation.
LDGNUM is INTEGER
The leading dimension of arrays DIFR, POLES and
GIVNUM, must be at least K.
POLES is REAL array, dimension ( LDGNUM, 2 )
On entry, POLES(1:K, 1) contains the new singular
values obtained from solving the secular equation, and
POLES(1:K, 2) is an array containing the poles in the secular
DIFL is REAL array, dimension ( K ).
On entry, DIFL(I) is the distance between I-th updated
(undeflated) singular value and the I-th (undeflated) old
singular value.
DIFR is REAL array, dimension ( LDGNUM, 2 ).
On entry, DIFR(I, 1) contains the distances between I-th
updated (undeflated) singular value and the I+1-th
(undeflated) old singular value. And DIFR(I, 2) is the
normalizing factor for the I-th right singular vector.
Z is REAL array, dimension ( K )
Contain the components of the deflation-adjusted updating row
K is INTEGER
Contains the dimension of the non-deflated matrix,
This is the order of the related secular equation. 1 <= K <=N.
C is REAL
C contains garbage if SQRE =0 and the C-value of a Givens
rotation related to the right null space if SQRE = 1.
S is REAL
S contains garbage if SQRE =0 and the S-value of a Givens
rotation related to the right null space if SQRE = 1.
WORK is REAL array, dimension ( K )
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
subroutine slalsa (integer ICOMPQ, integer SMLSIZ, integer N, integer NRHS, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldbx, * ) BX, integer LDBX, real, dimension( ldu, * ) U, integer
LDU, real, dimension( ldu, * ) VT, integer, dimension( * ) K, real, dimension( ldu, * ) DIFL, real, dimension( ldu, * ) DIFR, real, dimension( ldu, * ) Z, real, dimension( ldu, * ) POLES, integer,
dimension( * ) GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer LDGCOL, integer, dimension( ldgcol, * ) PERM, real, dimension( ldu, * ) GIVNUM, real, dimension( * ) C, real, dimension( * ) S,
real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
SLALSA is an itermediate step in solving the least squares problem
by computing the SVD of the coefficient matrix in compact form (The
singular vectors are computed as products of simple orthorgonal
If ICOMPQ = 0, SLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in
compact form by SLALSA.
ICOMPQ is INTEGER
Specifies whether the left or the right singular vector
matrix is involved.
= 0: Left singular vector matrix
= 1: Right singular vector matrix
SMLSIZ is INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N is INTEGER
The row and column dimensions of the upper bidiagonal matrix.
NRHS is INTEGER
The number of columns of B and BX. NRHS must be at least 1.
B is REAL array, dimension ( LDB, NRHS )
On input, B contains the right hand sides of the least
squares problem in rows 1 through M.
On output, B contains the solution X in rows 1 through N.
LDB is INTEGER
The leading dimension of B in the calling subprogram.
LDB must be at least max(1,MAX( M, N ) ).
BX is REAL array, dimension ( LDBX, NRHS )
On exit, the result of applying the left or right singular
vector matrix to B.
LDBX is INTEGER
The leading dimension of BX.
U is REAL array, dimension ( LDU, SMLSIZ ).
On entry, U contains the left singular vector matrices of all
subproblems at the bottom level.
LDU is INTEGER, LDU = > N.
The leading dimension of arrays U, VT, DIFL, DIFR,
POLES, GIVNUM, and Z.
VT is REAL array, dimension ( LDU, SMLSIZ+1 ).
On entry, VT**T contains the right singular vector matrices of
all subproblems at the bottom level.
K is INTEGER array, dimension ( N ).
DIFL is REAL array, dimension ( LDU, NLVL ).
where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
DIFR is REAL array, dimension ( LDU, 2 * NLVL ).
On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
distances between singular values on the I-th level and
singular values on the (I -1)-th level, and DIFR(*, 2 * I)
record the normalizing factors of the right singular vectors
matrices of subproblems on I-th level.
Z is REAL array, dimension ( LDU, NLVL ).
On entry, Z(1, I) contains the components of the deflation-
adjusted updating row vector for subproblems on the I-th
POLES is REAL array, dimension ( LDU, 2 * NLVL ).
On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
singular values involved in the secular equations on the I-th
GIVPTR is INTEGER array, dimension ( N ).
On entry, GIVPTR( I ) records the number of Givens
rotations performed on the I-th problem on the computation
GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
locations of Givens rotations performed on the I-th level on
the computation tree.
LDGCOL is INTEGER, LDGCOL = > N.
The leading dimension of arrays GIVCOL and PERM.
PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
On entry, PERM(*, I) records permutations done on the I-th
level of the computation tree.
GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ).
On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
values of Givens rotations performed on the I-th level on the
computation tree.
C is REAL array, dimension ( N ).
On entry, if the I-th subproblem is not square,
C( I ) contains the C-value of a Givens rotation related to
the right null space of the I-th subproblem.
S is REAL array, dimension ( N ).
On entry, if the I-th subproblem is not square,
S( I ) contains the S-value of a Givens rotation related to
the right null space of the I-th subproblem.
WORK is REAL array.
The dimension must be at least N.
IWORK is INTEGER array.
The dimension must be at least 3 * N
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
subroutine slalsd (character UPLO, integer SMLSIZ, integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB, real RCOND, integer RANK, real,
dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SLALSD uses the singular value decomposition of A to solve the least squares problem.
SLALSD uses the singular value decomposition of A to solve the least
squares problem of finding X to minimize the Euclidean norm of each
column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
are N-by-NRHS. The solution X overwrites B.
The singular values of A smaller than RCOND times the largest
singular value are treated as zero in solving the least squares
problem; in this case a minimum norm solution is returned.
The actual singular values are returned in D in ascending order.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
UPLO is CHARACTER*1
= ’U’: D and E define an upper bidiagonal matrix.
= ’L’: D and E define a lower bidiagonal matrix.
SMLSIZ is INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N is INTEGER
The dimension of the bidiagonal matrix. N >= 0.
NRHS is INTEGER
The number of columns of B. NRHS must be at least 1.
D is REAL array, dimension (N)
On entry D contains the main diagonal of the bidiagonal
matrix. On exit, if INFO = 0, D contains its singular values.
E is REAL array, dimension (N-1)
Contains the super-diagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
B is REAL array, dimension (LDB,NRHS)
On input, B contains the right hand sides of the least
squares problem. On output, B contains the solution X.
LDB is INTEGER
The leading dimension of B in the calling subprogram.
LDB must be at least max(1,N).
RCOND is REAL
The singular values of A less than or equal to RCOND times
the largest singular value are treated as zero in solving
the least squares problem. If RCOND is negative,
machine precision is used instead.
For example, if diag(S)*X=B were the least squares problem,
where diag(S) is a diagonal matrix of singular values, the
solution would be X(i) = B(i) / S(i) if S(i) is greater than
RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
RANK is INTEGER
The number of singular values of A greater than RCOND times
the largest singular value.
WORK is REAL array, dimension at least
(9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
IWORK is INTEGER array, dimension at least
(3*N*NLVL + 11*N)
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute a singular value while
working on the submatrix lying in rows and columns
INFO/(N+1) through MOD(INFO,N+1).
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Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
real function slansf (character NORM, character TRANSR, character UPLO, integer N, real, dimension( 0: * ) A, real, dimension( 0: * ) WORK)
SLANSF returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
real symmetric matrix A in RFP format.
SLANSF = ( max(abs(A(i,j))), NORM = ’M’ or ’m’
( norm1(A), NORM = ’1’, ’O’ or ’o’
( normI(A), NORM = ’I’ or ’i’
( normF(A), NORM = ’F’, ’f’, ’E’ or ’e’
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a matrix norm.
NORM is CHARACTER*1
Specifies the value to be returned in SLANSF as described
TRANSR is CHARACTER*1
Specifies whether the RFP format of A is normal or
transposed format.
= ’N’: RFP format is Normal;
= ’T’: RFP format is Transpose.
UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
= ’U’: RFP A came from an upper triangular matrix;
= ’L’: RFP A came from a lower triangular matrix.
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, SLANSF is
set to zero.
A is REAL array, dimension ( N*(N+1)/2 );
On entry, the upper (if UPLO = ’U’) or lower (if UPLO = ’L’)
part of the symmetric matrix A stored in RFP format. See the
"Notes" below for more details.
Unchanged on exit.
WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = ’I’ or ’1’ or ’O’; otherwise,
WORK is not referenced.
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Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine slarscl2 (integer M, integer N, real, dimension( * ) D, real, dimension( ldx, * ) X, integer LDX)
SLARSCL2 performs reciprocal diagonal scaling on a vector.
SLARSCL2 performs a reciprocal diagonal scaling on an vector:
x <-- inv(D) * x
where the diagonal matrix D is stored as a vector.
Eventually to be replaced by BLAS_sge_diag_scale in the new BLAS
M is INTEGER
The number of rows of D and X. M >= 0.
N is INTEGER
The number of columns of X. N >= 0.
D is REAL array, length M
Diagonal matrix D, stored as a vector of length M.
X is REAL array, dimension (LDX,N)
On entry, the vector X to be scaled by D.
On exit, the scaled vector.
LDX is INTEGER
The leading dimension of the vector X. LDX >= M.
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subroutine slarz (character SIDE, integer M, integer N, integer L, real, dimension( * ) V, integer INCV, real TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK)
SLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
SLARZ applies a real elementary reflector H to a real M-by-N
matrix C, from either the left or the right. H is represented in the
H = I - tau * v * v**T
where tau is a real scalar and v is a real vector.
If tau = 0, then H is taken to be the unit matrix.
H is a product of k elementary reflectors as returned by STZRZF.
SIDE is CHARACTER*1
= ’L’: form H * C
= ’R’: form C * H
M is INTEGER
The number of rows of the matrix C.
N is INTEGER
The number of columns of the matrix C.
L is INTEGER
The number of entries of the vector V containing
the meaningful part of the Householder vectors.
If SIDE = ’L’, M >= L >= 0, if SIDE = ’R’, N >= L >= 0.
V is REAL array, dimension (1+(L-1)*abs(INCV))
The vector v in the representation of H as returned by
STZRZF. V is not used if TAU = 0.
INCV is INTEGER
The increment between elements of v. INCV <> 0.
TAU is REAL
The value tau in the representation of H.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by the matrix H * C if SIDE = ’L’,
or C * H if SIDE = ’R’.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension
(N) if SIDE = ’L’
or (M) if SIDE = ’R’
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Further Details:
subroutine slarzb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * )
T, integer LDT, real, dimension( ldc, * ) C, integer LDC, real, dimension( ldwork, * ) WORK, integer LDWORK)
SLARZB applies a block reflector or its transpose to a general matrix.
SLARZB applies a real block reflector H or its transpose H**T to
a real distributed M-by-N C from the left or the right.
Currently, only STOREV = ’R’ and DIRECT = ’B’ are supported.
SIDE is CHARACTER*1
= ’L’: apply H or H**T from the Left
= ’R’: apply H or H**T from the Right
TRANS is CHARACTER*1
= ’N’: apply H (No transpose)
= ’C’: apply H**T (Transpose)
DIRECT is CHARACTER*1
Indicates how H is formed from a product of elementary
= ’F’: H = H(1) H(2) . . . H(k) (Forward, not supported yet)
= ’B’: H = H(k) . . . H(2) H(1) (Backward)
STOREV is CHARACTER*1
Indicates how the vectors which define the elementary
reflectors are stored:
= ’C’: Columnwise (not supported yet)
= ’R’: Rowwise
M is INTEGER
The number of rows of the matrix C.
N is INTEGER
The number of columns of the matrix C.
K is INTEGER
The order of the matrix T (= the number of elementary
reflectors whose product defines the block reflector).
L is INTEGER
The number of columns of the matrix V containing the
meaningful part of the Householder reflectors.
If SIDE = ’L’, M >= L >= 0, if SIDE = ’R’, N >= L >= 0.
V is REAL array, dimension (LDV,NV).
If STOREV = ’C’, NV = K; if STOREV = ’R’, NV = L.
LDV is INTEGER
The leading dimension of the array V.
If STOREV = ’C’, LDV >= L; if STOREV = ’R’, LDV >= K.
T is REAL array, dimension (LDT,K)
The triangular K-by-K matrix T in the representation of the
block reflector.
LDT is INTEGER
The leading dimension of the array T. LDT >= K.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (LDWORK,K)
LDWORK is INTEGER
The leading dimension of the array WORK.
If SIDE = ’L’, LDWORK >= max(1,N);
if SIDE = ’R’, LDWORK >= max(1,M).
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Further Details:
subroutine slarzt (character DIRECT, character STOREV, integer N, integer K, real, dimension( ldv, * ) V, integer LDV, real, dimension( * ) TAU, real, dimension( ldt, * ) T, integer LDT)
SLARZT forms the triangular factor T of a block reflector H = I - vtvH.
SLARZT forms the triangular factor T of a real block reflector
H of order > n, which is defined as a product of k elementary
If DIRECT = ’F’, H = H(1) H(2) . . . H(k) and T is upper triangular;
If DIRECT = ’B’, H = H(k) . . . H(2) H(1) and T is lower triangular.
If STOREV = ’C’, the vector which defines the elementary reflector
H(i) is stored in the i-th column of the array V, and
H = I - V * T * V**T
If STOREV = ’R’, the vector which defines the elementary reflector
H(i) is stored in the i-th row of the array V, and
H = I - V**T * T * V
Currently, only STOREV = ’R’ and DIRECT = ’B’ are supported.
DIRECT is CHARACTER*1
Specifies the order in which the elementary reflectors are
multiplied to form the block reflector:
= ’F’: H = H(1) H(2) . . . H(k) (Forward, not supported yet)
= ’B’: H = H(k) . . . H(2) H(1) (Backward)
STOREV is CHARACTER*1
Specifies how the vectors which define the elementary
reflectors are stored (see also Further Details):
= ’C’: columnwise (not supported yet)
= ’R’: rowwise
N is INTEGER
The order of the block reflector H. N >= 0.
K is INTEGER
The order of the triangular factor T (= the number of
elementary reflectors). K >= 1.
V is REAL array, dimension
(LDV,K) if STOREV = ’C’
(LDV,N) if STOREV = ’R’
The matrix V. See further details.
LDV is INTEGER
The leading dimension of the array V.
If STOREV = ’C’, LDV >= max(1,N); if STOREV = ’R’, LDV >= K.
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i).
T is REAL array, dimension (LDT,K)
The k by k triangular factor T of the block reflector.
If DIRECT = ’F’, T is upper triangular; if DIRECT = ’B’, T is
lower triangular. The rest of the array is not used.
LDT is INTEGER
The leading dimension of the array T. LDT >= K.
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A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored; the corresponding
array elements are modified but restored on exit. The rest of the
array is not used.
DIRECT = ’F’ and STOREV = ’C’: DIRECT = ’F’ and STOREV = ’R’:
( v1 v2 v3 ) / ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
( v1 v2 v3 )
. . .
. . .
1 . .
1 .
DIRECT = ’B’ and STOREV = ’C’: DIRECT = ’B’ and STOREV = ’R’:
1 / . 1 ( 1 . . . . v1 v1 v1 v1 v1 )
. . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
. . . ( . . 1 . . v3 v3 v3 v3 v3 )
. . .
( v1 v2 v3 )
( v1 v2 v3 )
V = ( v1 v2 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
subroutine slascl2 (integer M, integer N, real, dimension( * ) D, real, dimension( ldx, * ) X, integer LDX)
SLASCL2 performs diagonal scaling on a vector.
SLASCL2 performs a diagonal scaling on a vector:
x <-- D * x
where the diagonal matrix D is stored as a vector.
Eventually to be replaced by BLAS_sge_diag_scale in the new BLAS
M is INTEGER
The number of rows of D and X. M >= 0.
N is INTEGER
The number of columns of X. N >= 0.
D is REAL array, length M
Diagonal matrix D, stored as a vector of length M.
X is REAL array, dimension (LDX,N)
On entry, the vector X to be scaled by D.
On exit, the scaled vector.
LDX is INTEGER
The leading dimension of the vector X. LDX >= M.
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subroutine slatrz (integer M, integer N, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK)
SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
[ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
matrix and, R and A1 are M-by-M upper triangular matrices.
M is INTEGER
The number of rows of the matrix A. M >= 0.
N is INTEGER
The number of columns of the matrix A. N >= 0.
L is INTEGER
The number of columns of the matrix A containing the
meaningful part of the Householder vectors. N-M >= L >= 0.
A is REAL array, dimension (LDA,N)
On entry, the leading M-by-N upper trapezoidal part of the
array A must contain the matrix to be factorized.
On exit, the leading M-by-M upper triangular part of A
contains the upper triangular matrix R, and elements N-L+1 to
N of the first M rows of A, with the array TAU, represent the
orthogonal matrix Z as a product of M elementary reflectors.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (M)
The scalar factors of the elementary reflectors.
WORK is REAL array, dimension (M)
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Further Details:
The factorization is obtained by Householder’s method. The kth
transformation matrix, Z( k ), which is used to introduce zeros into
the ( m - k + 1 )th row of A, is given in the form
Z( k ) = ( I 0 ),
( 0 T( k ) )
T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
( 0 )
( z( k ) )
tau is a scalar and z( k ) is an l element vector. tau and z( k )
are chosen to annihilate the elements of the kth row of A2.
The scalar tau is returned in the kth element of TAU and the vector
u( k ) in the kth row of A2, such that the elements of z( k ) are
in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
the upper triangular part of A1.
Z is given by
Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
subroutine sopgtr (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) TAU, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WORK, integer INFO)
SOPGTR generates a real orthogonal matrix Q which is defined as the
product of n-1 elementary reflectors H(i) of order n, as returned by
SSPTRD using packed storage:
if UPLO = ’U’, Q = H(n-1) . . . H(2) H(1),
if UPLO = ’L’, Q = H(1) H(2) . . . H(n-1).
UPLO is CHARACTER*1
= ’U’: Upper triangular packed storage used in previous
call to SSPTRD;
= ’L’: Lower triangular packed storage used in previous
call to SSPTRD.
N is INTEGER
The order of the matrix Q. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The vectors which define the elementary reflectors, as
returned by SSPTRD.
TAU is REAL array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SSPTRD.
Q is REAL array, dimension (LDQ,N)
The N-by-N orthogonal matrix Q.
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
WORK is REAL array, dimension (N-1)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sopmtr (character SIDE, character UPLO, character TRANS, integer M, integer N, real, dimension( * ) AP, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension(
* ) WORK, integer INFO)
SOPMTR overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = ’L’ and nq = n if SIDE = ’R’. Q is defined as the product of
nq-1 elementary reflectors, as returned by SSPTRD using packed
if UPLO = ’U’, Q = H(nq-1) . . . H(2) H(1);
if UPLO = ’L’, Q = H(1) H(2) . . . H(nq-1).
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
UPLO is CHARACTER*1
= ’U’: Upper triangular packed storage used in previous
call to SSPTRD;
= ’L’: Lower triangular packed storage used in previous
call to SSPTRD.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
AP is REAL array, dimension
(M*(M+1)/2) if SIDE = ’L’
(N*(N+1)/2) if SIDE = ’R’
The vectors which define the elementary reflectors, as
returned by SSPTRD. AP is modified by the routine but
restored on exit.
TAU is REAL array, dimension (M-1) if SIDE = ’L’
or (N-1) if SIDE = ’R’
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SSPTRD.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension
(N) if SIDE = ’L’
(M) if SIDE = ’R’
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sorbdb (character TRANS, character SIGNS, integer M, integer P, integer Q, real, dimension( ldx11, * ) X11, integer LDX11, real, dimension( ldx12, * ) X12, integer LDX12, real, dimension(
ldx21, * ) X21, integer LDX21, real, dimension( ldx22, * ) X22, integer LDX22, real, dimension( * ) THETA, real, dimension( * ) PHI, real, dimension( * ) TAUP1, real, dimension( * ) TAUP2, real,
dimension( * ) TAUQ1, real, dimension( * ) TAUQ2, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORBDB simultaneously bidiagonalizes the blocks of an M-by-M
partitioned orthogonal matrix X:
[ B11 | B12 0 0 ]
[ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T
X = [-----------] = [---------] [----------------] [---------] .
[ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
[ 0 | 0 0 I ]
X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
not the case, then X must be transposed and/or permuted. This can be
done in constant time using the TRANS and SIGNS options. See SORCSD
for details.)
The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
represented implicitly by Householder vectors.
B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
implicitly by angles THETA, PHI.
TRANS is CHARACTER
= ’T’: X, U1, U2, V1T, and V2T are stored in row-major
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS is CHARACTER
= ’O’: The lower-left block is made nonpositive (the
"other" convention);
otherwise: The upper-right block is made nonpositive (the
"default" convention).
M is INTEGER
The number of rows and columns in X.
P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <=
X11 is REAL array, dimension (LDX11,Q)
On entry, the top-left block of the orthogonal matrix to be
reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the columns of tril(X11) specify reflectors for P1,
the rows of triu(X11,1) specify reflectors for Q1;
else TRANS = ’T’, and
the rows of triu(X11) specify reflectors for P1,
the columns of tril(X11,-1) specify reflectors for Q1.
LDX11 is INTEGER
The leading dimension of X11. If TRANS = ’N’, then LDX11 >=
P; else LDX11 >= Q.
X12 is REAL array, dimension (LDX12,M-Q)
On entry, the top-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the rows of triu(X12) specify the first P reflectors for
else TRANS = ’T’, and
the columns of tril(X12) specify the first P reflectors
for Q2.
LDX12 is INTEGER
The leading dimension of X12. If TRANS = ’N’, then LDX12 >=
P; else LDX11 >= M-Q.
X21 is REAL array, dimension (LDX21,Q)
On entry, the bottom-left block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the columns of tril(X21) specify reflectors for P2;
else TRANS = ’T’, and
the rows of triu(X21) specify reflectors for P2.
LDX21 is INTEGER
The leading dimension of X21. If TRANS = ’N’, then LDX21 >=
M-P; else LDX21 >= Q.
X22 is REAL array, dimension (LDX22,M-Q)
On entry, the bottom-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
M-P-Q reflectors for Q2,
else TRANS = ’T’, and
the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
M-P-Q reflectors for P2.
LDX22 is INTEGER
The leading dimension of X22. If TRANS = ’N’, then LDX22 >=
M-P; else LDX22 >= M-Q.
THETA is REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
TAUQ2 is REAL array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
WORK is REAL array, dimension (LWORK)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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Further Details:
The bidiagonal blocks B11, B12, B21, and B22 are represented
implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
lower bidiagonal. Every entry in each bidiagonal band is a product
of a sine or cosine of a THETA with a sine or cosine of a PHI. See
[1] or SORCSD for details.
P1, P2, Q1, and Q2 are represented as products of elementary
reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2
using SORGQR and SORGLQ.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine sorbdb1 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, real,
dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*) TAUQ1, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:
[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]
X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
M-P, or M-Q. Routines SORBDB2, SORBDB3, and SORBDB4 handle cases in
which Q is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by
angles THETA, PHI..fi
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P is INTEGER
The number of rows in X11. 0 <= P <= M.
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <=
X11 is REAL array, dimension (LDX11,Q)
On entry, the top block of the matrix X to be reduced. On
exit, the columns of tril(X11) specify reflectors for P1 and
the rows of triu(X11,1) specify reflectors for Q1.
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.
X21 is REAL array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.
LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.
THETA is REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
WORK is REAL array, dimension (LWORK)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine sorbdb2 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, real,
dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*) TAUQ1, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:
[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]
X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
Q, or M-Q. Routines SORBDB1, SORBDB3, and SORBDB4 handle cases in
which P is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
angles THETA, PHI..fi
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P is INTEGER
The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11 is REAL array, dimension (LDX11,Q)
On entry, the top block of the matrix X to be reduced. On
exit, the columns of tril(X11) specify reflectors for P1 and
the rows of triu(X11,1) specify reflectors for Q1.
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.
X21 is REAL array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.
LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.
THETA is REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
WORK is REAL array, dimension (LWORK)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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July 2012
Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine sorbdb3 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, real,
dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*) TAUQ1, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:
[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]
X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
Q, or M-Q. Routines SORBDB1, SORBDB2, and SORBDB4 handle cases in
which M-P is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
implicitly by angles THETA, PHI..fi
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P is INTEGER
The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11 is REAL array, dimension (LDX11,Q)
On entry, the top block of the matrix X to be reduced. On
exit, the columns of tril(X11) specify reflectors for P1 and
the rows of triu(X11,1) specify reflectors for Q1.
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.
X21 is REAL array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.
LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.
THETA is REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
WORK is REAL array, dimension (LWORK)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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July 2012
Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine sorbdb4 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, real,
dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*) TAUQ1, real, dimension(*) PHANTOM, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:
[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]
X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
M-P, or Q. Routines SORBDB1, SORBDB2, and SORBDB3 handle cases in
which M-Q is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
implicitly by angles THETA, PHI..fi
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P is INTEGER
The number of rows in X11. 0 <= P <= M.
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M and
M-Q <= min(P,M-P,Q).
X11 is REAL array, dimension (LDX11,Q)
On entry, the top block of the matrix X to be reduced. On
exit, the columns of tril(X11) specify reflectors for P1 and
the rows of triu(X11,1) specify reflectors for Q1.
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.
X21 is REAL array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.
LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.
THETA is REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
PHANTOM is REAL array, dimension (M)
The routine computes an M-by-1 column vector Y that is
orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
Y(P+1:M), respectively.
WORK is REAL array, dimension (LWORK)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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Univ. of California Berkeley
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NAG Ltd.
July 2012
Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine sorbdb5 (integer M1, integer M2, integer N, real, dimension(*) X1, integer INCX1, real, dimension(*) X2, integer INCX2, real, dimension(ldq1,*) Q1, integer LDQ1, real, dimension(ldq2,*)
Q2, integer LDQ2, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB5 orthogonalizes the column vector
X = [ X1 ]
[ X2 ]
with respect to the columns of
Q = [ Q1 ] .
[ Q2 ]
The columns of Q must be orthonormal.
If the projection is zero according to Kahan’s "twice is enough"
criterion, then some other vector from the orthogonal complement
is returned. This vector is chosen in an arbitrary but deterministic
M1 is INTEGER
The dimension of X1 and the number of rows in Q1. 0 <= M1.
M2 is INTEGER
The dimension of X2 and the number of rows in Q2. 0 <= M2.
N is INTEGER
The number of columns in Q1 and Q2. 0 <= N.
X1 is REAL array, dimension (M1)
On entry, the top part of the vector to be orthogonalized.
On exit, the top part of the projected vector.
INCX1 is INTEGER
Increment for entries of X1.
X2 is REAL array, dimension (M2)
On entry, the bottom part of the vector to be
orthogonalized. On exit, the bottom part of the projected
INCX2 is INTEGER
Increment for entries of X2.
Q1 is REAL array, dimension (LDQ1, N)
The top part of the orthonormal basis matrix.
LDQ1 is INTEGER
The leading dimension of Q1. LDQ1 >= M1.
Q2 is REAL array, dimension (LDQ2, N)
The bottom part of the orthonormal basis matrix.
LDQ2 is INTEGER
The leading dimension of Q2. LDQ2 >= M2.
WORK is REAL array, dimension (LWORK)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= N.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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subroutine sorbdb6 (integer M1, integer M2, integer N, real, dimension(*) X1, integer INCX1, real, dimension(*) X2, integer INCX2, real, dimension(ldq1,*) Q1, integer LDQ1, real, dimension(ldq2,*)
Q2, integer LDQ2, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB6 orthogonalizes the column vector
X = [ X1 ]
[ X2 ]
with respect to the columns of
Q = [ Q1 ] .
[ Q2 ]
The columns of Q must be orthonormal.
If the projection is zero according to Kahan’s "twice is enough"
criterion, then the zero vector is returned..fi
M1 is INTEGER
The dimension of X1 and the number of rows in Q1. 0 <= M1.
M2 is INTEGER
The dimension of X2 and the number of rows in Q2. 0 <= M2.
N is INTEGER
The number of columns in Q1 and Q2. 0 <= N.
X1 is REAL array, dimension (M1)
On entry, the top part of the vector to be orthogonalized.
On exit, the top part of the projected vector.
INCX1 is INTEGER
Increment for entries of X1.
X2 is REAL array, dimension (M2)
On entry, the bottom part of the vector to be
orthogonalized. On exit, the bottom part of the projected
INCX2 is INTEGER
Increment for entries of X2.
Q1 is REAL array, dimension (LDQ1, N)
The top part of the orthonormal basis matrix.
LDQ1 is INTEGER
The leading dimension of Q1. LDQ1 >= M1.
Q2 is REAL array, dimension (LDQ2, N)
The bottom part of the orthonormal basis matrix.
LDQ2 is INTEGER
The leading dimension of Q2. LDQ2 >= M2.
WORK is REAL array, dimension (LWORK)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= N.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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July 2012
recursive subroutine sorcsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, character SIGNS, integer M, integer P, integer Q, real, dimension( ldx11, * ) X11,
integer LDX11, real, dimension( ldx12, * ) X12, integer LDX12, real, dimension( ldx21, * ) X21, integer LDX21, real, dimension( ldx22, * ) X22, integer LDX22, real, dimension( * ) THETA, real,
dimension( ldu1, * ) U1, integer LDU1, real, dimension( ldu2, * ) U2, integer LDU2, real, dimension( ldv1t, * ) V1T, integer LDV1T, real, dimension( ldv2t, * ) V2T, integer LDV2T, real, dimension( *
) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)
SORCSD computes the CS decomposition of an M-by-M partitioned
orthogonal matrix X:
[ I 0 0 | 0 0 0 ]
[ 0 C 0 | 0 -S 0 ]
[ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
X = [-----------] = [---------] [---------------------] [---------] .
[ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
[ 0 S 0 | 0 C 0 ]
[ 0 0 I | 0 0 0 ]
X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
(M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
which R = MIN(P,M-P,Q,M-Q).
JOBU1 is CHARACTER
= ’Y’: U1 is computed;
otherwise: U1 is not computed.
JOBU2 is CHARACTER
= ’Y’: U2 is computed;
otherwise: U2 is not computed.
JOBV1T is CHARACTER
= ’Y’: V1T is computed;
otherwise: V1T is not computed.
JOBV2T is CHARACTER
= ’Y’: V2T is computed;
otherwise: V2T is not computed.
TRANS is CHARACTER
= ’T’: X, U1, U2, V1T, and V2T are stored in row-major
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS is CHARACTER
= ’O’: The lower-left block is made nonpositive (the
"other" convention);
otherwise: The upper-right block is made nonpositive (the
"default" convention).
M is INTEGER
The number of rows and columns in X.
P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11 is REAL array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).
X12 is REAL array, dimension (LDX12,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX12 is INTEGER
The leading dimension of X12. LDX12 >= MAX(1,P).
X21 is REAL array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX21 is INTEGER
The leading dimension of X11. LDX21 >= MAX(1,M-P).
X22 is REAL array, dimension (LDX22,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX22 is INTEGER
The leading dimension of X11. LDX22 >= MAX(1,M-P).
THETA is REAL array, dimension (R), in which R =
C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
U1 is REAL array, dimension (P)
If JOBU1 = ’Y’, U1 contains the P-by-P orthogonal matrix U1.
LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = ’Y’, LDU1 >=
U2 is REAL array, dimension (M-P)
If JOBU2 = ’Y’, U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.
LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = ’Y’, LDU2 >=
V1T is REAL array, dimension (Q)
If JOBV1T = ’Y’, V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.
LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = ’Y’, LDV1T >=
V2T is REAL array, dimension (M-Q)
If JOBV2T = ’Y’, V2T contains the (M-Q)-by-(M-Q) orthogonal
matrix V2**T.
LDV2T is INTEGER
The leading dimension of V2T. If JOBV2T = ’Y’, LDV2T >=
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
define the matrix in intermediate bidiagonal-block form
remaining after nonconvergence. INFO specifies the number
of nonzero PHI’s.
LWORK is INTEGER
The dimension of the array WORK.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the work array, and no error
message related to LWORK is issued by XERBLA.
IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: SBBCSD did not converge. See the description of WORK
above for details.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorcsd2by1 (character JOBU1, character JOBU2, character JOBV1T, integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer LDX11, real, dimension(ldx21,*) X21, integer LDX21,
real, dimension(*) THETA, real, dimension(ldu1,*) U1, integer LDU1, real, dimension(ldu2,*) U2, integer LDU2, real, dimension(ldv1t,*) V1T, integer LDV1T, real, dimension(*) WORK, integer LWORK,
integer, dimension(*) IWORK, integer INFO)
SORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
orthonormal columns that has been partitioned into a 2-by-1 block
[ I1 0 0 ]
[ 0 C 0 ]
[ X11 ] [ U1 | ] [ 0 0 0 ]
X = [-----] = [---------] [----------] V1**T .
[ X21 ] [ | U2 ] [ 0 0 0 ]
[ 0 S 0 ]
[ 0 0 I2]
X11 is P-by-Q. The orthogonal matrices U1, U2, and V1 are P-by-P,
(M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R
nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which
R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a
K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0)..fi
JOBU1 is CHARACTER
= ’Y’: U1 is computed;
otherwise: U1 is not computed.
JOBU2 is CHARACTER
= ’Y’: U2 is computed;
otherwise: U2 is not computed.
JOBV1T is CHARACTER
= ’Y’: V1T is computed;
otherwise: V1T is not computed.
M is INTEGER
The number of rows in X.
P is INTEGER
The number of rows in X11. 0 <= P <= M.
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11 is REAL array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).
X21 is REAL array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX21 is INTEGER
The leading dimension of X21. LDX21 >= MAX(1,M-P).
THETA is REAL array, dimension (R), in which R =
C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
U1 is REAL array, dimension (P)
If JOBU1 = ’Y’, U1 contains the P-by-P orthogonal matrix U1.
LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = ’Y’, LDU1 >=
U2 is REAL array, dimension (M-P)
If JOBU2 = ’Y’, U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.
LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = ’Y’, LDU2 >=
V1T is REAL array, dimension (Q)
If JOBV1T = ’Y’, V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.
LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = ’Y’, LDV1T >=
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
define the matrix in intermediate bidiagonal-block form
remaining after nonconvergence. INFO specifies the number
of nonzero PHI’s.
LWORK is INTEGER
The dimension of the array WORK.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the work array, and no error
message related to LWORK is issued by XERBLA.
IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: SBBCSD did not converge. See the description of WORK
above for details.
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
July 2012
subroutine sorg2l (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).
SORG2L generates an m by n real matrix Q with orthonormal columns,
which is defined as the last n columns of a product of k elementary
reflectors of order m
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. M >= N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the (n-k+i)-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by SGEQLF in the last k columns of its array
argument A.
On exit, the m by n matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
WORK is REAL array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorg2r (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
SORG2R generates an m by n real matrix Q with orthonormal columns,
which is defined as the first n columns of a product of k elementary
reflectors of order m
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. M >= N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the i-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by SGEQRF in the first k columns of its array
argument A.
On exit, the m-by-n matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
WORK is REAL array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorghr (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGHR generates a real orthogonal matrix Q which is defined as the
product of IHI-ILO elementary reflectors of order N, as returned by
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
N is INTEGER
The order of the matrix Q. N >= 0.
ILO is INTEGER
IHI is INTEGER
ILO and IHI must have the same values as in the previous call
of SGEHRD. Q is equal to the unit matrix except in the
submatrix Q(ilo+1:ihi,ilo+1:ihi).
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A is REAL array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by SGEHRD.
On exit, the N-by-N orthogonal matrix Q.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU is REAL array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEHRD.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= IHI-ILO.
For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorgl2 (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORGL2 generates an m by n real matrix Q with orthonormal rows,
which is defined as the first m rows of a product of k elementary
reflectors of order n
Q = H(k) . . . H(2) H(1)
as returned by SGELQF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. N >= M.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the i-th row must contain the vector which defines
the elementary reflector H(i), for i = 1,2,...,k, as returned
by SGELQF in the first k rows of its array argument A.
On exit, the m-by-n matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
WORK is REAL array, dimension (M)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorglq (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGLQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the first M rows of a product of K elementary
reflectors of order N
Q = H(k) . . . H(2) H(1)
as returned by SGELQF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. N >= M.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the i-th row must contain the vector which defines
the elementary reflector H(i), for i = 1,2,...,k, as returned
by SGELQF in the first k rows of its array argument A.
On exit, the M-by-N matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorgql (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGQL generates an M-by-N real matrix Q with orthonormal columns,
which is defined as the last N columns of a product of K elementary
reflectors of order M
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. M >= N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the (n-k+i)-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by SGEQLF in the last k columns of its array
argument A.
On exit, the M-by-N matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorgqr (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGQR generates an M-by-N real matrix Q with orthonormal columns,
which is defined as the first N columns of a product of K elementary
reflectors of order M
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. M >= N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the i-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by SGEQRF in the first k columns of its array
argument A.
On exit, the M-by-N matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorgr2 (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).
SORGR2 generates an m by n real matrix Q with orthonormal rows,
which is defined as the last m rows of a product of k elementary
reflectors of order n
Q = H(1) H(2) . . . H(k)
as returned by SGERQF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. N >= M.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the (m-k+i)-th row must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by SGERQF in the last k rows of its array argument
On exit, the m by n matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
WORK is REAL array, dimension (M)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorgrq (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGRQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the last M rows of a product of K elementary
reflectors of order N
Q = H(1) H(2) . . . H(k)
as returned by SGERQF.
M is INTEGER
The number of rows of the matrix Q. M >= 0.
N is INTEGER
The number of columns of the matrix Q. N >= M.
K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.
A is REAL array, dimension (LDA,N)
On entry, the (m-k+i)-th row must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by SGERQF in the last k rows of its array argument
On exit, the M-by-N matrix Q.
LDA is INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorgtr (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGTR generates a real orthogonal matrix Q which is defined as the
product of n-1 elementary reflectors of order N, as returned by
if UPLO = ’U’, Q = H(n-1) . . . H(2) H(1),
if UPLO = ’L’, Q = H(1) H(2) . . . H(n-1).
UPLO is CHARACTER*1
= ’U’: Upper triangle of A contains elementary reflectors
from SSYTRD;
= ’L’: Lower triangle of A contains elementary reflectors
from SSYTRD.
N is INTEGER
The order of the matrix Q. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by SSYTRD.
On exit, the N-by-N orthogonal matrix Q.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU is REAL array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SSYTRD.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N-1).
For optimum performance LWORK >= (N-1)*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorm2l (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer INFO)
SORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
SORM2L overwrites the general real m by n matrix C with
Q * C if SIDE = ’L’ and TRANS = ’N’, or
Q**T * C if SIDE = ’L’ and TRANS = ’T’, or
C * Q if SIDE = ’R’ and TRANS = ’N’, or
C * Q**T if SIDE = ’R’ and TRANS = ’T’,
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left
= ’R’: apply Q or Q**T from the Right
TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGEQLF in the last k columns of its array argument A.
A is modified by the routine but restored on exit.
LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDA >= max(1,M);
if SIDE = ’R’, LDA >= max(1,N).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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NAG Ltd.
December 2016
subroutine sorm2r (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
SORM2R overwrites the general real m by n matrix C with
Q * C if SIDE = ’L’ and TRANS = ’N’, or
Q**T* C if SIDE = ’L’ and TRANS = ’T’, or
C * Q if SIDE = ’R’ and TRANS = ’N’, or
C * Q**T if SIDE = ’R’ and TRANS = ’T’,
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left
= ’R’: apply Q or Q**T from the Right
TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGEQRF in the first k columns of its array argument A.
A is modified by the routine but restored on exit.
LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDA >= max(1,M);
if SIDE = ’R’, LDA >= max(1,N).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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NAG Ltd.
December 2016
subroutine sormbr (character VECT, character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
If VECT = ’Q’, SORMBR overwrites the general real M-by-N matrix C
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
If VECT = ’P’, SORMBR overwrites the general real M-by-N matrix C
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: P * C C * P
TRANS = ’T’: P**T * C C * P**T
Here Q and P**T are the orthogonal matrices determined by SGEBRD when
reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
P**T are defined as products of elementary reflectors H(i) and G(i)
Let nq = m if SIDE = ’L’ and nq = n if SIDE = ’R’. Thus nq is the
order of the orthogonal matrix Q or P**T that is applied.
If VECT = ’Q’, A is assumed to have been an NQ-by-K matrix:
if nq >= k, Q = H(1) H(2) . . . H(k);
if nq < k, Q = H(1) H(2) . . . H(nq-1).
If VECT = ’P’, A is assumed to have been a K-by-NQ matrix:
if k < nq, P = G(1) G(2) . . . G(k);
if k >= nq, P = G(1) G(2) . . . G(nq-1).
VECT is CHARACTER*1
= ’Q’: apply Q or Q**T;
= ’P’: apply P or P**T.
SIDE is CHARACTER*1
= ’L’: apply Q, Q**T, P or P**T from the Left;
= ’R’: apply Q, Q**T, P or P**T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q or P;
= ’T’: Transpose, apply Q**T or P**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
If VECT = ’Q’, the number of columns in the original
matrix reduced by SGEBRD.
If VECT = ’P’, the number of rows in the original
matrix reduced by SGEBRD.
K >= 0.
A is REAL array, dimension
(LDA,min(nq,K)) if VECT = ’Q’
(LDA,nq) if VECT = ’P’
The vectors which define the elementary reflectors H(i) and
G(i), whose products determine the matrices Q and P, as
returned by SGEBRD.
LDA is INTEGER
The leading dimension of the array A.
If VECT = ’Q’, LDA >= max(1,nq);
if VECT = ’P’, LDA >= max(1,min(nq,K)).
TAU is REAL array, dimension (min(nq,K))
TAU(i) must contain the scalar factor of the elementary
reflector H(i) or G(i) which determines Q or P, as returned
by SGEBRD in the array argument TAUQ or TAUP.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
or P*C or P**T*C or C*P or C*P**T.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For optimum performance LWORK >= N*NB if SIDE = ’L’, and
LWORK >= M*NB if SIDE = ’R’, where NB is the optimal
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
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NAG Ltd.
December 2016
subroutine sormhr (character SIDE, character TRANS, integer M, integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMHR overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = ’L’ and nq = n if SIDE = ’R’. Q is defined as the product of
IHI-ILO elementary reflectors, as returned by SGEHRD:
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
ILO is INTEGER
IHI is INTEGER
ILO and IHI must have the same values as in the previous call
of SGEHRD. Q is equal to the unit matrix except in the
submatrix Q(ilo+1:ihi,ilo+1:ihi).
If SIDE = ’L’, then 1 <= ILO <= IHI <= M, if M > 0, and
ILO = 1 and IHI = 0, if M = 0;
if SIDE = ’R’, then 1 <= ILO <= IHI <= N, if N > 0, and
ILO = 1 and IHI = 0, if N = 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’
(LDA,N) if SIDE = ’R’
The vectors which define the elementary reflectors, as
returned by SGEHRD.
LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,M) if SIDE = ’L’; LDA >= max(1,N) if SIDE = ’R’.
TAU is REAL array, dimension
(M-1) if SIDE = ’L’
(N-1) if SIDE = ’R’
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEHRD.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For optimum performance LWORK >= N*NB if SIDE = ’L’, and
LWORK >= M*NB if SIDE = ’R’, where NB is the optimal
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
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Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sorml2 (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer INFO)
SORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
SORML2 overwrites the general real m by n matrix C with
Q * C if SIDE = ’L’ and TRANS = ’N’, or
Q**T* C if SIDE = ’L’ and TRANS = ’T’, or
C * Q if SIDE = ’R’ and TRANS = ’N’, or
C * Q**T if SIDE = ’R’ and TRANS = ’T’,
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGELQF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left
= ’R’: apply Q or Q**T from the Right
TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGELQF in the first k rows of its array argument A.
A is modified by the routine but restored on exit.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
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NAG Ltd.
December 2016
subroutine sormlq (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMLQ overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGELQF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGELQF in the first k rows of its array argument A.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
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NAG Ltd.
December 2016
subroutine sormql (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMQL overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGEQLF in the last k columns of its array argument A.
LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDA >= max(1,M);
if SIDE = ’R’, LDA >= max(1,N).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sormqr (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMQR overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGEQRF in the first k columns of its array argument A.
LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDA >= max(1,M);
if SIDE = ’R’, LDA >= max(1,N).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
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Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sormr2 (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer INFO)
SORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).
SORMR2 overwrites the general real m by n matrix C with
Q * C if SIDE = ’L’ and TRANS = ’N’, or
Q**T* C if SIDE = ’L’ and TRANS = ’T’, or
C * Q if SIDE = ’R’ and TRANS = ’N’, or
C * Q**T if SIDE = ’R’ and TRANS = ’T’,
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGERQF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left
= ’R’: apply Q or Q**T from the Right
TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q’ (Transpose)
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGERQF in the last k rows of its array argument A.
A is modified by the routine but restored on exit.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine sormr3 (character SIDE, character TRANS, integer M, integer N, integer K, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer INFO)
SORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).
SORMR3 overwrites the general real m by n matrix C with
Q * C if SIDE = ’L’ and TRANS = ’N’, or
Q**T* C if SIDE = ’L’ and TRANS = ’C’, or
C * Q if SIDE = ’R’ and TRANS = ’N’, or
C * Q**T if SIDE = ’R’ and TRANS = ’C’,
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by STZRZF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left
= ’R’: apply Q or Q**T from the Right
TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
L is INTEGER
The number of columns of the matrix A containing
the meaningful part of the Householder reflectors.
If SIDE = ’L’, M >= L >= 0, if SIDE = ’R’, N >= L >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
STZRZF in the last k rows of its array argument A.
A is modified by the routine but restored on exit.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by STZRZF.
C is REAL array, dimension (LDC,N)
On entry, the m-by-n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sormrq (character SIDE, character TRANS, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMRQ overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGERQF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGERQF in the last k rows of its array argument A.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sormrz (character SIDE, character TRANS, integer M, integer N, integer K, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMRZ overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by STZRZF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.
L is INTEGER
The number of columns of the matrix A containing
the meaningful part of the Householder reflectors.
If SIDE = ’L’, M >= L >= 0, if SIDE = ’R’, N >= L >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
STZRZF in the last k rows of its array argument A.
A is modified by the routine but restored on exit.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by STZRZF.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sormtr (character SIDE, character UPLO, character TRANS, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer LDC,
real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMTR overwrites the general real M-by-N matrix C with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T
where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = ’L’ and nq = n if SIDE = ’R’. Q is defined as the product of
nq-1 elementary reflectors, as returned by SSYTRD:
if UPLO = ’U’, Q = H(nq-1) . . . H(2) H(1);
if UPLO = ’L’, Q = H(1) H(2) . . . H(nq-1).
SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A contains elementary reflectors
from SSYTRD;
= ’L’: Lower triangle of A contains elementary reflectors
from SSYTRD.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = ’L’
(LDA,N) if SIDE = ’R’
The vectors which define the elementary reflectors, as
returned by SSYTRD.
LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,M) if SIDE = ’L’; LDA >= max(1,N) if SIDE = ’R’.
TAU is REAL array, dimension
(M-1) if SIDE = ’L’
(N-1) if SIDE = ’R’
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SSYTRD.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For optimum performance LWORK >= N*NB if SIDE = ’L’, and
LWORK >= M*NB if SIDE = ’R’, where NB is the optimal
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine spbcon (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SPBCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite band matrix using the
Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
UPLO is CHARACTER*1
= ’U’: Upper triangular factor stored in AB;
= ’L’: Lower triangular factor stored in AB.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.
AB is REAL array, dimension (LDAB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A, stored in the
first KD+1 rows of the array. The j-th column of U or L is
stored in the j-th column of the array AB as follows:
if UPLO =’U’, AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
if UPLO =’L’, AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
ANORM is REAL
The 1-norm (or infinity-norm) of the symmetric band matrix A.
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine spbequ (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)
SPBEQU computes row and column scalings intended to equilibrate a
symmetric positive definite band matrix A and reduce its condition
number (with respect to the two-norm). S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
UPLO is CHARACTER*1
= ’U’: Upper triangular of A is stored;
= ’L’: Lower triangular of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.
AB is REAL array, dimension (LDAB,N)
The upper or lower triangle of the symmetric band matrix A,
stored in the first KD+1 rows of the array. The j-th column
of A is stored in the j-th column of the array AB as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
LDAB is INTEGER
The leading dimension of the array A. LDAB >= KD+1.
S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND is REAL
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i). If SCOND >= 0.1 and AMAX is neither too
large nor too small, it is not worth scaling by S.
AMAX is REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the i-th diagonal element is nonpositive.
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subroutine spbrfs (character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, real, dimension( ldb, * ) B,
integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SPBRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite
and banded, and provides error bounds and backward error estimates
for the solution.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AB is REAL array, dimension (LDAB,N)
The upper or lower triangle of the symmetric band matrix A,
stored in the first KD+1 rows of the array. The j-th column
of A is stored in the j-th column of the array AB as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
AFB is REAL array, dimension (LDAFB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A as computed by
SPBTRF, in the same storage format as A (see AB).
LDAFB is INTEGER
The leading dimension of the array AFB. LDAFB >= KD+1.
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SPBTRS.
On exit, the improved solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
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subroutine spbstf (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, integer INFO)
SPBSTF computes a split Cholesky factorization of a real
symmetric positive definite band matrix A.
This routine is designed to be used in conjunction with SSBGST.
The factorization has the form A = S**T*S where S is a band matrix
of the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.
AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first kd+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the factor S from the split Cholesky
factorization A = S**T*S. See Further Details.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.
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Further Details:
The band storage scheme is illustrated by the following example, when
N = 7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = ’U’, the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = ’L’, the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
Array elements marked * are not used by the routine.
subroutine spbtf2 (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, integer INFO)
SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
SPBTF2 computes the Cholesky factorization of a real symmetric
positive definite band matrix A.
The factorization has the form
A = U**T * U , if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix, U**T is the transpose of U, and
L is lower triangular.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of super-diagonals of the matrix A if UPLO = ’U’,
or the number of sub-diagonals if UPLO = ’L’. KD >= 0.
AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T of the band
matrix A, in the same storage format as A.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, the leading minor of order k is not
positive definite, and the factorization could not be
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Further Details:
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = ’U’:
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = ’L’ the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
subroutine spbtrf (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, integer INFO)
SPBTRF computes the Cholesky factorization of a real symmetric
positive definite band matrix A.
The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.
AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T of the band
matrix A, in the same storage format as A.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
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Further Details:
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = ’U’:
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = ’L’ the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
subroutine spbtrs (character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SPBTRS solves a system of linear equations A*X = B with a symmetric
positive definite band matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by SPBTRF.
UPLO is CHARACTER*1
= ’U’: Upper triangular factor stored in AB;
= ’L’: Lower triangular factor stored in AB.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AB is REAL array, dimension (LDAB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A, stored in the
first KD+1 rows of the array. The j-th column of U or L is
stored in the j-th column of the array AB as follows:
if UPLO =’U’, AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
if UPLO =’L’, AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine spftrf (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) A, integer INFO)
SPFTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A.
The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.
This is the block version of the algorithm, calling Level 3 BLAS.
TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.
UPLO is CHARACTER*1
= ’U’: Upper triangle of RFP A is stored;
= ’L’: Lower triangle of RFP A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension ( N*(N+1)/2 );
On entry, the symmetric matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = ’N’
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = ’T’ then RFP is
the transpose of RFP A as defined when
TRANSR = ’N’. The contents of RFP A are defined by UPLO as
follows: If UPLO = ’U’ the RFP A contains the NT elements of
upper packed A. If UPLO = ’L’ the RFP A contains the elements
of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
’T’. When TRANSR is ’N’ the LDA is N+1 when N is even and N
is odd. See the Note below for more details.
On exit, if INFO = 0, the factor U or L from the Cholesky
factorization RFP A = U**T*U or RFP A = L*L**T.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
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Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine spftri (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) A, integer INFO)
SPFTRI computes the inverse of a real (symmetric) positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by SPFTRF.
TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension ( N*(N+1)/2 )
On entry, the symmetric matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = ’N’
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = ’T’ then RFP is
the transpose of RFP A as defined when
TRANSR = ’N’. The contents of RFP A are defined by UPLO as
follows: If UPLO = ’U’ the RFP A contains the nt elements of
upper packed A. If UPLO = ’L’ the RFP A contains the elements
of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
’T’. When TRANSR is ’N’ the LDA is N+1 when N is even and N
is odd. See the Note below for more details.
On exit, the symmetric inverse of the original matrix, in the
same storage format.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the (i,i) element of the factor U or L is
zero, and the inverse could not be computed.
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Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine spftrs (character TRANSR, character UPLO, integer N, integer NRHS, real, dimension( 0: * ) A, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SPFTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by SPFTRF.
TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.
UPLO is CHARACTER*1
= ’U’: Upper triangle of RFP A is stored;
= ’L’: Lower triangle of RFP A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A is REAL array, dimension ( N*(N+1)/2 )
The triangular factor U or L from the Cholesky factorization
of RFP A = U**H*U or RFP A = L*L**T, as computed by SPFTRF.
See note below for more details about RFP A.
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine sppcon (character UPLO, integer N, real, dimension( * ) AP, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SPPCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite packed matrix using
the Cholesky factorization A = U**T*U or A = L*L**T computed by
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, packed columnwise in a linear
array. The j-th column of U or L is stored in the array AP
as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
ANORM is REAL
The 1-norm (or infinity-norm) of the symmetric matrix A.
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sppequ (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)
SPPEQU computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A in packed storage and reduce
its condition number (with respect to the two-norm). S contains the
scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
This choice of S puts the condition number of B within a factor N of
the smallest possible condition number over all possible diagonal
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND is REAL
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i). If SCOND >= 0.1 and AMAX is neither too
large nor too small, it is not worth scaling by S.
AMAX is REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element is nonpositive.
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subroutine spprfs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( * ) AFP, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX,
real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SPPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite
and packed, and provides error bounds and backward error estimates
for the solution.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
AFP is REAL array, dimension (N*(N+1)/2)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by SPPTRF/CPPTRF,
packed columnwise in a linear array in the same format as A
(see AP).
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SPPTRS.
On exit, the improved solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
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subroutine spptrf (character UPLO, integer N, real, dimension( * ) AP, integer INFO)
SPPTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A stored in packed format.
The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
See below for further details.
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T, in the same
storage format as A.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
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Further Details:
The packed storage scheme is illustrated by the following example
when N = 4, UPLO = ’U’:
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
subroutine spptri (character UPLO, integer N, real, dimension( * ) AP, integer INFO)
SPPTRI computes the inverse of a real symmetric positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by SPPTRF.
UPLO is CHARACTER*1
= ’U’: Upper triangular factor is stored in AP;
= ’L’: Lower triangular factor is stored in AP.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T, packed columnwise as
a linear array. The j-th column of U or L is stored in the
array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
On exit, the upper or lower triangle of the (symmetric)
inverse of A, overwriting the input factor U or L.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the (i,i) element of the factor U or L is
zero, and the inverse could not be computed.
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subroutine spptrs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SPPTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A in packed storage using the Cholesky
factorization A = U**T*U or A = L*L**T computed by SPPTRF.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, packed columnwise in a linear
array. The j-th column of U or L is stored in the array AP
as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine spstf2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, real TOL, real, dimension( 2*n ) WORK, integer INFO)
SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
SPSTF2 computes the Cholesky factorization with complete
pivoting of a real symmetric positive semidefinite matrix A.
The factorization has the form
P**T * A * P = U**T * U , if UPLO = ’U’,
P**T * A * P = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular, and
P is stored as vector PIV.
This algorithm does not attempt to check that A is positive
semidefinite. This version of the algorithm calls level 2 BLAS.
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= ’U’: Upper triangular
= ’L’: Lower triangular
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = ’U’, the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the factor U or L from the Cholesky
factorization as above.
PIV is INTEGER array, dimension (N)
PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
RANK is INTEGER
The rank of A given by the number of steps the algorithm
TOL is REAL
User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
will be used. The algorithm terminates at the (K-1)st step
if the pivot <= TOL.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
WORK is REAL array, dimension (2*N)
Work space.
INFO is INTEGER
< 0: If INFO = -K, the K-th argument had an illegal value,
= 0: algorithm completed successfully, and
> 0: the matrix A is either rank deficient with computed rank
as returned in RANK, or is not positive semidefinite. See
Section 7 of LAPACK Working Note #161 for further
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subroutine spstrf (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, real TOL, real, dimension( 2*n ) WORK, integer INFO)
SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
SPSTRF computes the Cholesky factorization with complete
pivoting of a real symmetric positive semidefinite matrix A.
The factorization has the form
P**T * A * P = U**T * U , if UPLO = ’U’,
P**T * A * P = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular, and
P is stored as vector PIV.
This algorithm does not attempt to check that A is positive
semidefinite. This version of the algorithm calls level 3 BLAS.
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= ’U’: Upper triangular
= ’L’: Lower triangular
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = ’U’, the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the factor U or L from the Cholesky
factorization as above.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
PIV is INTEGER array, dimension (N)
PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
RANK is INTEGER
The rank of A given by the number of steps the algorithm
TOL is REAL
User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
will be used. The algorithm terminates at the (K-1)st step
if the pivot <= TOL.
WORK is REAL array, dimension (2*N)
Work space.
INFO is INTEGER
< 0: If INFO = -K, the K-th argument had an illegal value,
= 0: algorithm completed successfully, and
> 0: the matrix A is either rank deficient with computed rank
as returned in RANK, or is not positive semidefinite. See
Section 7 of LAPACK Working Note #161 for further
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subroutine ssbgst (character VECT, character UPLO, integer N, integer KA, integer KB, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldbb, * ) BB, integer LDBB, real, dimension( ldx, *
) X, integer LDX, real, dimension( * ) WORK, integer INFO)
SSBGST reduces a real symmetric-definite banded generalized
eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y,
such that C has the same bandwidth as A.
B must have been previously factorized as S**T*S by SPBSTF, using a
split Cholesky factorization. A is overwritten by C = X**T*A*X, where
X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
bandwidth of A.
VECT is CHARACTER*1
= ’N’: do not form the transformation matrix X;
= ’V’: form X.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrices A and B. N >= 0.
KA is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KA >= 0.
KB is INTEGER
The number of superdiagonals of the matrix B if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KA >= KB >= 0.
AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first ka+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
On exit, the transformed matrix X**T*A*X, stored in the same
format as A.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KA+1.
BB is REAL array, dimension (LDBB,N)
The banded factor S from the split Cholesky factorization of
B, as returned by SPBSTF, stored in the first KB+1 rows of
the array.
LDBB is INTEGER
The leading dimension of the array BB. LDBB >= KB+1.
X is REAL array, dimension (LDX,N)
If VECT = ’V’, the n-by-n matrix X.
If VECT = ’N’, the array X is not referenced.
LDX is INTEGER
The leading dimension of the array X.
LDX >= max(1,N) if VECT = ’V’; LDX >= 1 otherwise.
WORK is REAL array, dimension (2*N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
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subroutine ssbtrd (character VECT, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldq, * ) Q,
integer LDQ, real, dimension( * ) WORK, integer INFO)
SSBTRD reduces a real symmetric band matrix A to symmetric
tridiagonal form T by an orthogonal similarity transformation:
Q**T * A * Q = T.
VECT is CHARACTER*1
= ’N’: do not form Q;
= ’V’: form Q;
= ’U’: update a matrix X, by forming X*Q.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KD >= 0.
AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = ’U’) or the
first subdiagonal (if UPLO = ’L’) are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T.
E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = T(i,i+1) if UPLO = ’U’; E(i) = T(i+1,i) if UPLO = ’L’.
Q is REAL array, dimension (LDQ,N)
On entry, if VECT = ’U’, then Q must contain an N-by-N
matrix X; if VECT = ’N’ or ’V’, then Q need not be set.
On exit:
if VECT = ’V’, Q contains the N-by-N orthogonal matrix Q;
if VECT = ’U’, Q contains the product X*Q;
if VECT = ’N’, the array Q is not referenced.
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1, and LDQ >= N if VECT = ’V’ or ’U’.
WORK is REAL array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
Modified by Linda Kaufman, Bell Labs.
subroutine ssfrk (character TRANSR, character UPLO, character TRANS, integer N, integer K, real ALPHA, real, dimension( lda, * ) A, integer LDA, real BETA, real, dimension( * ) C)
SSFRK performs a symmetric rank-k operation for matrix in RFP format.
Level 3 BLAS like routine for C in RFP Format.
SSFRK performs one of the symmetric rank--k operations
C := alpha*A*A**T + beta*C,
C := alpha*A**T*A + beta*C,
where alpha and beta are real scalars, C is an n--by--n symmetric
matrix and A is an n--by--k matrix in the first case and a k--by--n
matrix in the second case.
TRANSR is CHARACTER*1
= ’N’: The Normal Form of RFP A is stored;
= ’T’: The Transpose Form of RFP A is stored.
UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the array C is to be referenced as
UPLO = ’U’ or ’u’ Only the upper triangular part of C
is to be referenced.
UPLO = ’L’ or ’l’ Only the lower triangular part of C
is to be referenced.
Unchanged on exit.
TRANS is CHARACTER*1
On entry, TRANS specifies the operation to be performed as
TRANS = ’N’ or ’n’ C := alpha*A*A**T + beta*C.
TRANS = ’T’ or ’t’ C := alpha*A**T*A + beta*C.
Unchanged on exit.
N is INTEGER
On entry, N specifies the order of the matrix C. N must be
at least zero.
Unchanged on exit.
K is INTEGER
On entry with TRANS = ’N’ or ’n’, K specifies the number
of columns of the matrix A, and on entry with TRANS = ’T’
or ’t’, K specifies the number of rows of the matrix A. K
must be at least zero.
Unchanged on exit.
ALPHA is REAL
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
A is REAL array of DIMENSION (LDA,ka)
where KA
is K when TRANS = ’N’ or ’n’, and is N otherwise. Before
entry with TRANS = ’N’ or ’n’, the leading N--by--K part of
the array A must contain the matrix A, otherwise the leading
K--by--N part of the array A must contain the matrix A.
Unchanged on exit.
LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. When TRANS = ’N’ or ’n’
then LDA must be at least max( 1, n ), otherwise LDA must
be at least max( 1, k ).
Unchanged on exit.
BETA is REAL
On entry, BETA specifies the scalar beta.
Unchanged on exit.
C is REAL array, dimension (NT)
NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
Format. RFP Format is described by TRANSR, UPLO and N.
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subroutine sspcon (character UPLO, integer N, real, dimension( * ) AP, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SSPCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric packed matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by SSPTRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= ’U’: Upper triangular, form is A = U*D*U**T;
= ’L’: Lower triangular, form is A = L*D*L**T.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSPTRF, stored as a
packed triangular matrix.
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSPTRF.
ANORM is REAL
The 1-norm of the original matrix A.
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK is REAL array, dimension (2*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sspgst (integer ITYPE, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) BP, integer INFO)
SSPGST reduces a real symmetric-definite generalized eigenproblem
to standard form, using packed storage.
If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
B must have been previously factorized as U**T*U or L*L**T by SPPTRF.
ITYPE is INTEGER
= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
= 2 or 3: compute U*A*U**T or L**T*A*L.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored and B is factored as
= ’L’: Lower triangle of A is stored and B is factored as
N is INTEGER
The order of the matrices A and B. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.
BP is REAL array, dimension (N*(N+1)/2)
The triangular factor from the Cholesky factorization of B,
stored in the same format as A, as returned by SPPTRF.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine ssprfs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( * ) AFP, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension(
ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SSPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite
and packed, and provides error bounds and backward error estimates
for the solution.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
AFP is REAL array, dimension (N*(N+1)/2)
The factored form of the matrix A. AFP contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**T or
A = L*D*L**T as computed by SSPTRF, stored as a packed
triangular matrix.
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSPTRF.
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SSPTRS.
On exit, the improved solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
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subroutine ssptrd (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, integer INFO)
SSPTRD reduces a real symmetric matrix A stored in packed form to
symmetric tridiagonal form T by an orthogonal similarity
transformation: Q**T * A * Q = T.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = ’U’, the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= ’L’, the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = ’U’, E(i) = A(i+1,i) if UPLO = ’L’.
TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
If UPLO = ’U’, the matrix Q is represented as a product of elementary
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
If UPLO = ’L’, the matrix Q is represented as a product of elementary
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).
subroutine ssptrf (character UPLO, integer N, real, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO)
SSPTRF computes the factorization of a real symmetric matrix A stored
in packed format using the Bunch-Kaufman diagonal pivoting method:
A = U*D*U**T or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L, stored as a packed triangular
matrix overwriting A (see below for further details).
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ’U’ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ’L’ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
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Further Details:
5-96 - Based on modifications by J. Lewis, Boeing Computer Services
If UPLO = ’U’, then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = ’L’, then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
subroutine ssptri (character UPLO, integer N, real, dimension( * ) AP, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer INFO)
SSPTRI computes the inverse of a real symmetric indefinite matrix
A in packed storage using the factorization A = U*D*U**T or
A = L*D*L**T computed by SSPTRF.
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= ’U’: Upper triangular, form is A = U*D*U**T;
= ’L’: Lower triangular, form is A = L*D*L**T.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by SSPTRF,
stored as a packed triangular matrix.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix, stored as a packed triangular matrix. The j-th column
of inv(A) is stored in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
if UPLO = ’L’,
AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSPTRF.
WORK is REAL array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
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subroutine ssptrs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SSPTRS solves a system of linear equations A*X = B with a real
symmetric matrix A stored in packed format using the factorization
A = U*D*U**T or A = L*D*L**T computed by SSPTRF.
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= ’U’: Upper triangular, form is A = U*D*U**T;
= ’L’: Lower triangular, form is A = L*D*L**T.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSPTRF, stored as a
packed triangular matrix.
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSPTRF.
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine sstegr (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * )
W, real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
SSTEGR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
a well defined set of pairwise different real eigenvalues, the corresponding
real eigenvectors are pairwise orthogonal.
The spectrum may be computed either completely or partially by specifying
either an interval (VL,VU] or a range of indices IL:IU for the desired
SSTEGR is a compatibility wrapper around the improved SSTEMR routine.
See SSTEMR for further details.
One important change is that the ABSTOL parameter no longer provides any
benefit and hence is no longer used.
Note : SSTEGR and SSTEMR work only on machines which follow
IEEE-754 floating-point standard in their handling of infinities and
NaNs. Normal execution may create these exceptiona values and hence
may abort due to a floating point exception in environments which
do not conform to the IEEE-754 standard.
JOBZ is CHARACTER*1
= ’N’: Compute eigenvalues only;
= ’V’: Compute eigenvalues and eigenvectors.
RANGE is CHARACTER*1
= ’A’: all eigenvalues will be found.
= ’V’: all eigenvalues in the half-open interval (VL,VU]
will be found.
= ’I’: the IL-th through IU-th eigenvalues will be found.
N is INTEGER
The order of the matrix. N >= 0.
D is REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal matrix
T. On exit, D is overwritten.
E is REAL array, dimension (N)
On entry, the (N-1) subdiagonal elements of the tridiagonal
matrix T in elements 1 to N-1 of E. E(N) need not be set on
input, but is used internally as workspace.
On exit, E is overwritten.
VL is REAL
If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.
VU is REAL
If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.
IL is INTEGER
If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.
IU is INTEGER
If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.
ABSTOL is REAL
Unused. Was the absolute error tolerance for the
eigenvalues/eigenvectors in previous versions.
M is INTEGER
The total number of eigenvalues found. 0 <= M <= N.
If RANGE = ’A’, M = N, and if RANGE = ’I’, M = IU-IL+1.
W is REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z is REAL array, dimension (LDZ, max(1,M) )
If JOBZ = ’V’, and if INFO = 0, then the first M columns of Z
contain the orthonormal eigenvectors of the matrix T
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and an upper bound must be used.
Supplying N columns is always safe.
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = ’V’, then LDZ >= max(1,N).
ISUPPZ is INTEGER ARRAY, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th computed eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is relevant in the case when the matrix
is split. ISUPPZ is only accessed when JOBZ is ’V’ and N > 0.
WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
(and minimal) LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,18*N)
if JOBZ = ’V’, and LWORK >= max(1,12*N) if JOBZ = ’N’.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK is INTEGER array, dimension (LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= max(1,10*N)
if the eigenvectors are desired, and LIWORK >= max(1,8*N)
if only the eigenvalues are to be computed.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO is INTEGER
On exit, INFO
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = 1X, internal error in SLARRE,
if INFO = 2X, internal error in SLARRV.
Here, the digit X = ABS( IINFO ) < 10, where IINFO is
the nonzero error code returned by SLARRE or
SLARRV, respectively.
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NAG Ltd.
June 2016
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, LBNL/NERSC, USA
subroutine sstein (integer N, real, dimension( * ) D, real, dimension( * ) E, integer M, real, dimension( * ) W, integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, real, dimension( ldz,
* ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
SSTEIN computes the eigenvectors of a real symmetric tridiagonal
matrix T corresponding to specified eigenvalues, using inverse
The maximum number of iterations allowed for each eigenvector is
specified by an internal parameter MAXITS (currently set to 5).
N is INTEGER
The order of the matrix. N >= 0.
D is REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E is REAL array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix
T, in elements 1 to N-1.
M is INTEGER
The number of eigenvectors to be found. 0 <= M <= N.
W is REAL array, dimension (N)
The first M elements of W contain the eigenvalues for
which eigenvectors are to be computed. The eigenvalues
should be grouped by split-off block and ordered from
smallest to largest within the block. ( The output array
W from SSTEBZ with ORDER = ’B’ is expected here. )
IBLOCK is INTEGER array, dimension (N)
The submatrix indices associated with the corresponding
eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
the first submatrix from the top, =2 if W(i) belongs to
the second submatrix, etc. ( The output array IBLOCK
from SSTEBZ is expected here. )
ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to
ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
through ISPLIT( 2 ), etc.
( The output array ISPLIT from SSTEBZ is expected here. )
Z is REAL array, dimension (LDZ, M)
The computed eigenvectors. The eigenvector associated
with the eigenvalue W(i) is stored in the i-th column of
Z. Any vector which fails to converge is set to its current
iterate after MAXITS iterations.
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
WORK is REAL array, dimension (5*N)
IWORK is INTEGER array, dimension (N)
IFAIL is INTEGER array, dimension (M)
On normal exit, all elements of IFAIL are zero.
If one or more eigenvectors fail to converge after
MAXITS iterations, then their indices are stored in
array IFAIL.
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, then i eigenvectors failed to converge
in MAXITS iterations. Their indices are stored in
array IFAIL.
Internal Parameters:
MAXITS INTEGER, default = 5
The maximum number of iterations performed.
EXTRA INTEGER, default = 2
The number of iterations performed after norm growth
criterion is satisfied, should be at least 1.
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subroutine sstemr (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, integer M, real, dimension( * ) W, real,
dimension( ldz, * ) Z, integer LDZ, integer NZC, integer, dimension( * ) ISUPPZ, logical TRYRAC, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
SSTEMR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
a well defined set of pairwise different real eigenvalues, the corresponding
real eigenvectors are pairwise orthogonal.
The spectrum may be computed either completely or partially by specifying
either an interval (VL,VU] or a range of indices IL:IU for the desired
Depending on the number of desired eigenvalues, these are computed either
by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
computed by the use of various suitable L D L^T factorizations near clusters
of close eigenvalues (referred to as RRRs, Relatively Robust
Representations). An informal sketch of the algorithm follows.
For each unreduced block (submatrix) of T,
(a) Compute T - sigma I = L D L^T, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
the corresponding eigenvector by forming a rank revealing twisted
factorization. Go back to (c) for any clusters that remain.
For more details, see:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
2004. Also LAPACK Working Note 154.
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem",
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.
Further Details
1.SSTEMR works only on machines which follow IEEE-754
floating-point standard in their handling of infinities and NaNs.
This permits the use of efficient inner loops avoiding a check for
zero divisors.
JOBZ is CHARACTER*1
= ’N’: Compute eigenvalues only;
= ’V’: Compute eigenvalues and eigenvectors.
RANGE is CHARACTER*1
= ’A’: all eigenvalues will be found.
= ’V’: all eigenvalues in the half-open interval (VL,VU]
will be found.
= ’I’: the IL-th through IU-th eigenvalues will be found.
N is INTEGER
The order of the matrix. N >= 0.
D is REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal matrix
T. On exit, D is overwritten.
E is REAL array, dimension (N)
On entry, the (N-1) subdiagonal elements of the tridiagonal
matrix T in elements 1 to N-1 of E. E(N) need not be set on
input, but is used internally as workspace.
On exit, E is overwritten.
VL is REAL
If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.
VU is REAL
If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.
IL is INTEGER
If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.
IU is INTEGER
If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.
M is INTEGER
The total number of eigenvalues found. 0 <= M <= N.
If RANGE = ’A’, M = N, and if RANGE = ’I’, M = IU-IL+1.
W is REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z is REAL array, dimension (LDZ, max(1,M) )
If JOBZ = ’V’, and if INFO = 0, then the first M columns of Z
contain the orthonormal eigenvectors of the matrix T
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and can be computed with a workspace
query by setting NZC = -1, see below.
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = ’V’, then LDZ >= max(1,N).
NZC is INTEGER
The number of eigenvectors to be held in the array Z.
If RANGE = ’A’, then NZC >= max(1,N).
If RANGE = ’V’, then NZC >= the number of eigenvalues in (VL,VU].
If RANGE = ’I’, then NZC >= IU-IL+1.
If NZC = -1, then a workspace query is assumed; the
routine calculates the number of columns of the array Z that
are needed to hold the eigenvectors.
This value is returned as the first entry of the Z array, and
no error message related to NZC is issued by XERBLA.
ISUPPZ is INTEGER ARRAY, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th computed eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is relevant in the case when the matrix
is split. ISUPPZ is only accessed when JOBZ is ’V’ and N > 0.
TRYRAC is LOGICAL
If TRYRAC.EQ..TRUE., indicates that the code should check whether
the tridiagonal matrix defines its eigenvalues to high relative
accuracy. If so, the code uses relative-accuracy preserving
algorithms that might be (a bit) slower depending on the matrix.
If the matrix does not define its eigenvalues to high relative
accuracy, the code can uses possibly faster algorithms.
If TRYRAC.EQ..FALSE., the code is not required to guarantee
relatively accurate eigenvalues and can use the fastest possible
On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
does not define its eigenvalues to high relative accuracy.
WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
(and minimal) LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,18*N)
if JOBZ = ’V’, and LWORK >= max(1,12*N) if JOBZ = ’N’.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK is INTEGER array, dimension (LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= max(1,10*N)
if the eigenvectors are desired, and LIWORK >= max(1,8*N)
if only the eigenvalues are to be computed.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO is INTEGER
On exit, INFO
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = 1X, internal error in SLARRE,
if INFO = 2X, internal error in SLARRV.
Here, the digit X = ABS( IINFO ) < 10, where IINFO is
the nonzero error code returned by SLARRE or
SLARRV, respectively.
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NAG Ltd.
June 2016
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
subroutine stbcon (character NORM, character UPLO, character DIAG, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real RCOND, real, dimension( * ) WORK, integer, dimension( * )
IWORK, integer INFO)
STBCON estimates the reciprocal of the condition number of a
triangular band matrix A, in either the 1-norm or the infinity-norm.
The norm of A is computed and an estimate is obtained for
norm(inv(A)), then the reciprocal of the condition number is
computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals or subdiagonals of the
triangular band matrix A. KD >= 0.
AB is REAL array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first kd+1 rows of the array. The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine stbrfs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldb, * ) B, integer LDB, real,
dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
STBRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular band
coefficient matrix.
The solution matrix X must be computed by STBTRS or some other
means before entering this routine. STBRFS does not do iterative
refinement because doing so cannot improve the backward error.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals or subdiagonals of the
triangular band matrix A. KD >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AB is REAL array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first kd+1 rows of the array. The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is REAL array, dimension (LDX,NRHS)
The solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine stbtrs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldb, * ) B, integer LDB, integer
STBTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular band matrix of order N, and B is an
N-by NRHS matrix. A check is made to verify that A is nonsingular.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
TRANS is CHARACTER*1
Specifies the form the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals or subdiagonals of the
triangular band matrix A. KD >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AB is REAL array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first kd+1 rows of AB. The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the
solutions X have not been computed.
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subroutine stfsm (character TRANSR, character SIDE, character UPLO, character TRANS, character DIAG, integer M, integer N, real ALPHA, real, dimension( 0: * ) A, real, dimension( 0: ldb−1, 0: * ) B,
integer LDB)
STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
Level 3 BLAS like routine for A in RFP Format.
STFSM solves the matrix equation
op( A )*X = alpha*B or X*op( A ) = alpha*B
where alpha is a scalar, X and B are m by n matrices, A is a unit, or
non-unit, upper or lower triangular matrix and op( A ) is one of
op( A ) = A or op( A ) = A**T.
A is in Rectangular Full Packed (RFP) Format.
The matrix X is overwritten on B.
TRANSR is CHARACTER*1
= ’N’: The Normal Form of RFP A is stored;
= ’T’: The Transpose Form of RFP A is stored.
SIDE is CHARACTER*1
On entry, SIDE specifies whether op( A ) appears on the left
or right of X as follows:
SIDE = ’L’ or ’l’ op( A )*X = alpha*B.
SIDE = ’R’ or ’r’ X*op( A ) = alpha*B.
Unchanged on exit.
UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
UPLO = ’U’ or ’u’ RFP A came from an upper triangular matrix
UPLO = ’L’ or ’l’ RFP A came from a lower triangular matrix
Unchanged on exit.
TRANS is CHARACTER*1
On entry, TRANS specifies the form of op( A ) to be used
in the matrix multiplication as follows:
TRANS = ’N’ or ’n’ op( A ) = A.
TRANS = ’T’ or ’t’ op( A ) = A’.
Unchanged on exit.
DIAG is CHARACTER*1
On entry, DIAG specifies whether or not RFP A is unit
triangular as follows:
DIAG = ’U’ or ’u’ A is assumed to be unit triangular.
DIAG = ’N’ or ’n’ A is not assumed to be unit
Unchanged on exit.
M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.
N is INTEGER
On entry, N specifies the number of columns of B. N must be
at least zero.
Unchanged on exit.
ALPHA is REAL
On entry, ALPHA specifies the scalar alpha. When alpha is
zero then A is not referenced and B need not be set before
Unchanged on exit.
A is REAL array, dimension (NT)
NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
RFP Format is described by TRANSR, UPLO and N as follows:
If TRANSR=’N’ then RFP A is (0:N,0:K-1) when N is even;
K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
TRANSR = ’T’ then RFP is the transpose of RFP A as
defined when TRANSR = ’N’. The contents of RFP A are defined
by UPLO as follows: If UPLO = ’U’ the RFP A contains the NT
elements of upper packed A either in normal or
transpose Format. If UPLO = ’L’ the RFP A contains
the NT elements of lower packed A either in normal or
transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = ’T’. When TRANSR is ’N’ the LDA is N+1 when N is
even and is N when is odd.
See the Note below for more details. Unchanged on exit.
B is REAL array, DIMENSION (LDB,N)
Before entry, the leading m by n part of the array B must
contain the right-hand side matrix B, and on exit is
overwritten by the solution matrix X.
LDB is INTEGER
On entry, LDB specifies the first dimension of B as declared
in the calling (sub) program. LDB must be at least
max( 1, m ).
Unchanged on exit.
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Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine stftri (character TRANSR, character UPLO, character DIAG, integer N, real, dimension( 0: * ) A, integer INFO)
STFTRI computes the inverse of a triangular matrix A stored in RFP
This is a Level 3 BLAS version of the algorithm.
TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (NT);
NT=N*(N+1)/2. On entry, the triangular factor of a Hermitian
Positive Definite matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = ’N’
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = ’T’ then RFP is
the transpose of RFP A as defined when
TRANSR = ’N’. The contents of RFP A are defined by UPLO as
follows: If UPLO = ’U’ the RFP A contains the nt elements of
upper packed A; If UPLO = ’L’ the RFP A contains the nt
elements of lower packed A. The LDA of RFP A is (N+1)/2 when
TRANSR = ’T’. When TRANSR is ’N’ the LDA is N+1 when N is
even and N is odd. See the Note below for more details.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, A(i,i) is exactly zero. The triangular
matrix is singular and its inverse can not be computed.
Univ. of Tennessee
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NAG Ltd.
December 2016
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine stfttp (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) ARF, real, dimension( 0: * ) AP, integer INFO)
STFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
STFTTP copies a triangular matrix A from rectangular full packed
format (TF) to standard packed format (TP).
TRANSR is CHARACTER*1
= ’N’: ARF is in Normal format;
= ’T’: ARF is in Transpose format;
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
N is INTEGER
The order of the matrix A. N >= 0.
ARF is REAL array, dimension ( N*(N+1)/2 ),
On entry, the upper or lower triangular matrix A stored in
RFP format. For a further discussion see Notes below.
AP is REAL array, dimension ( N*(N+1)/2 ),
On exit, the upper or lower triangular matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
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NAG Ltd.
December 2016
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine stfttr (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) ARF, real, dimension( 0: lda−1, 0: * ) A, integer LDA, integer INFO)
STFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
STFTTR copies a triangular matrix A from rectangular full packed
format (TF) to standard full format (TR).
TRANSR is CHARACTER*1
= ’N’: ARF is in Normal format;
= ’T’: ARF is in Transpose format.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
N is INTEGER
The order of the matrices ARF and A. N >= 0.
ARF is REAL array, dimension (N*(N+1)/2).
On entry, the upper (if UPLO = ’U’) or lower (if UPLO = ’L’)
matrix A in RFP format. See the "Notes" below for more
A is REAL array, dimension (LDA,N)
On exit, the triangular matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine stgsen (integer IJOB, logical WANTQ, logical WANTZ, logical, dimension( * ) SELECT, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real,
dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, integer M, real PL, real PR, real,
dimension( * ) DIF, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
STGSEN reorders the generalized real Schur decomposition of a real
matrix pair (A, B) (in terms of an orthonormal equivalence trans-
formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
appears in the leading diagonal blocks of the upper quasi-triangular
matrix A and the upper triangular B. The leading columns of Q and
Z form orthonormal bases of the corresponding left and right eigen-
spaces (deflating subspaces). (A, B) must be in generalized real
Schur canonical form (as returned by SGGES), i.e. A is block upper
triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
STGSEN also computes the generalized eigenvalues
w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
of the reordered matrix pair (A, B).
Optionally, STGSEN computes the estimates of reciprocal condition
numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
(A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
between the matrix pairs (A11, B11) and (A22,B22) that correspond to
the selected cluster and the eigenvalues outside the cluster, resp.,
and norms of "projections" onto left and right eigenspaces w.r.t.
the selected cluster in the (1,1)-block.
IJOB is INTEGER
Specifies whether condition numbers are required for the
cluster of eigenvalues (PL and PR) or the deflating subspaces
(Difu and Difl):
=0: Only reorder w.r.t. SELECT. No extras.
=1: Reciprocal of norms of "projections" onto left and right
eigenspaces w.r.t. the selected cluster (PL and PR).
=2: Upper bounds on Difu and Difl. F-norm-based estimate
=3: Estimate of Difu and Difl. 1-norm-based estimate
About 5 times as expensive as IJOB = 2.
=4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
version to get it all.
=5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
N is INTEGER
The order of the matrices A and B. N >= 0.
A is REAL array, dimension(LDA,N)
On entry, the upper quasi-triangular matrix A, with (A, B) in
generalized real Schur canonical form.
On exit, A is overwritten by the reordered matrix A.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension(LDB,N)
On entry, the upper triangular matrix B, with (A, B) in
generalized real Schur canonical form.
On exit, B is overwritten by the reordered matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
ALPHAR is REAL array, dimension (N)
ALPHAI is REAL array, dimension (N)
BETA is REAL array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
and BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real generalized Schur form of (A,B) were further reduced
to triangular form using complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Q is REAL array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
On exit, Q has been postmultiplied by the left orthogonal
transformation matrix which reorder (A, B); The leading M
columns of Q form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTQ = .FALSE., Q is not referenced.
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1;
and if WANTQ = .TRUE., LDQ >= N.
Z is REAL array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
On exit, Z has been postmultiplied by the left orthogonal
transformation matrix which reorder (A, B); The leading M
columns of Z form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTZ = .FALSE., Z is not referenced.
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1;
If WANTZ = .TRUE., LDZ >= N.
M is INTEGER
The dimension of the specified pair of left and right eigen-
spaces (deflating subspaces). 0 <= M <= N.
PL is REAL
PR is REAL
If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
reciprocal of the norm of "projections" onto left and right
eigenspaces with respect to the selected cluster.
0 < PL, PR <= 1.
If M = 0 or M = N, PL = PR = 1.
If IJOB = 0, 2 or 3, PL and PR are not referenced.
DIF is REAL array, dimension (2).
If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
estimates of Difu and Difl.
If M = 0 or N, DIF(1:2) = F-norm([A, B]).
If IJOB = 0 or 1, DIF is not referenced.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 4*N+16.
If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= 1.
If IJOB = 1, 2 or 4, LIWORK >= N+6.
If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO is INTEGER
=0: Successful exit.
<0: If INFO = -i, the i-th argument had an illegal value.
=1: Reordering of (A, B) failed because the transformed
matrix pair (A, B) would be too far from generalized
Schur form; the problem is very ill-conditioned.
(A, B) may have been partially reordered.
If requested, 0 is returned in DIF(*), PL and PR.
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Further Details:
STGSEN first collects the selected eigenvalues by computing
orthogonal U and W that move them to the top left corner of (A, B).
In other words, the selected eigenvalues are the eigenvalues of
(A11, B11) in:
U**T*(A, B)*W = (A11 A12) (B11 B12) n1
( 0 A22),( 0 B22) n2
n1 n2 n1 n2
where N = n1+n2 and U**T means the transpose of U. The first n1 columns
of U and W span the specified pair of left and right eigenspaces
(deflating subspaces) of (A, B).
If (A, B) has been obtained from the generalized real Schur
decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
reordered generalized real Schur form of (C, D) is given by
(C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
and the first n1 columns of Q*U and Z*W span the corresponding
deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
Note that if the selected eigenvalue is sufficiently ill-conditioned,
then its value may differ significantly from its value before
The reciprocal condition numbers of the left and right eigenspaces
spanned by the first n1 columns of U and W (or Q*U and Z*W) may
be returned in DIF(1:2), corresponding to Difu and Difl, resp.
The Difu and Difl are defined as:
Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
where sigma-min(Zu) is the smallest singular value of the
(2*n1*n2)-by-(2*n1*n2) matrix
Zu = [ kron(In2, A11) -kron(A22**T, In1) ]
[ kron(In2, B11) -kron(B22**T, In1) ].
Here, Inx is the identity matrix of size nx and A22**T is the
transpose of A22. kron(X, Y) is the Kronecker product between
the matrices X and Y.
When DIF(2) is small, small changes in (A, B) can cause large changes
in the deflating subspace. An approximate (asymptotic) bound on the
maximum angular error in the computed deflating subspaces is
EPS * norm((A, B)) / DIF(2),
where EPS is the machine precision.
The reciprocal norm of the projectors on the left and right
eigenspaces associated with (A11, B11) may be returned in PL and PR.
They are computed as follows. First we compute L and R so that
P*(A, B)*Q is block diagonal, where
P = ( I -L ) n1 Q = ( I R ) n1
( 0 I ) n2 and ( 0 I ) n2
n1 n2 n1 n2
and (L, R) is the solution to the generalized Sylvester equation
A11*R - L*A22 = -A12
B11*R - L*B22 = -B12
Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is
EPS * norm((A, B)) / PL.
There are also global error bounds which valid for perturbations up
to a certain restriction: A lower bound (x) on the smallest
F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
(i.e. (A + E, B + F), is
x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
An approximate bound on x can be computed from DIF(1:2), PL and PR.
If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
(L’, R’) and unperturbed (L, R) left and right deflating subspaces
associated with the selected cluster in the (1,1)-blocks can be
bounded as
max-angle(L, L’) <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
max-angle(R, R’) <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
See LAPACK User’s Guide section 4.11 or the following references
for more information.
Note that if the default method for computing the Frobenius-norm-
based estimate DIF is not wanted (see SLATDF), then the parameter
IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF
(IJOB = 2 will be used)). See STGSYL for more details.
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software,
Report UMINF - 94.04, Department of Computing Science, Umea
University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
subroutine stgsja (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, integer K, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B,
integer LDB, real TOLA, real TOLB, real, dimension( * ) ALPHA, real, dimension( * ) BETA, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldq, * )
Q, integer LDQ, real, dimension( * ) WORK, integer NCYCLE, integer INFO)
STGSJA computes the generalized singular value decomposition (GSVD)
of two real upper triangular (or trapezoidal) matrices A and B.
On entry, it is assumed that matrices A and B have the following
forms, which may be obtained by the preprocessing subroutine SGGSVP
from a general M-by-N matrix A and P-by-N matrix B:
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
B = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal.
On exit,
U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
where U, V and Q are orthogonal matrices.
R is a nonsingular upper triangular matrix, and D1 and D2 are
‘‘diagonal’’ matrices, which are of the following structures:
If M-K-L >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )
K L
D2 = L ( 0 S )
P-L ( 0 0 )
N-K-L K L
( 0 R ) = K ( 0 R11 R12 ) K
L ( 0 0 R22 ) L
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )
K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )
N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.
R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.
The computation of the orthogonal transformation matrices U, V or Q
is optional. These matrices may either be formed explicitly, or they
may be postmultiplied into input matrices U1, V1, or Q1.
JOBU is CHARACTER*1
= ’U’: U must contain an orthogonal matrix U1 on entry, and
the product U1*U is returned;
= ’I’: U is initialized to the unit matrix, and the
orthogonal matrix U is returned;
= ’N’: U is not computed.
JOBV is CHARACTER*1
= ’V’: V must contain an orthogonal matrix V1 on entry, and
the product V1*V is returned;
= ’I’: V is initialized to the unit matrix, and the
orthogonal matrix V is returned;
= ’N’: V is not computed.
JOBQ is CHARACTER*1
= ’Q’: Q must contain an orthogonal matrix Q1 on entry, and
the product Q1*Q is returned;
= ’I’: Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= ’N’: Q is not computed.
M is INTEGER
The number of rows of the matrix A. M >= 0.
P is INTEGER
The number of rows of the matrix B. P >= 0.
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
K is INTEGER
L is INTEGER
K and L specify the subblocks in the input matrices A and B:
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
of A and B, whose GSVD is going to be computed by STGSJA.
See Further Details.
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
matrix R or part of R. See Purpose for details.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
a part of R. See Purpose for details.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).
TOLA is REAL
TOLB is REAL
TOLA and TOLB are the convergence criteria for the Jacobi-
Kogbetliantz iteration procedure. Generally, they are the
same as used in the preprocessing step, say
TOLA = max(M,N)*norm(A)*MACHEPS,
TOLB = max(P,N)*norm(B)*MACHEPS.
ALPHA is REAL array, dimension (N)
BETA is REAL array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
ALPHA(1:K) = 1,
BETA(1:K) = 0,
and if M-K-L >= 0,
ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(S),
or if M-K-L < 0,
ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
Furthermore, if K+L < N,
ALPHA(K+L+1:N) = 0 and
BETA(K+L+1:N) = 0.
U is REAL array, dimension (LDU,M)
On entry, if JOBU = ’U’, U must contain a matrix U1 (usually
the orthogonal matrix returned by SGGSVP).
On exit,
if JOBU = ’I’, U contains the orthogonal matrix U;
if JOBU = ’U’, U contains the product U1*U.
If JOBU = ’N’, U is not referenced.
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = ’U’; LDU >= 1 otherwise.
V is REAL array, dimension (LDV,P)
On entry, if JOBV = ’V’, V must contain a matrix V1 (usually
the orthogonal matrix returned by SGGSVP).
On exit,
if JOBV = ’I’, V contains the orthogonal matrix V;
if JOBV = ’V’, V contains the product V1*V.
If JOBV = ’N’, V is not referenced.
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = ’V’; LDV >= 1 otherwise.
Q is REAL array, dimension (LDQ,N)
On entry, if JOBQ = ’Q’, Q must contain a matrix Q1 (usually
the orthogonal matrix returned by SGGSVP).
On exit,
if JOBQ = ’I’, Q contains the orthogonal matrix Q;
if JOBQ = ’Q’, Q contains the product Q1*Q.
If JOBQ = ’N’, Q is not referenced.
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = ’Q’; LDQ >= 1 otherwise.
WORK is REAL array, dimension (2*N)
NCYCLE is INTEGER
The number of cycles required for convergence.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles.
Internal Parameters
MAXIT specifies the total loops that the iterative procedure
may take. If after MAXIT cycles, the routine fails to
converge, we return INFO = 1..fi
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Further Details:
STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
matrix B13 to the form:
U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
of Z. C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an L-by-L nonsingular upper triangular matrix.
subroutine stgsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension(
ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) S, real, dimension( * ) DIF, integer MM, integer M, real, dimension( * ) WORK, integer LWORK, integer,
dimension( * ) IWORK, integer INFO)
STGSNA estimates reciprocal condition numbers for specified
eigenvalues and/or eigenvectors of a matrix pair (A, B) in
generalized real Schur canonical form (or of any matrix pair
(Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
Z**T denotes the transpose of Z.
(A, B) must be in generalized real Schur form (as returned by SGGES),
i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
blocks. B is upper triangular.
JOB is CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (DIF):
= ’E’: for eigenvalues only (S);
= ’V’: for eigenvectors only (DIF);
= ’B’: for both eigenvalues and eigenvectors (S and DIF).
HOWMNY is CHARACTER*1
= ’A’: compute condition numbers for all eigenpairs;
= ’S’: compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT is LOGICAL array, dimension (N)
If HOWMNY = ’S’, SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers
for the eigenpair corresponding to a real eigenvalue w(j),
SELECT(j) must be set to .TRUE.. To select condition numbers
corresponding to a complex conjugate pair of eigenvalues w(j)
and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
set to .TRUE..
If HOWMNY = ’A’, SELECT is not referenced.
N is INTEGER
The order of the square matrix pair (A, B). N >= 0.
A is REAL array, dimension (LDA,N)
The upper quasi-triangular matrix A in the pair (A,B).
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension (LDB,N)
The upper triangular matrix B in the pair (A,B).
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
VL is REAL array, dimension (LDVL,M)
If JOB = ’E’ or ’B’, VL must contain left eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns of VL, as returned by STGEVC.
If JOB = ’V’, VL is not referenced.
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1.
If JOB = ’E’ or ’B’, LDVL >= N.
VR is REAL array, dimension (LDVR,M)
If JOB = ’E’ or ’B’, VR must contain right eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns ov VR, as returned by STGEVC.
If JOB = ’V’, VR is not referenced.
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1.
If JOB = ’E’ or ’B’, LDVR >= N.
S is REAL array, dimension (MM)
If JOB = ’E’ or ’B’, the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus
S(j), DIF(j), and the j-th columns of VL and VR all
correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If JOB = ’V’, S is not referenced.
DIF is REAL array, dimension (MM)
If JOB = ’V’ or ’B’, the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of DIF are set to the same value. If
the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
is set to 0; this can only occur when the true value would be
very small anyway.
If JOB = ’E’, DIF is not referenced.
MM is INTEGER
The number of elements in the arrays S and DIF. MM >= M.
M is INTEGER
The number of elements of the arrays S and DIF used to store
the specified condition numbers; for each selected real
eigenvalue one element is used, and for each selected complex
conjugate pair of eigenvalues, two elements are used.
If HOWMNY = ’A’, M is set to N.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If JOB = ’V’ or ’B’ LWORK >= 2*N*(N+2)+16.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK is INTEGER array, dimension (N + 6)
If JOB = ’E’, IWORK is not referenced.
INFO is INTEGER
=0: Successful exit
<0: If INFO = -i, the i-th argument had an illegal value
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Further Details:
The reciprocal of the condition number of a generalized eigenvalue
w = (a, b) is defined as
S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
where u and v are the left and right eigenvectors of (A, B)
corresponding to w; |z| denotes the absolute value of the complex
number, and norm(u) denotes the 2-norm of the vector u.
The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
of the matrix pair (A, B). If both a and b equal zero, then (A B) is
singular and S(I) = -1 is returned.
An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact
eigenvalue lambda is
chord(w, lambda) <= EPS * norm(A, B) / S(I)
where EPS is the machine precision.
The reciprocal of the condition number DIF(i) of right eigenvector u
and left eigenvector v corresponding to the generalized eigenvalue w
is defined as follows:
a) If the i-th eigenvalue w = (a,b) is real
Suppose U and V are orthogonal transformations such that
U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
( 0 S22 ),( 0 T22 ) n-1
1 n-1 1 n-1
Then the reciprocal condition number DIF(i) is
Difl((a, b), (S22, T22)) = sigma-min( Zl ),
where sigma-min(Zl) denotes the smallest singular value of the
2(n-1)-by-2(n-1) matrix
Zl = [ kron(a, In-1) -kron(1, S22) ]
[ kron(b, In-1) -kron(1, T22) ] .
Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
Kronecker product between the matrices X and Y.
Note that if the default method for computing DIF(i) is wanted
(see SLATDF), then the parameter DIFDRI (see below) should be
changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
See STGSYL for more details.
b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
Suppose U and V are orthogonal transformations such that
U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
( 0 S22 ),( 0 T22) n-2
2 n-2 2 n-2
and (S11, T11) corresponds to the complex conjugate eigenvalue
pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
( 0 s22 ) ( 0 t22 )
where the generalized eigenvalues w = s11/t11 and
conjg(w) = s22/t22.
Then the reciprocal condition number DIF(i) is bounded by
min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
Z1 is the complex 2-by-2 matrix
Z1 = [ s11 -s22 ]
[ t11 -t22 ],
This is done by computing (using real arithmetic) the
roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
where Z1**T denotes the transpose of Z1 and det(X) denotes
the determinant of X.
and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ]
[ kron(T11**T, In-2) -kron(I2, T22) ]
Note that if the default method for computing DIF is wanted (see
SLATDF), then the parameter DIFDRI (see below) should be changed
from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
for more details.
For each eigenvalue/vector specified by SELECT, DIF stores a
Frobenius norm-based estimate of Difl.
An approximate error bound for the i-th computed eigenvector VL(i) or
VR(i) is given by
EPS * norm(A, B) / DIF(i).
See ref. [2-3] for more details and further references.
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software,
Report UMINF - 94.04, Department of Computing Science, Umea
University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
No 1, 1996.
subroutine stpcon (character NORM, character UPLO, character DIAG, integer N, real, dimension( * ) AP, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
STPCON estimates the reciprocal of the condition number of a packed
triangular matrix A, in either the 1-norm or the infinity-norm.
The norm of A is computed and an estimate is obtained for
norm(inv(A)), then the reciprocal of the condition number is
computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array. The j-th column of A is stored in the array
AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine stpmqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer L, integer NB, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real,
dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, integer INFO)
STPMQRT applies a real orthogonal matrix Q obtained from a
"triangular-pentagonal" real block reflector H to a general
real matrix C, which consists of two blocks A and B.
SIDE is CHARACTER*1
= ’L’: apply Q or Q^T from the Left;
= ’R’: apply Q or Q^T from the Right.
TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q^T.
M is INTEGER
The number of rows of the matrix B. M >= 0.
N is INTEGER
The number of columns of the matrix B. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
L is INTEGER
The order of the trapezoidal part of V.
K >= L >= 0. See Further Details.
NB is INTEGER
The block size used for the storage of T. K >= NB >= 1.
This must be the same value of NB used to generate T
in CTPQRT.
V is REAL array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
CTPQRT in B. See Further Details.
LDV is INTEGER
The leading dimension of the array V.
If SIDE = ’L’, LDV >= max(1,M);
if SIDE = ’R’, LDV >= max(1,N).
T is REAL array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by CTPQRT, stored as a NB-by-K matrix.
LDT is INTEGER
The leading dimension of the array T. LDT >= NB.
A is REAL array, dimension
(LDA,N) if SIDE = ’L’ or
(LDA,K) if SIDE = ’R’
On entry, the K-by-N or M-by-K matrix A.
On exit, A is overwritten by the corresponding block of
Q*C or Q^T*C or C*Q or C*Q^T. See Further Details.
LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDC >= max(1,K);
If SIDE = ’R’, LDC >= max(1,M).
B is REAL array, dimension (LDB,N)
On entry, the M-by-N matrix B.
On exit, B is overwritten by the corresponding block of
Q*C or Q^T*C or C*Q or C*Q^T. See Further Details.
LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,M).
WORK is REAL array. The dimension of WORK is
N*NB if SIDE = ’L’, or M*NB if SIDE = ’R’.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
The columns of the pentagonal matrix V contain the elementary reflectors
H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
trapezoidal block V2:
V = [V1]
The size of the trapezoidal block V2 is determined by the parameter L,
where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L
rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular;
if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
If SIDE = ’L’: C = [A] where A is K-by-N, B is M-by-N and V is M-by-K.
If SIDE = ’R’: C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K.
The real orthogonal matrix Q is formed from V and T.
If TRANS=’N’ and SIDE=’L’, C is on exit replaced with Q * C.
If TRANS=’T’ and SIDE=’L’, C is on exit replaced with Q^T * C.
If TRANS=’N’ and SIDE=’R’, C is on exit replaced with C * Q.
If TRANS=’T’ and SIDE=’R’, C is on exit replaced with C * Q^T.
subroutine stpqrt (integer M, integer N, integer L, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldt, * ) T, integer LDT, real,
dimension( * ) WORK, integer INFO)
STPQRT computes a blocked QR factorization of a real
"triangular-pentagonal" matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.
M is INTEGER
The number of rows of the matrix B.
M >= 0.
N is INTEGER
The number of columns of the matrix B, and the order of the
triangular matrix A.
N >= 0.
L is INTEGER
The number of rows of the upper trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
NB is INTEGER
The block size to be used in the blocked QR. N >= NB >= 1.
A is REAL array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first M-L rows
are rectangular, and the last L rows are upper trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
T is REAL array, dimension (LDT,N)
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See Further Details.
LDT is INTEGER
The leading dimension of the array T. LDT >= NB.
WORK is REAL array, dimension (NB*N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
The input matrix C is a (N+M)-by-N matrix
C = [ A ]
[ B ]
where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:
B = [ B1 ] <- (M-L)-by-N rectangular
[ B2 ] <- L-by-N upper trapezoidal.
The upper trapezoidal matrix B2 consists of the first L rows of a
N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is upper triangular.
The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C
C = [ A ] <- upper triangular N-by-N
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ] <- identity, N-by-N
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
V = [ V1 ] <- (M-L)-by-N rectangular
[ V2 ] <- L-by-N upper trapezoidal.
The columns of V represent the vectors which define the H(i)’s.
The number of blocks is B = ceiling(N/NB), where each
block is of order NB except for the last block, which is of order
IB = N - (B-1)*NB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
for the last block) T’s are stored in the NB-by-N matrix T as
T = [T1 T2 ... TB].
subroutine stpqrt2 (integer M, integer N, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldt, * ) T, integer LDT, integer INFO)
STPQRT2 computes a QR factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
STPQRT2 computes a QR factorization of a real "triangular-pentagonal"
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.
M is INTEGER
The total number of rows of the matrix B.
M >= 0.
N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.
L is INTEGER
The number of rows of the upper trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
A is REAL array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first M-L rows
are rectangular, and the last L rows are upper trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
T is REAL array, dimension (LDT,N)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
The input matrix C is a (N+M)-by-N matrix
C = [ A ]
[ B ]
where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:
B = [ B1 ] <- (M-L)-by-N rectangular
[ B2 ] <- L-by-N upper trapezoidal.
The upper trapezoidal matrix B2 consists of the first L rows of a
N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is upper triangular.
The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C
C = [ A ] <- upper triangular N-by-N
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ] <- identity, N-by-N
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
V = [ V1 ] <- (M-L)-by-N rectangular
[ V2 ] <- L-by-N upper trapezoidal.
The columns of V represent the vectors which define the H(i)’s.
The (M+N)-by-(M+N) block reflector H is then given by
H = I - W * T * W^H
where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.
subroutine stprfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer
LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
STPRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular packed
coefficient matrix.
The solution matrix X must be computed by STPTRS or some other
means before entering this routine. STPRFS does not do iterative
refinement because doing so cannot improve the backward error.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array. The j-th column of A is stored in the array
AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is REAL array, dimension (LDX,NRHS)
The solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine stptri (character UPLO, character DIAG, integer N, real, dimension( * ) AP, integer INFO)
STPTRI computes the inverse of a real upper or lower triangular
matrix A stored in packed format.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangular matrix A, stored
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
See below for further details.
On exit, the (triangular) inverse of the original matrix, in
the same packed storage format.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, A(i,i) is exactly zero. The triangular
matrix is singular and its inverse can not be computed.
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Further Details:
A triangular matrix A can be transferred to packed storage using one
of the following program segments:
UPLO = ’U’: UPLO = ’L’:
JC = 1 JC = 1
DO 2 J = 1, N DO 2 J = 1, N
DO 1 I = 1, J DO 1 I = J, N
AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J)
1 CONTINUE 1 CONTINUE
JC = JC + J JC = JC + N - J + 1
2 CONTINUE 2 CONTINUE
subroutine stptrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( ldb, * ) B, integer LDB, integer INFO)
STPTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular matrix of order N stored in packed format,
and B is an N-by-NRHS matrix. A check is made to verify that A is
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array. The j-th column of A is stored in the array
AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the
solutions X have not been computed.
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subroutine stpttf (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) AP, real, dimension( 0: * ) ARF, integer INFO)
STPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
STPTTF copies a triangular matrix A from standard packed format (TP)
to rectangular full packed format (TF).
TRANSR is CHARACTER*1
= ’N’: ARF in Normal format is wanted;
= ’T’: ARF in Conjugate-transpose format is wanted.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension ( N*(N+1)/2 ),
On entry, the upper or lower triangular matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
ARF is REAL array, dimension ( N*(N+1)/2 ),
On exit, the upper or lower triangular matrix A stored in
RFP format. For a further discussion see Notes below.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine stpttr (character UPLO, integer N, real, dimension( * ) AP, real, dimension( lda, * ) A, integer LDA, integer INFO)
STPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
STPTTR copies a triangular matrix A from standard packed format (TP)
to standard full format (TR).
UPLO is CHARACTER*1
= ’U’: A is upper triangular.
= ’L’: A is lower triangular.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension ( N*(N+1)/2 ),
On entry, the upper or lower triangular matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
A is REAL array, dimension ( LDA, N )
On exit, the triangular matrix A. If UPLO = ’U’, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine strcon (character NORM, character UPLO, character DIAG, integer N, real, dimension( lda, * ) A, integer LDA, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer
STRCON estimates the reciprocal of the condition number of a
triangular matrix A, in either the 1-norm or the infinity-norm.
The norm of A is computed and an estimate is obtained for
norm(inv(A)), then the reciprocal of the condition number is
computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
The triangular matrix A. If UPLO = ’U’, the leading N-by-N
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = ’L’, the leading N-by-N lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = ’U’, the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine strevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension(
ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * ) WORK, integer INFO)
STREVC computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix: A = Q*T*Q**T, as computed by SHSEQR.
The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
T*x = w*x, (y**H)*T = w*(y**H)
where y**H denotes the conjugate transpose of y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.
This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the orthogonal factor that reduces a matrix
A to Schur form T, then Q*X and Q*Y are the matrices of right and
left eigenvectors of A.
SIDE is CHARACTER*1
= ’R’: compute right eigenvectors only;
= ’L’: compute left eigenvectors only;
= ’B’: compute both right and left eigenvectors.
HOWMNY is CHARACTER*1
= ’A’: compute all right and/or left eigenvectors;
= ’B’: compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= ’S’: compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.
SELECT is LOGICAL array, dimension (N)
If HOWMNY = ’S’, SELECT specifies the eigenvectors to be
If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector is
computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
Not referenced if HOWMNY = ’A’ or ’B’.
N is INTEGER
The order of the matrix T. N >= 0.
T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = ’L’ or ’B’ and HOWMNY = ’B’, VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = ’L’ or ’B’, VL contains:
if HOWMNY = ’A’, the matrix Y of left eigenvectors of T;
if HOWMNY = ’B’, the matrix Q*Y;
if HOWMNY = ’S’, the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = ’R’.
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1, and if
SIDE = ’L’ or ’B’, LDVL >= N.
VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = ’R’ or ’B’ and HOWMNY = ’B’, VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = ’R’ or ’B’, VR contains:
if HOWMNY = ’A’, the matrix X of right eigenvectors of T;
if HOWMNY = ’B’, the matrix Q*X;
if HOWMNY = ’S’, the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = ’L’.
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = ’R’ or ’B’, LDVR >= N.
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = ’A’ or ’B’, M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
WORK is REAL array, dimension (3*N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.
Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.
subroutine strevc3 (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension
( ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * ) WORK, integer LWORK, integer INFO)
STREVC3 computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix: A = Q*T*Q**T, as computed by SHSEQR.
The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
T*x = w*x, (y**H)*T = w*(y**H)
where y**H denotes the conjugate transpose of y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.
This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the orthogonal factor that reduces a matrix
A to Schur form T, then Q*X and Q*Y are the matrices of right and
left eigenvectors of A.
This uses a Level 3 BLAS version of the back transformation.
SIDE is CHARACTER*1
= ’R’: compute right eigenvectors only;
= ’L’: compute left eigenvectors only;
= ’B’: compute both right and left eigenvectors.
HOWMNY is CHARACTER*1
= ’A’: compute all right and/or left eigenvectors;
= ’B’: compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= ’S’: compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.
SELECT is LOGICAL array, dimension (N)
If HOWMNY = ’S’, SELECT specifies the eigenvectors to be
If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector is
computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
Not referenced if HOWMNY = ’A’ or ’B’.
N is INTEGER
The order of the matrix T. N >= 0.
T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = ’L’ or ’B’ and HOWMNY = ’B’, VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = ’L’ or ’B’, VL contains:
if HOWMNY = ’A’, the matrix Y of left eigenvectors of T;
if HOWMNY = ’B’, the matrix Q*Y;
if HOWMNY = ’S’, the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = ’R’.
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1, and if SIDE = ’L’ or ’B’, LDVL >= N.
VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = ’R’ or ’B’ and HOWMNY = ’B’, VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = ’R’ or ’B’, VR contains:
if HOWMNY = ’A’, the matrix X of right eigenvectors of T;
if HOWMNY = ’B’, the matrix Q*X;
if HOWMNY = ’S’, the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = ’L’.
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1, and if SIDE = ’R’ or ’B’, LDVR >= N.
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = ’A’ or ’B’, M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
WORK is REAL array, dimension (MAX(1,LWORK))
LWORK is INTEGER
The dimension of array WORK. LWORK >= max(1,3*N).
For optimum performance, LWORK >= N + 2*N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.
Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.
subroutine strexc (character COMPQ, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldq, * ) Q, integer LDQ, integer IFST, integer ILST, real, dimension( * ) WORK, integer INFO)
STREXC reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that the diagonal block of T with row index IFST is
moved to row ILST.
The real Schur form T is reordered by an orthogonal similarity
transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
is updated by postmultiplying it with Z.
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.
COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.
N is INTEGER
The order of the matrix T. N >= 0.
If N == 0 arguments ILST and IFST may be any value.
T is REAL array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
Schur canonical form.
On exit, the reordered upper quasi-triangular matrix, again
in Schur canonical form.
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q is REAL array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
orthogonal transformation matrix Z which reorders T.
If COMPQ = ’N’, Q is not referenced.
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1, and if
COMPQ = ’V’, LDQ >= max(1,N).
IFST is INTEGER
ILST is INTEGER
Specify the reordering of the diagonal blocks of T.
The block with row index IFST is moved to row ILST, by a
sequence of transpositions between adjacent blocks.
On exit, if IFST pointed on entry to the second row of a
2-by-2 block, it is changed to point to the first row; ILST
always points to the first row of the block in its final
position (which may differ from its input value by +1 or -1).
1 <= IFST <= N; 1 <= ILST <= N.
WORK is REAL array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: two adjacent blocks were too close to swap (the problem
is very ill-conditioned); T may have been partially
reordered, and ILST points to the first row of the
current position of the block being moved.
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subroutine strrfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx,
* ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
STRRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular
coefficient matrix.
The solution matrix X must be computed by STRTRS or some other
means before entering this routine. STRRFS does not do iterative
refinement because doing so cannot improve the backward error.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A is REAL array, dimension (LDA,N)
The triangular matrix A. If UPLO = ’U’, the leading N-by-N
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = ’L’, the leading N-by-N lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = ’U’, the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is REAL array, dimension (LDX,NRHS)
The solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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subroutine strsen (character JOB, character COMPQ, logical, dimension( * ) SELECT, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * )
WR, real, dimension( * ) WI, integer M, real S, real SEP, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
STRSEN reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
the leading diagonal blocks of the upper quasi-triangular matrix T,
and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.
Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.
JOB is CHARACTER*1
Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= ’N’: none;
= ’E’: for eigenvalues only (S);
= ’V’: for invariant subspace only (SEP);
= ’B’: for both eigenvalues and invariant subspace (S and
COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.
SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
N is INTEGER
The order of the matrix T. N >= 0.
T is REAL array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form.
On exit, T is overwritten by the reordered matrix T, again in
Schur canonical form, with the selected eigenvalues in the
leading diagonal blocks.
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q is REAL array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
orthogonal transformation matrix which reorders T; the
leading M columns of Q form an orthonormal basis for the
specified invariant subspace.
If COMPQ = ’N’, Q is not referenced.
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = ’V’, LDQ >= N.
WR is REAL array, dimension (N)
WI is REAL array, dimension (N)
The real and imaginary parts, respectively, of the reordered
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T, with WR(i) = T(i,i) and, if
T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
WI(i+1) = -WI(i). Note that if a complex eigenvalue is
sufficiently ill-conditioned, then its value may differ
significantly from its value before reordering.
M is INTEGER
The dimension of the specified invariant subspace.
0 < = M <= N.
S is REAL
If JOB = ’E’ or ’B’, S is a lower bound on the reciprocal
condition number for the selected cluster of eigenvalues.
S cannot underestimate the true reciprocal condition number
by more than a factor of sqrt(N). If M = 0 or N, S = 1.
If JOB = ’N’ or ’V’, S is not referenced.
SEP is REAL
If JOB = ’V’ or ’B’, SEP is the estimated reciprocal
condition number of the specified invariant subspace. If
M = 0 or N, SEP = norm(T).
If JOB = ’N’ or ’E’, SEP is not referenced.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If JOB = ’N’, LWORK >= max(1,N);
if JOB = ’E’, LWORK >= max(1,M*(N-M));
if JOB = ’V’ or ’B’, LWORK >= max(1,2*M*(N-M)).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK is INTEGER
The dimension of the array IWORK.
If JOB = ’N’ or ’E’, LIWORK >= 1;
if JOB = ’V’ or ’B’, LIWORK >= max(1,M*(N-M)).
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too
close to separate (the problem is very ill-conditioned);
T may have been partially reordered, and WR and WI
contain the eigenvalues in the same order as in T; S and
SEP (if requested) are set to zero.
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Further Details:
STRSEN first collects the selected eigenvalues by computing an
orthogonal transformation Z to move them to the top left corner of T.
In other words, the selected eigenvalues are the eigenvalues of T11
Z**T * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2
where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
of Z span the specified invariant subspace of T.
If T has been obtained from the real Schur factorization of a matrix
A = Q*T*Q**T, then the reordered real Schur factorization of A is given
by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
the corresponding invariant subspace of A.
The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that
P = ( I R ) n1
( 0 0 ) n2
n1 n2
is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:
T11*R - R*T22 = T12.
Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound
(1 + F-norm(R)**2)**(-1/2)
on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
An approximate error bound for the computed average of the
eigenvalues of T11 is
EPS * norm(T) / S
where EPS is the machine precision.
The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:
sep( T11, T22 ) = sigma-min( C )
where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix
C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is
EPS * norm(T) / SEP
subroutine strsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension(
ldvr, * ) VR, integer LDVR, real, dimension( * ) S, real, dimension( * ) SEP, integer MM, integer M, real, dimension( ldwork, * ) WORK, integer LDWORK, integer, dimension( * ) IWORK, integer INFO)
STRSNA estimates reciprocal condition numbers for specified
eigenvalues and/or right eigenvectors of a real upper
quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.
JOB is CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (SEP):
= ’E’: for eigenvalues only (S);
= ’V’: for eigenvectors only (SEP);
= ’B’: for both eigenvalues and eigenvectors (S and SEP).
HOWMNY is CHARACTER*1
= ’A’: compute condition numbers for all eigenpairs;
= ’S’: compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT is LOGICAL array, dimension (N)
If HOWMNY = ’S’, SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers
for the eigenpair corresponding to a real eigenvalue w(j),
SELECT(j) must be set to .TRUE.. To select condition numbers
corresponding to a complex conjugate pair of eigenvalues w(j)
and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
set to .TRUE..
If HOWMNY = ’A’, SELECT is not referenced.
N is INTEGER
The order of the matrix T. N >= 0.
T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T, in Schur canonical form.
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL is REAL array, dimension (LDVL,M)
If JOB = ’E’ or ’B’, VL must contain left eigenvectors of T
(or of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VL, as returned by
SHSEIN or STREVC.
If JOB = ’V’, VL is not referenced.
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = ’E’ or ’B’, LDVL >= N.
VR is REAL array, dimension (LDVR,M)
If JOB = ’E’ or ’B’, VR must contain right eigenvectors of T
(or of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VR, as returned by
SHSEIN or STREVC.
If JOB = ’V’, VR is not referenced.
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = ’E’ or ’B’, LDVR >= N.
S is REAL array, dimension (MM)
If JOB = ’E’ or ’B’, the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus
S(j), SEP(j), and the j-th columns of VL and VR all
correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If JOB = ’V’, S is not referenced.
SEP is REAL array, dimension (MM)
If JOB = ’V’ or ’B’, the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of SEP are set to the same value. If
the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
is set to 0; this can only occur when the true value would be
very small anyway.
If JOB = ’E’, SEP is not referenced.
MM is INTEGER
The number of elements in the arrays S (if JOB = ’E’ or ’B’)
and/or SEP (if JOB = ’V’ or ’B’). MM >= M.
M is INTEGER
The number of elements of the arrays S and/or SEP actually
used to store the estimated condition numbers.
If HOWMNY = ’A’, M is set to N.
WORK is REAL array, dimension (LDWORK,N+6)
If JOB = ’E’, WORK is not referenced.
LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = ’V’ or ’B’, LDWORK >= N.
IWORK is INTEGER array, dimension (2*(N-1))
If JOB = ’E’, IWORK is not referenced.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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Further Details:
The reciprocal of the condition number of an eigenvalue lambda is
defined as
S(lambda) = |v**T*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v**T denotes the transpose of v, and norm(u)
denotes the Euclidean norm. These reciprocal condition numbers always
lie between zero (very badly conditioned) and one (very well
conditioned). If n = 1, S(lambda) is defined to be 1.
An approximate error bound for a computed eigenvalue W(i) is given by
EPS * norm(T) / S(i)
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u
corresponding to lambda is defined as follows. Suppose
T = ( lambda c )
( 0 T22 )
Then the reciprocal condition number is
SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
where sigma-min denotes the smallest singular value. We approximate
the smallest singular value by the reciprocal of an estimate of the
one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
defined to be abs(T(1,1)).
An approximate error bound for a computed right eigenvector VR(i)
is given by
EPS * norm(T) / SEP(i)
subroutine strti2 (character UPLO, character DIAG, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)
STRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
STRTI2 computes the inverse of a real upper or lower triangular
This is the Level 2 BLAS version of the algorithm.
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= ’U’: Upper triangular
= ’L’: Lower triangular
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= ’N’: Non-unit triangular
= ’U’: Unit triangular
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the triangular matrix A. If UPLO = ’U’, the
leading n by n upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading n by n lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If DIAG = ’U’, the
diagonal elements of A are also not referenced and are
assumed to be 1.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
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subroutine strtri (character UPLO, character DIAG, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)
STRTRI computes the inverse of a real upper or lower triangular
matrix A.
This is the Level 3 BLAS version of the algorithm.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the triangular matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If DIAG = ’U’, the
diagonal elements of A are also not referenced and are
assumed to be 1.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, A(i,i) is exactly zero. The triangular
matrix is singular and its inverse can not be computed.
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subroutine strtrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer INFO)
STRTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular matrix of order N, and B is an N-by-NRHS
matrix. A check is made to verify that A is nonsingular.
UPLO is CHARACTER*1
= ’U’: A is upper triangular;
= ’L’: A is lower triangular.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)
DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A is REAL array, dimension (LDA,N)
The triangular matrix A. If UPLO = ’U’, the leading N-by-N
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = ’L’, the leading N-by-N lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = ’U’, the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the solutions
X have not been computed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine strttf (character TRANSR, character UPLO, integer N, real, dimension( 0: lda−1, 0: * ) A, integer LDA, real, dimension( 0: * ) ARF, integer INFO)
STRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
STRTTF copies a triangular matrix A from standard full format (TR)
to rectangular full packed format (TF) .
TRANSR is CHARACTER*1
= ’N’: ARF in Normal form is wanted;
= ’T’: ARF in Transpose form is wanted.
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N).
On entry, the triangular matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced.
LDA is INTEGER
The leading dimension of the matrix A. LDA >= max(1,N).
ARF is REAL array, dimension (NT).
NT=N*(N+1)/2. On exit, the triangular matrix A in RFP format.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.
RFP A RFP A
Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
subroutine strttp (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) AP, integer INFO)
STRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
STRTTP copies a triangular matrix A from full format (TR) to standard
packed format (TP).
UPLO is CHARACTER*1
= ’U’: A is upper triangular.
= ’L’: A is lower triangular.
N is INTEGER
The order of the matrices AP and A. N >= 0.
A is REAL array, dimension (LDA,N)
On exit, the triangular matrix A. If UPLO = ’U’, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AP is REAL array, dimension (N*(N+1)/2
On exit, the upper or lower triangular matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
December 2016
subroutine stzrzf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
STZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
to upper triangular form by means of orthogonal transformations.
The upper trapezoidal matrix A is factored as
A = ( R 0 ) * Z,
where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
triangular matrix.
M is INTEGER
The number of rows of the matrix A. M >= 0.
N is INTEGER
The number of columns of the matrix A. N >= M.
A is REAL array, dimension (LDA,N)
On entry, the leading M-by-N upper trapezoidal part of the
array A must contain the matrix to be factorized.
On exit, the leading M-by-M upper triangular part of A
contains the upper triangular matrix R, and elements M+1 to
N of the first M rows of A, with the array TAU, represent the
orthogonal matrix Z as a product of M elementary reflectors.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU is REAL array, dimension (M)
The scalar factors of the elementary reflectors.
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
April 2012
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
The N-by-N matrix Z can be computed by
Z = Z(1)*Z(2)* ... *Z(M)
where each N-by-N Z(k) is given by
Z(k) = I - tau(k)*v(k)*v(k)**T
with v(k) is the kth row vector of the M-by-N matrix
V = ( I A(:,M+1:N) )
I is the M-by-M identity matrix, A(:,M+1:N)
is the output stored in A on exit from DTZRZF,
and tau(k) is the kth element of the array TAU.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"http://man.sourcentral.org/debian-stretch/3+spptrf","timestamp":"2024-11-13T15:51:56Z","content_type":"text/html","content_length":"714593","record_id":"<urn:uuid:35424f95-13d1-4be1-85dc-b009eef0440e>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00385.warc.gz"} |
Create and Solve Contextual Problems
6th Grade - Create and Solve Contextual Problems
Create and solve contextual problems that lead naturally to division of fractions. 0606.2.3
Links verified on 7/7/2014
1. Divide fractions - using circles, will show visually how the divisor will fit into the dividend a whole number of times
2. Dividing fractions - Fraction Bar -This is a very versatile tool that can be used to illustrate a variety of number operations.
3. Divide Two Fractions - This page will show you how to divide two fractions. There are three combinations of this. 1) Dividing two "normal" fractions, 2) Dividing a mixed number by a fraction,
and 3) Dividing two mixed numbers. Fill in the boxes for the type of problem you need below, then click "Divide."
4. Divide Fractions with Lines - instruction and practice in dividing fractions. Each example will show visually how the divisor will fit into the dividend a whole number of times.
5. Divide Fractions with Circles - instruction and practice in dividing fractions. Each example will show visually how the divisor will fit into the dividend a whole number of times.
6. Divide Fractions-Strict - will give more instruction and practice in dividing fractions. Each quotient might be a number less than one or a mixed number larger than one. The quotient must be
entered in lowest terms.
7. Dividing Fractions - animated explanation from Math is Fun
8. Dividing Fractions - The Math Page provides eight examples of dividing fractions
9. Dividing Fractions - index of lessons on dividing fractions
10. Dividing Fractions by Fractions - explanation, interactive practice and a timed quiz at AAA Math
11. Dividing Fractions - Practice Problems - Thirty-four practice problems from the Math Page.
12. Lesson plan - Dividing Fractions in Word Problems
13. Multiplying and Dividing Fractions - five multiple choice questions
14. Putting out bowls of potpourri - Solve this word problem - Video lesson
│ │ video format | interactive lesson | a quiz | │
│ │ │ | {"url":"https://www.internet4classrooms.com/grade_level_help/divide_fractions_math_sixth_6th_grade.htm","timestamp":"2024-11-03T16:59:19Z","content_type":"text/html","content_length":"30993","record_id":"<urn:uuid:95ea8eab-797e-4a79-a159-b28f227118c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00566.warc.gz"} |
Tag archives: puzzle
By Louise Mayor
Have you got what it takes to crack the third conundrum in the Physics World at 25 Puzzle? You can catch up on the previous two instalments here. #PW25puzzle
You are trying to find a phrase with the pattern 3, 6, 2, 8, 8. The puzzle answer is the six-letter word. We hope you enjoy the joke.
By Louise Mayor
Welcome to the second instalment of the Physics World at 25 Puzzle. The first puzzle was released last week and your second challenge lies below. #PW25puzzle
Is Schrödinger’s cat alive or dead?
1. Schrödinger’s cat is alive.
2. Schrödinger’s cat is dead.
3. Exactly one of statements 6 and 9 is true.
4. Exactly one of statements 2 and 6 is false.
5. Statements 4, 5 and 10 are all false.
6. Exactly one of statements 1 and 10 is false.
7. Exactly 5 statements are true.
8. Exactly one of statements 3 and 10 is false.
9. Exactly one of statements 6 and 10 is true.
10. Exactly one of statements 1 and 2 is false.
11. Statements 1, 8 and 11 are all false.
Enter your answer as a list, in numerical order, of the number(s) of the statements that are definitely true, as a single string with no spaces, such as, for example, 25811.
By Louise Mayor
This month is the 25th anniversary of Physics World – the member magazine of the Institute of Physics – and in addition to a special celebratory issue, we’ve decided to set you a challenge.
In fact, we have teamed up with GCHQ – one of the UK’s three Intelligence Agencies and home to some of the country’s hottest code-breaking talent – to create with us a set of five physics-themed
puzzles. The puzzles have been devised by three GCHQ members of staff, who today we still know only as Colin, Nick and Pete. (Thank you, guys!)
Below is Puzzle 1, the first of the five. The rest will be released on successive Tuesdays throughout October on this blog. The first is the easiest – they only get harder from here on in! | {"url":"https://blog.physicsworld.com/tag/puzzle/page/2/","timestamp":"2024-11-10T01:23:47Z","content_type":"application/xhtml+xml","content_length":"33272","record_id":"<urn:uuid:a49dfe2c-e8a9-4f0b-8481-d78dc772365c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00130.warc.gz"} |
A class of self-similar solutions for a high-temperature axisymmetric jet
A one-parameter self-similar problem for the high-temperature flow region of a nonisothermal axisymmetric point jet is formulated in the context of the theory of a compressible boundary layer,
assuming a power-law temperature dependence of the dynamic viscosity and heat conductivity. The problem is solved numerically and, in certain cases, analytically; asymptotic solutions are obtained in
several cases. The conditions under which a self-similar representation of the solution is appropriate are defined.
PMTF Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki
Pub Date:
August 1985
□ Axisymmetric Flow;
□ Computational Fluid Dynamics;
□ High Temperature Fluids;
□ Jet Flow;
□ Asymptotic Methods;
□ Compressible Boundary Layer;
□ Conductive Heat Transfer;
□ Temperature Dependence;
□ Viscous Flow;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1985PMTF........46B/abstract","timestamp":"2024-11-05T20:41:53Z","content_type":"text/html","content_length":"34141","record_id":"<urn:uuid:899e470f-01ea-4204-80ab-8a186fb9784e>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00177.warc.gz"} |
The Classification of the Finite Simple Groups, Number 10: Part V, Chapters 9–17: Theorem $C_6$ and Theorem $C^*_4$, Case Asearch
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The Classification of the Finite Simple Groups, Number 10: Part V, Chapters 9–17: Theorem $C_6$ and Theorem $C^*_4$, Case A
Softcover ISBN: 978-1-4704-7553-6
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Click above image for expanded view
The Classification of the Finite Simple Groups, Number 10: Part V, Chapters 9–17: Theorem $C_6$ and Theorem $C^*_4$, Case A
Softcover ISBN: 978-1-4704-7553-6
Product Code: SURV/40.10
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN: 978-1-4704-7566-6
Product Code: SURV/40.10.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN: 978-1-4704-7553-6
eBook ISBN: 978-1-4704-7566-6
Product Code: SURV/40.10.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
• Mathematical Surveys and Monographs
Volume: 40; 2023; 570 pp
MSC: Primary 20
This book is the tenth in a series of volumes whose aim is to provide a complete proof of the classification theorem for the finite simple groups based on a fairly short and clearly enumerated
set of background results. Specifically, this book completes our identification of the simple groups of bicharacteristic type begun in the ninth volume of the series (see Mathematical Surveys and
Monographs, Volume 40.9). This is a fascinating set of simple groups which have properties in common with matrix groups (or, more generally, groups of Lie type) defined both over fields of
characteristic 2 and over fields of characteristic 3. This set includes 11 of the celebrated 26 sporadic simple groups along with several of their large simple subgroups. Together with SURV/40.9,
this volume provides the first unified treatment of this class of simple groups.
Graduate students and researchers interested in the theory of finite groups.
□ Chapters
□ General group-theoretic lemmas
□ Theorem $\mathscr {C}_6$ and $\mathscr {C}_6^*$
□ Theorems $\mathscr {C}_4$ and $\mathscr {C}_4^*$: Introduction
□ Theorem $\mathscr {C}_4^*$: Stage A1. First steps
□ Theorem $\mathscr {C}_4^*$: Stage A2. Nonconstrained $p$-rank 3 centralizers
□ Theorem $\mathscr {C}_4^*$: Stage A3. $KM$-singularities
□ Theorem $\mathscr {C}_4^*$: Stage A4. Setups for recognizing $G$
□ Theorem $\mathscr {C}_4^*$: Stage A5. Recognition
□ Properties of $\mathscr {K}$-groups
• Book Details
• Table of Contents
• Additional Material
• Requests
Volume: 40; 2023; 570 pp
MSC: Primary 20
This book is the tenth in a series of volumes whose aim is to provide a complete proof of the classification theorem for the finite simple groups based on a fairly short and clearly enumerated set of
background results. Specifically, this book completes our identification of the simple groups of bicharacteristic type begun in the ninth volume of the series (see Mathematical Surveys and
Monographs, Volume 40.9). This is a fascinating set of simple groups which have properties in common with matrix groups (or, more generally, groups of Lie type) defined both over fields of
characteristic 2 and over fields of characteristic 3. This set includes 11 of the celebrated 26 sporadic simple groups along with several of their large simple subgroups. Together with SURV/40.9,
this volume provides the first unified treatment of this class of simple groups.
Graduate students and researchers interested in the theory of finite groups.
• Chapters
• General group-theoretic lemmas
• Theorem $\mathscr {C}_6$ and $\mathscr {C}_6^*$
• Theorems $\mathscr {C}_4$ and $\mathscr {C}_4^*$: Introduction
• Theorem $\mathscr {C}_4^*$: Stage A1. First steps
• Theorem $\mathscr {C}_4^*$: Stage A2. Nonconstrained $p$-rank 3 centralizers
• Theorem $\mathscr {C}_4^*$: Stage A3. $KM$-singularities
• Theorem $\mathscr {C}_4^*$: Stage A4. Setups for recognizing $G$
• Theorem $\mathscr {C}_4^*$: Stage A5. Recognition
• Properties of $\mathscr {K}$-groups
You may be interested in...
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Time: 2023-06-08T13:57:58+00:00
write an introductory paragraph for the following 2 problems:
problem 1:
Use the following data:
Time, t (s) 0 2 4 6 8 10 12 14 16
Position, x (m) 1 1.8 8.9 10.3 9.2 13.2 9.3 12 16.8
a. To find the velocity at all times using the two-point central differencing formula (and where
appropriate the three-point forward or backward differencing formulae); these should yield
results correct to O(h^2).
b. To find the acceleration at all times using the three-point central differencing formula (and
where appropriate the four-point forward or backward differencing formulae); these should also
yield results correct to O(h^2)
c. Plot the position, velocity, and acceleration with respect to time
problem 2:
The objective of this problem is to compare forward, central, and backward finite-difference
approximations, correct to O(h2), of the first derivative of a function to the actual value of the derivative.
This will be done for
𝑓(𝑥) = e^𝑥 + 3𝑥^2
a. Use calculus to determine the correct value of the derivative at 𝑥 = 3.
b. To evaluate the central finite-difference approximations, start with 𝛥𝑥 = 0.2, and then increase
in increments of 0.05 up to a maximum value of 𝛥𝑥 = 1; compute the % true relative error in
each case.
c. Repeat b) for three-point forward finite-difference approximations.
d. Repeat b) for three-point backward finite-difference approximations.
e. Plot the errors computed in b), c), and d) versus 𝛥𝑥 on the same plot for comparison
an example of what is expected is as follows:
The last problem uses curve fitting to fit the given set of data with the equation y = ax emx. The equation is changed to its linear form to find the transformed unknown parameters. The standard
deviations of each transformed parameter are then calculated using the weighted version of linear least squares regression. Afterwards, the original parameters with their respective deviations are
found by using the transformed parameters and deviations found previously. The data given and the fitted curved are plotted with the adjusted observations y . These observations are estimated and
standard deviations for those are calculated
In this series of problems, we will be exploring the concepts of velocity, acceleration, and the finite-difference method applied to position data. In the first problem, we will utilize a given data
set of time and position in order to calculate the velocity and acceleration at various time points using different differencing formulas. This problem will entail plotting the position, velocity,
and acceleration with respect to time in order to observe their relationships. In the second problem, we will compare forward, central, and backward finite-difference approximations in the context of
approximating the first derivative of a mathematical function. By evaluating various 𝛥𝑥 values and calculating the corresponding true relative error, we will analyze the effectiveness of these
finite-difference methods. Lastly, we will present a visual comparison of the errors from each method in a single plot. Through these problems, we will develop an understanding of the underlying
concepts and applications of finite-difference methods and their use in analyzing physical and mathematical scenarios.
now write the introductory paragraph for the last problem:
The amount of mass transported via a pipe over a period of time can be computed as
𝑀 = ∫ 𝑄(𝑡)𝑐(𝑡)𝑑𝑡
where M = mass (mg), t1 = initial time (min), t2 = final time (min), Q(t) = flow rate (m3/min), and c(t) =
concentration (mg/m3). The following functional representations define the temporal variations in flow
and concentration:
𝑄(𝑡) = 9 + 4 cos2(0.4𝑡)
𝑐(𝑡) = 5𝑒−0.5𝑡 + 2𝑒0.15𝑡
Using Romberg integration to a tolerance of 0.1%, determine the mass transported between:
a. t1 = 1 and t2 = 4 minutes
b. t1 = 4 and t2 = 8 minutes
c. t1 = 1 and t2 = 8 minutes
In this final problem, we will be investigating the mass transportation through a pipe system over varying periods of time. By utilizing the given formulas for flow rate and concentration, and
implementing Romberg integration, we will compute the mass transported while maintaining a tolerance of 0.1%. We will conduct this analysis for three distinct time spans: from 1 to 4 minutes, from 4
to 8 minutes, and from 1 to 8 minutes. Through this exploration, we will gain insights into the integral calculus application in the engineering context, particularly how mass transportation is
influenced by time and the variations in the flow rate and concentration parameters. | {"url":"https://wildvisualizer.com/conversation/wildchat/789773","timestamp":"2024-11-06T21:50:44Z","content_type":"text/html","content_length":"17875","record_id":"<urn:uuid:9c77482e-e7c9-42e3-b4b3-61969da9bf4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00780.warc.gz"} |
Arithmetic Circuit Primer | Brevis-SDK Doc
Arithmetic Circuit Primer
An arithmetic circuit (or just "circuit") is a core piece in ZK proofs. The circuit takes some inputs and it allows us to express "how the computation is done" in a verifiable way. If you have no
prior experience with arithmetic circuits, it is recommended to read Vitalik's intro to arithmetic programs. Though one can go very deep into this topic, surprisingly, there aren't many concepts you
need to know before you can write circuit applications. Here are some key points to remember:
Circuits are "Fixtures"
When you compile and setup a circuit, you get a proving key (PK) and a verifying key (VK). These keys are unique per circuit. It means if you change any circuit logic and recompile, these won't be
the same. Normally in the blockchain space, you want your contract to hold the verifying key.
Just like how you can be sure that an Ethereum transaction is signed correctly by someone using their private key if you can recover their public key (address) from it, if you can verify a set of
inputs against a proof using your VK, it means the proof is for sure generated by someone using the set of inputs in the circuit that corresponds to your VK.
By using the PK and the circuit, it allows us to prove that because a set of inputs satisfy a fixed circuit, the inputs are correct in terms of that circuit.
Variables Have an "Upper Limit"
Much like how you can only use up to 2^64 - 1 for uint64 in normal programming languages, in circuits, we also have a limit for circuit variables. This upper limit is dependent on which elliptic
curve the underlying proving system uses, but in our framework it's always the max scalar field number of BLS12_377. It is ok to not know what this is as long as we keep in mind the max limit is 2^
252 (or 248 bits if we round down to the nearest byte, which is 31 bytes). Because actual integer values on Ethereum rarely reach their type limit (uint256), this means for most cases, we can
represent Solidity uint256 numbers as circuit variables. But hashes (bytes32) are often truly 32 bytes, and cannot be fit into 31 bytes (our upper limit), how do we represent them in circuit? The
Brevis SDK simply "chops off" the bytes32 at a certain point and puts them into two circuit variables. That's it.
All Operations are Arithmetic
This might be the most counter-intuitive part. The circuits we write are not the same as the programs we use to do the writing. The biggest difference comes from that any meaningful "logic" is
implemented through arithmetic gates. For example, there is no comparison operator (a == b), instead, such checks can be done through subtracting b from a and checking whether the result is 0.
There are No "Dynamic" Circuits
Just like what you would expect when you solder a physical circuit, you wouldn't expect the wires to magically desolder themselves and jump around at runtime. Our arithmetic circuits are the same. If
you wire some variables together through some arithmetic gates, the wires are soldered once you compile, and your if statements will not make them desolder themselves and jump around at runtime. | {"url":"https://docs.brevis.network/developer-resources/arithmetic-circuit-primer","timestamp":"2024-11-07T21:46:11Z","content_type":"text/html","content_length":"131196","record_id":"<urn:uuid:18ed1737-2860-4c06-a84f-e3af5858e98e>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00716.warc.gz"} |
t-Test, Chi-Square, ANOVA, Regression, Correlation...
Two-way ANOVA (without repeated measures)
Load ANOVA data set
What is a two-way ANOVA?
Two-way (or two factor) analysis of variance tests whether there is a difference between more than two independent samples split between two variables or factors.
What is a factor?
A factor is, for example, the gender of a person with the characteristics male and female, the form of therapy used for a disease with therapy A, B and C or the field of study with, for example,
medicine, business administration, psychology and math.
In the case of variance analysis, a factor is a categorical variable. You use an analysis of variance whenever you want to test whether these categories have an influence on the so-called dependent
For example, you could test whether gender has an influence on salary, whether therapy has an influence on blood pressure, or whether the field of study has an influence on the duration of studies.
Salary, blood pressure or study duration are then the dependent variables. In all these cases you can check whether the factor has an influence on the dependent variable.
Since you only have one factor in these cases, you would use a one-way analysis of variance in these cases (except of course for the gender, there we have a variable with only two expressions, there
we would use the t-test for independent samples).
Two factors
Now you may have another categorical variable that you want to include as well. You might be interested in whether:
• in addition to gender, the highest level of education also has an influence on salary.
• besides therapy, gender also has an influence on blood pressure.
• in addition to the field of study, the university attended also has an influence on the duration of studies.
In all three cases you would not have one factor, but two factors each. And since you now have two factors, you must use the two-way analysis of variance.
Using the two-way analysis of variance, you can now answer three things:
• Does factor 1 have an effect on the dependent variable?
• Does factor 2 have an effect on the dependent variable?
• Is there an interaction between factor 1 and factor 2?
Therefore, in the case of one-way analysis of variance, we have one factor from which we create the groups. In the case of two-way analysis of variance, the groups result from the combination of the
expressions of the two factors.
Example two-way ANOVA
Here's an example dataset for a two-way ANOVA in medicine. Let's say we are interested in studying the effect of two factors, "Treatment" and "Gender," on the response variable "Blood Pressure."
Load data set
In this example, we have two levels of the "Treatment" factor (A and B) and two levels of the "Gender" factor (Male and Female). The "Blood Pressure" measurements are recorded for each participant
based on their treatment and gender.
To perform a two-way ANOVA on this dataset, we would test the null hypothesis that there is no interaction between the "Treatment" and "Gender" factors and no main effects of each factor on the
"Blood Pressure" response variable.
Three statements can be tested with the 2 way ANOVA, so there are 3 null hypotheses and therefore 3 alternative hypotheses.
Null hypotheses H[0] Alternative hypotheses H[1]
There are no significant differences in the mean between the groups (factor levels) of the first There is a significant difference in the mean between the groups (factor levels) of the first
factor. factor.
There are no significant differences in the mean between the groups (factor levels) of the second There is a significant difference in the mean between the groups (factor levels) of the second
factor. factor.
One factor has no effect on the effect of the other factor. One factor has an effect on the effect of the other factor.
For a two-way analysis of variance to be calculated without repeated measures, the following assumptions must be met:
• The scale level of the dependent variable should be metric, and that of the independent variables (factors) nominal.
• Independence: The measurements should be independent, i.e. the measured value of one group should not be influenced by the measured value of another group. If this were the case, we would need an
analysis of variance with repeated measures.
• Homogeneity: The variances in each group should be approximately equal. This can be checked with Levene's test.
• Normal distribution: The data within the groups should be normally distributed.
So the dependent variable could be, for example, salary, blood pressure, and study duration. These are all metric variables. And the independent variable should be nominally or ordinally scaled. For
example, gender, highest level of education, or a type of therapy. Note, however, that rank order is not used with ordinal variables, so this information is lost.
Calculate two-way ANOVA
To calculate a two-way ANOVA, the following formulas are needed. Let's look at this with an example.
Let's say you work in the marketing department of a bank and you want to find out if gender and the fact that a person has studied or not have an influence on their attitude towards retirement
In this example, your two independent variables (factors) are gender (male or female) and study (yes or no). Your dependent variable is attitude toward retirement planning, where 1 means "not
important" and 10 means "very important."
After all, is attitude toward retirement planning really a metric variable? Let's just assume that attitude toward retirement planning was measured using a Likert scale and thus we consider the
resulting variable to be metric.
Mean values
In the first step we calculate the mean values of the individual groups, i.e. of male and not studied, which is 5.8 then of male and studied, which is 5.4, we now do the same for female.
Then we calculate the mean of all male and female and of not studied and studied respectively. Finally, we calculate the overall mean as 5.4.
Sums of squares
With this, we can now calculate the required sums of squares. SS[tot] is the sum of squares of each individual value minus the overall mean.
SS[btw] results from the sum of squares of the group means minus the overall mean multiplied by the number of values in the groups.
The sums of squares of the factors SS[A] and SS[B] result from the sum of squares of the means of the factor levels minus the total mean.
Now we can calculate the sum of squares for the interaction. These are obtained by calculating SS[btw] minus SS[A] minus SS[B].
Finally, we calculate the sum of squares for the error. This will calculate similar to the total sum of squares, so again we use each individual value. Only in this case, instead of subtracting the
overall mean from each value, we subtract the respective group mean from each value.
Degrees of freedom
The required degrees of freedom are as follows:
Mean squares or variance
Together with the sums of squares and the degrees of freedom, the variance can now be calculated:
And now we can calculate the F-values. These are obtained by dividing the variance of factor A, factor B or the interaction AB by the error variance.
To calculate the p-value, we need the F-value, the degrees of freedom and the F-distribution. We use the F-distribution p-value calculator on DATAtab. Of course, you can also just calculate the
example completely with DATAtab, more about that in the next section.
This gives us a p-value of 0.323 for Factor A, a p-value of 0.686 for Factor B, and a p-value of 0.55 for the interaction. None of these p-values is less than 0.05 and thus we retain the respective
null hypotheses.
Calculating two-way ANOVA with DATAtab
We take the same example from above. The data is now arranged in the form so that your statistics software can do something with it. In each row is a respondent.
Attitude towards retirement planning Studied Gender
6 no male
4 no male
5 no female
... ... ...
5 yes female
9 yes female
2 yes female
3 yes female
This example consists of only 20 cases, which of course is not much, giving us very low test power, but as an example it should fit.
To calculate a two-way analysis of variance online, simply visit datatab.com and copy your own data into this table.
Then click on Hypothesis tests. Under this tab you will find a lot of hypothesis tests and depending on which variable you select, you will get an appropriate hypothesis test suggested.
When you copy your data into the table, the variables appear under the table, if the correct scale level is not automatically detected, you can simply change it by clicking on the scale level itself.
We want to know if gender and whether you have studied or not has an impact on your attitude towards retirement planning. So we just click on all three variables.
DATAtab will now automatically calculate a two-way analysis of variance without repeated measures. DATAtab outputs the three null and the three alternative hypotheses, then the descriptive statistics
and the Levene test of equality of variance. With the Levene test you can check if the variances within the groups are equal. The p-value is greater than 0.05, so we assume equality of variance
within groups for these data.
Next come the results of the two-way ANOVA.
Interpreting two-way ANOVA
The most important in this table are the three marked rows. With these three rows, you can test whether the 3 null hypotheses we made earlier are kept or rejected.
The first row tests you null hypothesis of whether studied or not studied has an effect on attitude towards retirement planning. The second row tests whether gender has an effect on attitude. Finally
the third row tests, the interaction between studied and gender.
You can read the p-value in each case right at the last column. Let's say we set the significance level at 5%. If our calculated p-value is less than 0.05, then the null hypothesis is rejected, and
if the calculated p-value is greater than 0.05, the null hypothesis is not rejected.
Thus, in this case, we see that all three p-values are greater than 0.05 and thus we cannot reject any of the three null hypotheses.
Therefore, neither whether one has studied or not nor gender has a significant effect on attitudes toward retirement planning. And there is also no significant interaction between studied and gender
in terms of attitudes toward retirement planning.
If you don't know exactly how to interpret the results, you can also just click on Summary in Words. In addition, it is important to check in advance whether the assumptions for the analysis of
variance are met at all.
Interaction effect
But what exactly does interaction mean? Let us first have a look at this diagram.
The dependent variable is plotted on the y axis, in our example: the attitude towards retirement provision. On the x axis, one of the two factors is plotted, let's just take gender. The other factor
is represented by lines with different colors. Green is studied and red is not studied.
The endpoints of the lines are the mean values of the groups, e.g. male and not studied.
In this diagram, one can see that both gender and the variable of having studied or not have an influence on attitudes toward retirement planning. Females have a higher value than males and studied
have a higher value than not studied.
But now finally to the interaction effects, for that we compare these two graphs.
In the first case, we said there is no interaction effect. If a person has studied, he has a value that is, say, 1.5 higher than a person who has not studied.
This increase of 1.5 is independent of whether the person is male or female.
It is different in this case, here studied persons also have a higher value, but how much higher the value is depends on whether one is male or female. If I am male, there is a difference of, let's
say for example 0.5 and if I am female, there is a difference of 3.5.
So in this case we clearly have an interaction between gender and study because the two variables affect each other. It makes a difference how strong the influence from studying is depending on
whether I am male or female.
In this case, we do have an interaction effect, but the direction still remains the same. So females have higher scores than males and studied have higher scores than non-studied.
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Concepts & Steps of Operations Research April 2014 Exam Assignment Problems
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Steps –
1. Take Dummy (if required) & then convert in Regret matrix (if required)
2. Do Row minimization and Column minimization.
3. Cover all Zeroes in the Table with Minimum possible lines. (Start from maximum zeroes, either row-wise or column-wise).
4. If optimal, find assignment.
5. If not optimal, write next table, change values & check again with Minimum possible lines.
6. Continue the iterations till optimal solution is reached.
An Assignment problem involves assignment or matching of two things. E. g. matching of workers and jobs or matching of salesmen and areas etc.
The basic principle in Assignment problem is that the matching is on a one to one basis.
i. e. One worker can do only one job or one salesman can operate in only one area.
The method used for solving Assignment problem is called “Hungarian method”.
1. The Assignment problem can be Balanced or Unbalanced problem.
A Balanced problem means the no. of rows and no. of columns in the problem are equal. E. g. if the problem contains 4 workers and 4 jobs, then it is balanced.
Where as, an Unbalanced problem means the no. of rows and no. of columns are not equal. E. g. if the problem contains 4 workers and 3 jobs it is not balanced. Then first we need to balance the
problem by taking a Dummy job (imaginary job).
2. Dummy:
A Dummy is an imaginary entity. The purpose of Dummy is to balance the problem. Since the Dummy is imaginary, all values for Dummy are always zero. Dummy can come as row or column, depending on
3. The Assignment problem can be of Minimization type or Maximization type.
A Minimization Assignment problem involves cost, time or distance data. The objective of solution is to minimize the final answer.
A Maximization Assignment problem involves sales, revenue or profit data. The objective of solution is maximization of the final answer. We need to first convert the Maximization problem in
Minimization problem. This conversion is called Regret matrix. From the original profit values, we find out the highest profit value. From this highest profit, we subtract all profit values. The
resulting matrix is Regret matrix.
4. Prohibited or Restricted problem:
A Prohibited problem is the one in which there are one or more restrictions. E. g. say there are 4 contractors – C1, C2, C3 & C4. And there are 4 roads to be repaired – R1, R2, R3 & R4. But
contractor C2 cannot or is not allowed to work on R3. This is a prohibited problem
Then we assign a very high or infinite value (represented by M) to C2-R3 and proceed with solution. Throughout the solution steps, M does not change. Since M is infinity, no assignment is possible in
1. Multiple optimal solutions in an assignment problem.(Apr 2002) (Apr 2005)
An Assignment problem can have more than one optimal solution, which is called multiple optimal solutions. The meaning of multiple optimal solutions is – The total cost or total profit will remain
same for different sets or combinations of allocations. It means we have the flexibility of assigning different allocations while still maintaining Minimum (Optimal) cost or Maximum (Optimal) profit.
We can detect multiple optimal solutions when there are multiple zeroes in any columns or rows in the final (Optimal) table in the Assignment problem.
2. State the algorithm of solving an Assignment problem.(Apr 2002) (Apr 2005)
Algorithm of solving an Assignment problem is as follows –
1] Check if the problem is Balanced or Unbalanced. If no. of rows = no. of columns, problem is balanced. If unbalanced, take Dummy row or column as required. All values for Dummy =0.
2] Check if the problem is of Minimization type (cost) or Maximization type (profit). If Maximization, convert to Minimization by finding Regret matrix.
3] Do row minima. Find minimum value in each row and subtract it from all values in that row.
4] Do column minima. Find minimum value in each column and subtract it from all values in that column.
5] Check for optimality. Draw minimum no. of straight lines to cover all zeroes in the matrix. If Minimum no. of lines= Size of matrix (e. g. 4 x 4, 5 x 5 etc.) then solution is optimal. If not, do
6] Iteration – A} Find minimum uncovered value in the matrix.
B} Subtract it from all uncovered values.
C} Add it to all double covered values.
D} Other values remain same.
7] Again check for optimality. Continue procedure till we get optimal solution.
3. Restricted assignment problem, which is an unbalanced problem. (Apr 2003)
A restricted assignment problem is the one in which one or more allocations are prohibited or not possible. For such allocations we assign “M”, which is infinitely high cost. No allocation is given
in M.
An unbalanced problem is one in which number of rows is not equal to number of columns. Then we need to introduce dummy row or dummy column as required. All values for dummy are zero.
Hence, in an unbalanced restricted problem, in the first table we will introduce dummy for balancing the problem and “M” to prohibited or restricted allocations.
4. Unbalanced Assignment Problem. (Oct 2003) (Oct 2005)
An unbalanced Assignment problem is the one in which number of rows is not equal to number of columns. Dummy row or column is required as applicable for balancing the problem. All values in Dummy are
zero. Then the problem is solved as a normal assignment problem. Step one will be Row minima.
5. Optimality in an Assignment problem. (Oct 2006)
An Assignment problem is optimal when minimum number of lines required to cover all zeroes in the matrix are equal to size of the matrix. Size of the matrix means number of rows or number of columns.
E. g. Size can be 4 x 4 or 5 x 5.
Once optimality is detected then we can do allocations in the matrix. Allocations are done in the zero (0) values.
We test the optimality after doing Row minimization and Column minimization.
6. Explain ‘Hungarian method’ of solving assignment problem:
Whenever the pay off matrix of any assignment problem is not a square matrix i.e.no. of rows not equal to number of columns the problem is called unbalanced assignment problem. Hungarian method
of solving such problem is as follows :
1. Insert row or column with all values zero such that pay off matrix become
square matrix.
2. Row minima : Subtract smallest element of each row from corresponding
element of that row.
3. Column minima : Subtract smallest element of each column from
corresponding element of that column.
4. Draw minimum number of vertical and horizontal lines which can cover all
zeros of pay off matrix.
5. Total number of lines drawn is not equal to size of pay off matrix then select
smallest uncovered number, subtract it from every uncovered number, add it
to point of intersection and no change in other element. Repeat this step till
we get optimum matrix.
6. This second phase of solution. Select any row or column with single zero and
make allocation cancel out other zeros in corresponding row and column.
7. Explain regret matrix in assignment problem.
For solving the maximization assignment problem by Hungarian method, given problem is converted in minimization. Every element of given matrix is subtracted from highest element of given matrix,
resultant matrix is called regret matrix. It is also known as opportunity loss matrix.
Regret matrix in assignment problem of maximization type. It is used for maximization of profit, income, revenue, sales etc.
8. Explain reduced matrix in assignment problem.
1. Reduced matrix is a part of Hungarian method of solving assignment problem. If given assignment problem is balanced problem then following two steps are performed.
Step 1 : (Row minima) Subtract smallest element of each row from corresponding elements of that row.
Step 2 : (Column minima) Subtract smallest element of each column from corresponding elements of that column.
Matrix obtained after execution of above steps is called ‘reduced matrix’.
3 Comments
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1. In assignment problem,if for 10 marks.
incase the full problem is solved till optimal.and incase there are two options with the same answer in the end. how many marks will be deducted if youve written only one option. The question
doesnt state about asking if there is any alternate solution. just basic.
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1. Marks are provided for steps. So for every wrong step, marks will be deducted.
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Alg2.4 Exponential Functions and Equations
In this unit, students build on their understanding of exponential functions from an earlier course. Previously, they saw functions whose domain is the integers. Here, they write, interpret, and
evaluate exponential functions whose domain is the real numbers. In the second half of the unit, students learn about logarithms in base 2 and 10 as a way to express the exponent that makes an
exponential equation true. They then use logarithms to solve exponential equations and to answer questions about exponential functions. During this time, students encounter the constant \(e\) and
learn that it is used to model situations with continuous growth rates, leading to working with the natural logarithm. The unit ends with an exposure to logarithmic functions.
Read More
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Orbital Parameters and Coordinate Frames
Perigee and Apogee
The perigee and apogee are respectively the closest and furthers points of the orbit from the Earth's surface. This apparently simple definition has some complications.
In the strictest meaning of the words, the perigee and apogee are points in an elliptical orbit at which the orbiting object has the smallest and largest radius vectors respectively. They lie at
opposite ends of the semi-major axis of the ellipse.
We often refer to the perigee and apogee heights; i.e. the height of the perigee and apogee points above the planetary surface. Loosely, we drop the word `height' and refer to these values simply as
the perigee and apogee.
But complications arise when you start thinking about the fact that the Earth is not actually a sphere, but (to a much better approximation) an oblate spheroid - its cross-section is also an ellipse.
Depending on the orientation of the orbit, the two ellipses are misaligned. In the figure below, a satellite orbit in red is drawn around an exaggerated Earth in blue. In this case, the smallest
height of the orbit around the spheroid Earth is not necessarily the height of the perigee point P. In fact in the example shown, the point of smallest height is about 40 degrees further around the
Worse yet, the flattening of the Earth means that the orbit is not actually an ellipse. The instantaneous (`osculating') orbital parameters change with time as you go round the orbit, so the true
perigee and apogee are different from ones obtained from time-averaged `mean elements'.
It is conventional in space situational awareness contexts to quote perigee and apogee heights relative to a fictitious perfectly spherical Earth with radius equal to the equatorial radius. There are
a lot of reasons that this makes good sense, but the reader should remember that the actual height above the Earth may be considerably higher if the perigee is over the polar regions.
The Friends of Perigee and Apogee
The `gee' in apogee and perigee is from the Greek word for the Earth. For orbits around other bodies, special names are in use. Most of these are falling out of favour and are being replaced by the
terms periapsis and apoapsis, referring to the `apsis' (plural, apsides), the generic name for the extreme points in the orbit. There are several terms used for selenocentric orbits. I list terms
that I've seen in relatively recent use; there are many others. I'm torn between the poetry of them and the fact that they are totally unnecessary.
│Central body│ Terms │ Plural │
│Still in use │
│Earth │perigee, apogee │-gees │
│Sun │perihelion, aphelion │-helia │
│Moon │perilune, apolune │-lunes │
│Moon │periselene, aposelene │-selenes? │
│Mars │periares, apoares │-ares │
│Jupiter │perijove, apojove │-joves │
│Star │periastron, apoastron │-astrons │
│Galaxy │perigalacticon, apogalacticon │-icones? (not used)│
│Probably obsolescent │
│Moon │pericynthion, apocynthion │-cynthia │
│Venus │pericytherion, apocytherion │ │
│Saturn │perikrone, apokrone │ │
│Black hole │peribothron, apobothron │-bothroi │
Coordinate Frames and Units
The orbital inclination (and the node and argument of perigee, which are not given in this release of GCAT) depends on the equator and pole of your coordinate frame.
In GCAT, angles are always expressed in degrees and distances are usually in km (but see below for heliocentric orbits).
• Earth orbits are given with respect to the equator of date (strictly, TEME equator). Note that astronomical positions are often given with respect to the (inertially fixed) ICRF frame. The Earth
wobbles due to precession, so the equator of date changes with time relative to the ICRF. Distances are always given in km.
• Heliocentric orbits are given with respect to the J2000 ecliptic. Distances (aphelion and perihelion) are given in astronomical units (AU). 1 AU = 149597870.700 km.
• Orbits around other central bodies (Moon, Mars, etc.) are given with respect to the IAU equator of that body. Where no IAU equator is yet defined in JPL Horizons, the ecliptic is used instead.
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< 1 >
This development of the Lambert W function also applies to complex values of x
Taylor series
The Taylor series of W[0] around 0 can be calculated with the inverse Lagrange theorem and is
The radius of convergence is −1/e, which you can determine in the ratio test. The function defined by this sequence can be extended to a holomorphic function defined for all complex numbers with a
branching point along the interval (−∞, −1/e]. This holomorphic function is the main branch of the Lambert W function.
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Time Series Variables
Basic Concepts
Time series variables are data pieces inside an agent that keep track of that agent's previous values. Since they must hold multiple values in a series, these variables are windowed values, storing
data for a specific number of time steps. Modellers can specify how many time steps a time series variable should hold.
Once created, a time series variable will hold values of one specific agent attribute for a specific number of time steps. The default value of each tick in the time series variable is 0. Modellers
can use the offset parameter of the time series variable creation methods to determine how many historical values should be kept at the same time. The variable will only store the latest offset
For example, suppose a model saves historical values corresponding to the number of ticks that have been run so far. Assume further that this model contains a time series variable with an offset of
3, and the model has run for 2 ticks. At this point, since the model has run for 2 ticks, the values in the time series variable will be 0,1,2. The 0 represents an uninitialized variable, while the 2
represents the most recent tick. If the model runs for another tick, the time series variable will now contain the values 1,2,3. The uninitialized value has been replaced with the next most recent
value of 1, while the new value of 3 has been added into the variable.
The Simudyne SDK allows modellers to create time series variables with the WindowedDouble and WindowedLong classes.
// Create new windowed value, storing 10 ticks of data.
WindowedDouble mortgages = createWindowedDouble(10);
WindowedLong balances = createWindowedLong(10);
The time series variable objects allow modellers to access specific ticks.
WindowedLong balance = createWindowedLong(10);
// Retrieves the value of balance from 2 ticks ago.
// Both of these calls retrieve the current value of the balance.
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