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The problem is to approximate the integral of a function f as the average of the function evaluated at a set of points x1, ..., xN: ∫ s f ( u ) d u ≈ 1 N ∑ i = 1 N f ( x i ) . {\displaystyle \int _{^{s}}f(u)\,{\rm {d}}u\approx {\frac {1}{N}}\,\sum _{i=1}^{N}f(x_{i}).} Since we are integrating over the s-dimensional unit cube, each xi is a vector of s elements. | https://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method |
The difference between quasi-Monte Carlo and Monte Carlo is the way the xi are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Monte Carlo uses a pseudorandom sequence. The advantage of using low-discrepancy sequences is a faster rate of convergence. | https://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method |
Quasi-Monte Carlo has a rate of convergence close to O(1/N), whereas the rate for the Monte Carlo method is O(N−0.5).The Quasi-Monte Carlo method recently became popular in the area of mathematical finance or computational finance. In these areas, high-dimensional numerical integrals, where the integral should be evaluated within a threshold ε, occur frequently. Hence, the Monte Carlo method and the quasi-Monte Carlo method are beneficial in these situations. | https://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method |
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method. However, the secant method predates Newton's method by over 3000 years. | https://en.wikipedia.org/wiki/Secant_Method |
In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem. It involves finding solutions to the initial value problem for different initial conditions until one finds the solution that also satisfies the boundary conditions of the boundary value problem. In layman's terms, one "shoots" out trajectories in different directions from one boundary until one finds the trajectory that "hits" the other boundary condition. | https://en.wikipedia.org/wiki/Shooting_method |
In numerical analysis, the singular boundary method (SBM) belongs to a family of meshless boundary collocation techniques which include the method of fundamental solutions (MFS), boundary knot method (BKM), regularized meshless method (RMM), boundary particle method (BPM), modified MFS, and so on. This family of strong-form collocation methods is designed to avoid singular numerical integration and mesh generation in the traditional boundary element method (BEM) in the numerical solution of boundary value problems with boundary nodes, in which a fundamental solution of the governing equation is explicitly known. The salient feature of the SBM is to overcome the fictitious boundary in the method of fundamental solution, while keeping all merits of the latter. | https://en.wikipedia.org/wiki/Singular_boundary_method |
The method offers several advantages over the classical domain or boundary discretization methods, among which are: meshless. The method requires neither domain nor boundary meshing but boundary-only discretization points; integration-free. The numerical integration of singular or nearly singular kernels could be otherwise troublesome, expensive, and complicated, as in the case, for example, the boundary element method; boundary-only discretization for homogeneous problems. The SBM shares all the advantages of the BEM over domain discretization methods such as the finite element or finite difference methods; to overcome the perplexing fictitious boundary in the method of fundamental solutions (see Figs. 1 and 2), thanks to the introduction of the concept of the origin intensity factor, which isolates the singularity of the fundamental solutions.The SBM provides a significant and promising alternative to popular boundary-type methods such as the BEM and MFS, in particular, for infinite domain, wave, thin-walled structures, and inverse problems. | https://en.wikipedia.org/wiki/Singular_boundary_method |
In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately (see below). | https://en.wikipedia.org/wiki/Split-step_method |
Second, it is necessary to Fourier transform back and forth because the linear step is made in the frequency domain while the nonlinear step is made in the time domain. An example of usage of this method is in the field of light pulse propagation in optical fibers, where the interaction of linear and nonlinear mechanisms makes it difficult to find general analytical solutions. However, the split-step method provides a numerical solution to the problem. Another application of the split-step method that has been gaining a lot of traction since the 2010s is the simulation of Kerr frequency comb dynamics in optical microresonators. The relative ease of implementation of the Lugiato–Lefever equation with reasonable numerical cost, along with its success in reproducing experimental spectra as well as predicting soliton behavior in these microresonators has made the method very popular. | https://en.wikipedia.org/wiki/Split-step_method |
In numerical analysis, the uniform theory of diffraction (UTD) is a high-frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more than one dimension at the same point. UTD is an extension of Joseph Keller's geometrical theory of diffraction (GTD). The uniform theory of diffraction approximates near field electromagnetic fields as quasi optical and uses knife-edge diffraction to determine diffraction coefficients for each diffracting object-source combination. These coefficients are then used to calculate the field strength and phase for each direction away from the diffracting point. These fields are then added to the incident fields and reflected fields to obtain a total solution. | https://en.wikipedia.org/wiki/Geometrical_theory_of_diffraction |
In numerical analysis, transfinite interpolation is a means to construct functions over a planar domain in such a way that they match a given function on the boundary. This method is applied in geometric modelling and in the field of finite element method.The transfinite interpolation method, first introduced by William J. Gordon and Charles A. Hall, receives its name due to how a function belonging to this class is able to match the primitive function at a nondenumerable number of points. In the authors' words: We use the term ‘transfinite’ to describe the general class of interpolation schemes studied herein since, unlike the classical methods of higher dimensional interpolation which match the primitive function F at a finite number of distinct points, these methods match F at a non-denumerable (transfinite) number of points. Transfinite interpolation is similar to the Coons patch, invented in 1967. | https://en.wikipedia.org/wiki/Transfinite_interpolation |
In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial differential equations. The analysis is based on the Fourier decomposition of numerical error and was developed at Los Alamos National Laboratory after having been briefly described in a 1947 article by British researchers Crank and Nicolson. This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time. Later, the method was given a more rigorous treatment in an article co-authored by John von Neumann. | https://en.wikipedia.org/wiki/Von_Neumann_stability_analysis |
In numerical calculation, the days of the week are represented as weekday numbers. If Monday is the first day of the week, the days may be coded 1 to 7, for Monday through Sunday, as is practiced in ISO 8601. The day designated with 7 may also be counted as 0, by applying the arithmetic modulo 7, which calculates the remainder of a number after division by 7. Thus, the number 7 is treated as 0, the number 8 as 1, the number 9 as 2, the number 18 as 4, and so on. | https://en.wikipedia.org/wiki/Determination_of_the_day_of_the_week |
If Sunday is counted as day 1, then 7 days later (i.e. day 8) is also a Sunday, and day 18 is the same as day 4, which is a Wednesday since this falls three days after Sunday (i.e. 18 mod 7 = 4). The basic approach of nearly all of the methods to calculate the day of the week begins by starting from an 'anchor date': a known pair (such as 1 January 1800 as a Wednesday), determining the number of days between the known day and the day that you are trying to determine, and using arithmetic modulo 7 to find a new numerical day of the week. One standard approach is to look up (or calculate, using a known rule) the value of the first day of the week of a given century, look up (or calculate, using a method of congruence) an adjustment for the month, calculate the number of leap years since the start of the century, and then add these together along with the number of years since the start of the century, and the day number of the month. | https://en.wikipedia.org/wiki/Determination_of_the_day_of_the_week |
Eventually, one ends up with a day-count to which one applies modulo 7 to determine the day of the week of the date.Some methods do all the additions first and then cast out sevens, whereas others cast them out at each step, as in Lewis Carroll's method. Either way is quite viable: the former is easier for calculators and computer programs, the latter for mental calculation (it is quite possible to do all the calculations in one's head with a little practice). None of the methods given here perform range checks, so unreasonable dates will produce erroneous results. | https://en.wikipedia.org/wiki/Determination_of_the_day_of_the_week |
In numerical computation, pseudocode often consists of mathematical notation, typically from set and matrix theory, mixed with the control structures of a conventional programming language, and perhaps also natural language descriptions. This is a compact and often informal notation that can be understood by a wide range of mathematically trained people, and is frequently used as a way to describe mathematical algorithms. For example, the sum operator (capital-sigma notation) or the product operator (capital-pi notation) may represent a for-loop and a selection structure in one expression: Return ∑ k ∈ S x k {\displaystyle \sum _{k\in S}x_{k}} Normally non-ASCII typesetting is used for the mathematical equations, for example by means of markup languages, such as TeX or MathML, or proprietary formula editors. Mathematical style pseudocode is sometimes referred to as pidgin code, for example pidgin ALGOL (the origin of the concept), pidgin Fortran, pidgin BASIC, pidgin Pascal, pidgin C, and pidgin Lisp. | https://en.wikipedia.org/wiki/Pseudo_code |
In numerical computation, the numerical solution may not satisfy Gauss's law for magnetism due to the discretization errors of the numerical methods. However, in many cases, e.g., for magnetohydrodynamics, it is important to preserve Gauss's law for magnetism precisely (up to the machine precision). Violation of Gauss's law for magnetism on the discrete level will introduce a strong non-physical force. In view of energy conservation, violation of this condition leads to a non-conservative energy integral, and the error is proportional to the divergence of the magnetic field.There are various ways to preserve Gauss's law for magnetism in numerical methods, including the divergence-cleaning techniques, the constrained transport method, potential-based formulations and de Rham complex based finite element methods where stable and structure-preserving algorithms are constructed on unstructured meshes with finite element differential forms. | https://en.wikipedia.org/wiki/Gauss'_law_for_magnetism |
In numerical control machining, the programming of the NC tool exploits the fact that the Minkowski sum of the cutting piece with its trajectory gives the shape of the cut in the material. | https://en.wikipedia.org/wiki/Minkowski_difference |
In numerical control systems, the position of the tool is defined by a set of instructions called the part program. Positioning control is handled using either an open-loop or a closed-loop system. In an open-loop system, communication takes place in one direction only: from the controller to the motor. | https://en.wikipedia.org/wiki/Numeric_control |
In a closed-loop system, feedback is provided to the controller so that it can correct for errors in position, velocity, and acceleration, which can arise due to variations in load or temperature. Open-loop systems are generally cheaper but less accurate. Stepper motors can be used in both types of systems, while servo motors can only be used in closed systems. | https://en.wikipedia.org/wiki/Numeric_control |
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads In German and some other languages, it is named after Johannes Kepler, who derived it in 1615 after seeing it used for wine barrels (barrel rule, Keplersche Fassregel). The approximate equality in the rule becomes exact if f is a polynomial up to and including 3rd degree. If the 1/3 rule is applied to n equal subdivisions of the integration range , one obtains the composite Simpson's 1/3 rule. | https://en.wikipedia.org/wiki/Simpson's_Rule |
Points inside the integration range are given alternating weights 4/3 and 2/3. Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve on order of the error. | https://en.wikipedia.org/wiki/Simpson's_Rule |
If the 3/8 rule is applied to n equal subdivisions of the integration range , one obtains the composite Simpson's 3/8 rule. Simpson's 1/3 and 3/8 rules are two special cases of closed Newton–Cotes formulas. In naval architecture and ship stability estimation, there also exists Simpson's third rule, which has no special importance in general numerical analysis, see Simpson's rules (ship stability). | https://en.wikipedia.org/wiki/Simpson's_Rule |
In numerical integration, methods such as the trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is likely to be within those error bars. The problem Monte Carlo integration addresses is the computation of a multidimensional definite integral I = ∫ Ω f ( x ¯ ) d x ¯ {\displaystyle I=\int _{\Omega }f({\overline {\mathbf {x} }})\,d{\overline {\mathbf {x} }}} where Ω, a subset of Rm, has volume V = ∫ Ω d x ¯ {\displaystyle V=\int _{\Omega }d{\overline {\mathbf {x} }}} The naive Monte Carlo approach is to sample points uniformly on Ω: given N uniform samples, x ¯ 1 , ⋯ , x ¯ N ∈ Ω , {\displaystyle {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in \Omega ,} I can be approximated by I ≈ Q N ≡ V 1 N ∑ i = 1 N f ( x ¯ i ) = V ⟨ f ⟩ {\displaystyle I\approx Q_{N}\equiv V{\frac {1}{N}}\sum _{i=1}^{N}f({\overline {\mathbf {x} }}_{i})=V\langle f\rangle } .This is because the law of large numbers ensures that lim N → ∞ Q N = I {\displaystyle \lim _{N\to \infty }Q_{N}=I} .Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. | https://en.wikipedia.org/wiki/MISER_algorithm |
V a r ( f ) ≡ σ N 2 = 1 N − 1 ∑ i = 1 N ( f ( x ¯ i ) − ⟨ f ⟩ ) 2 . {\displaystyle \mathrm {Var} (f)\equiv \sigma _{N}^{2}={\frac {1}{N-1}}\sum _{i=1}^{N}\left(f({\overline {\mathbf {x} }}_{i})-\langle f\rangle \right)^{2}.} which leads to V a r ( Q N ) = V 2 N 2 ∑ i = 1 N V a r ( f ) = V 2 V a r ( f ) N = V 2 σ N 2 N {\displaystyle \mathrm {Var} (Q_{N})={\frac {V^{2}}{N^{2}}}\sum _{i=1}^{N}\mathrm {Var} (f)=V^{2}{\frac {\mathrm {Var} (f)}{N}}=V^{2}{\frac {\sigma _{N}^{2}}{N}}} .As long as the sequence { σ 1 2 , σ 2 2 , σ 3 2 , … } {\displaystyle \left\{\sigma _{1}^{2},\sigma _{2}^{2},\sigma _{3}^{2},\ldots \right\}} is bounded, this variance decreases asymptotically to zero as 1/N. | https://en.wikipedia.org/wiki/MISER_algorithm |
The estimation of the error of QN is thus δ Q N ≈ V a r ( Q N ) = V σ N N , {\displaystyle \delta Q_{N}\approx {\sqrt {\mathrm {Var} (Q_{N})}}=V{\frac {\sigma _{N}}{\sqrt {N}}},} which decreases as 1 N {\displaystyle {\tfrac {1}{\sqrt {N}}}} . This is standard error of the mean multiplied with V {\displaystyle V} . This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. | https://en.wikipedia.org/wiki/MISER_algorithm |
It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem-specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher-dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub-domains—and importance sampling—sampling from non-uniform distributions—are two examples of such techniques. | https://en.wikipedia.org/wiki/MISER_algorithm |
In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory. | https://en.wikipedia.org/wiki/Givens_rotation |
In numerical linear algebra, a Jacobi rotation is a rotation, Qkℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A, when applied as a similarity transformation: A ↦ Q k ℓ T A Q k ℓ = A ′ . {\displaystyle A\mapsto Q_{k\ell }^{T}AQ_{k\ell }=A'.\,\!} → . | https://en.wikipedia.org/wiki/Jacobi_rotation |
{\displaystyle {\begin{bmatrix}{*}&&&\cdots &&&*\\&\ddots &&&&&\\&&a_{kk}&\cdots &a_{k\ell }&&\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\&&a_{\ell k}&\cdots &a_{\ell \ell }&&\\&&&&&\ddots &\\{*}&&&\cdots &&&*\end{bmatrix}}\to {\begin{bmatrix}{*}&&&\cdots &&&*\\&\ddots &&&&&\\&&a'_{kk}&\cdots &0&&\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\&&0&\cdots &a'_{\ell \ell }&&\\&&&&&\ddots &\\{*}&&&\cdots &&&*\end{bmatrix}}.} It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors. Only rows k and ℓ and columns k and ℓ of A will be affected, and that A′ will remain symmetric. | https://en.wikipedia.org/wiki/Jacobi_rotation |
Also, an explicit matrix for Qkℓ is rarely computed; instead, auxiliary values are computed and A is updated in an efficient and numerically stable way. However, for reference, we may write the matrix as Q k ℓ = . {\displaystyle Q_{k\ell }={\begin{bmatrix}1&&&&&&\\&\ddots &&&&0&\\&&c&\cdots &s&&\\&&\vdots &\ddots &\vdots &&\\&&-s&\cdots &c&&\\&0&&&&\ddots &\\&&&&&&1\end{bmatrix}}.} | https://en.wikipedia.org/wiki/Jacobi_rotation |
That is, Qkℓ is an identity matrix except for four entries, two on the diagonal (qkk and qℓℓ, both equal to c) and two symmetrically placed off the diagonal (qkℓ and qℓk, equal to s and −s, respectively). Here c = cos θ and s = sin θ for some angle θ; but to apply the rotation, the angle itself is not required. Using Kronecker delta notation, the matrix entries can be written q i j = δ i j + ( δ i k δ j k + δ i ℓ δ j ℓ ) ( c − 1 ) + ( δ i k δ j ℓ − δ i ℓ δ j k ) s . | https://en.wikipedia.org/wiki/Jacobi_rotation |
{\displaystyle q_{ij}=\delta _{ij}+(\delta _{ik}\delta _{jk}+\delta _{i\ell }\delta _{j\ell })(c-1)+(\delta _{ik}\delta _{j\ell }-\delta _{i\ell }\delta _{jk})s.\,\!} Suppose h is an index other than k or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically, a h k ′ = a k h ′ = c a h k − s a h ℓ {\displaystyle a'_{hk}=a'_{kh}=ca_{hk}-sa_{h\ell }\,\!} | https://en.wikipedia.org/wiki/Jacobi_rotation |
a h ℓ ′ = a ℓ h ′ = c a h ℓ + s a h k {\displaystyle a'_{h\ell }=a'_{\ell h}=ca_{h\ell }+sa_{hk}\,\!} a k ℓ ′ = a ℓ k ′ = ( c 2 − s 2 ) a k ℓ + s c ( a k k − a ℓ ℓ ) = 0 {\displaystyle a'_{k\ell }=a'_{\ell k}=(c^{2}-s^{2})a_{k\ell }+sc(a_{kk}-a_{\ell \ell })=0\,\!} a k k ′ = c 2 a k k + s 2 a ℓ ℓ − 2 s c a k ℓ {\displaystyle a'_{kk}=c^{2}a_{kk}+s^{2}a_{\ell \ell }-2sca_{k\ell }\,\!} a ℓ ℓ ′ = s 2 a k k + c 2 a ℓ ℓ + 2 s c a k ℓ . {\displaystyle a'_{\ell \ell }=s^{2}a_{kk}+c^{2}a_{\ell \ell }+2sca_{k\ell }.\,\!} | https://en.wikipedia.org/wiki/Jacobi_rotation |
In numerical linear algebra, an incomplete LU factorization (abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as a preconditioner. | https://en.wikipedia.org/wiki/Incomplete_LU_factorization |
In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation A X − X B = C {\displaystyle AX-XB=C} . Developed by R.H. Bartels and G.W. | https://en.wikipedia.org/wiki/Bartels–Stewart_algorithm |
Stewart in 1971, it was the first numerically stable method that could be systematically applied to solve such equations. The algorithm works by using the real Schur decompositions of A {\displaystyle A} and B {\displaystyle B} to transform A X − X B = C {\displaystyle AX-XB=C} into a triangular system that can then be solved using forward or backward substitution. In 1979, G. Golub, C. Van Loan and S. Nash introduced an improved version of the algorithm, known as the Hessenberg–Schur algorithm. It remains a standard approach for solving Sylvester equations when X {\displaystyle X} is of small to moderate size. | https://en.wikipedia.org/wiki/Bartels–Stewart_algorithm |
In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee, is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth. The reverse Cuthill–McKee algorithm (RCM) due to Alan George and Joseph Liu is the same algorithm but with the resulting index numbers reversed. In practice this generally results in less fill-in than the CM ordering when Gaussian elimination is applied.The Cuthill McKee algorithm is a variant of the standard breadth-first search algorithm used in graph algorithms. | https://en.wikipedia.org/wiki/Cuthill–McKee_algorithm |
It starts with a peripheral node and then generates levels R i {\displaystyle R_{i}} for i = 1 , 2 , . . {\displaystyle i=1,2,..} until all nodes are exhausted. The set R i + 1 {\displaystyle R_{i+1}} is created from set R i {\displaystyle R_{i}} by listing all vertices adjacent to all nodes in R i {\displaystyle R_{i}} . These nodes are ordered according to predecessors and degree. | https://en.wikipedia.org/wiki/Cuthill–McKee_algorithm |
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either strictly diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. A publication was not delivered before 1874 by Seidel. | https://en.wikipedia.org/wiki/Gauss–Seidel_method |
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is named after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but only became widely used in the 1950s with the advent of computers. | https://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm |
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. | https://en.wikipedia.org/wiki/Jacobi_method |
The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after Carl Gustav Jacob Jacobi. | https://en.wikipedia.org/wiki/Jacobi_method |
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently. The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate. | https://en.wikipedia.org/wiki/QR_algorithm |
In numerical linear algebra, the Rayleigh–Ritz method is commonly applied to approximate an eigenvalue problem for the matrix A ∈ C N × N {\displaystyle A\in \mathbb {C} ^{N\times N}} of size N {\displaystyle N} using a projected matrix of a smaller size m < N {\displaystyle m | https://en.wikipedia.org/wiki/Rayleigh–Ritz_method |
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions. It is an example of an operator splitting method. | https://en.wikipedia.org/wiki/Alternating_direction_implicit_method |
In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method. Unlike the original BiCG method, it doesn't require multiplication by the transpose of the system matrix. | https://en.wikipedia.org/wiki/Biconjugate_gradient_stabilized_method |
In numerical linear algebra, the conjugate gradient method is an iterative method for numerically solving the linear system A x = b {\displaystyle {\boldsymbol {Ax}}={\boldsymbol {b}}} where A {\displaystyle {\boldsymbol {A}}} is symmetric positive-definite. The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. The intent of this article is to document the important steps in these derivations. | https://en.wikipedia.org/wiki/Derivation_of_the_conjugate_gradient_method |
In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging iterative process. It was devised simultaneously by David M. Young Jr. | https://en.wikipedia.org/wiki/Successive_over-relaxation |
and by Stanley P. Frankel in 1950 for the purpose of automatically solving linear systems on digital computers. Over-relaxation methods had been used before the work of Young and Frankel. An example is the method of Lewis Fry Richardson, and the methods developed by R. V. Southwell. However, these methods were designed for computation by human calculators, requiring some expertise to ensure convergence to the solution which made them inapplicable for programming on digital computers. These aspects are discussed in the thesis of David M. Young Jr. | https://en.wikipedia.org/wiki/Successive_over-relaxation |
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as a i x i − 1 + b i x i + c i x i + 1 = d i , {\displaystyle a_{i}x_{i-1}+b_{i}x_{i}+c_{i}x_{i+1}=d_{i},} where a 1 = 0 {\displaystyle a_{1}=0} and c n = 0 {\displaystyle c_{n}=0} . = . {\displaystyle {\begin{bmatrix}b_{1}&c_{1}&&&0\\a_{2}&b_{2}&c_{2}&&\\&a_{3}&b_{3}&\ddots &\\&&\ddots &\ddots &c_{n-1}\\0&&&a_{n}&b_{n}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}d_{1}\\d_{2}\\d_{3}\\\vdots \\d_{n}\end{bmatrix}}.} | https://en.wikipedia.org/wiki/Thomas_algorithm |
For such systems, the solution can be obtained in O ( n ) {\displaystyle O(n)} operations instead of O ( n 3 ) {\displaystyle O(n^{3})} required by Gaussian elimination. A first sweep eliminates the a i {\displaystyle a_{i}} 's, and then an (abbreviated) backward substitution produces the solution. Examples of such matrices commonly arise from the discretization of 1D Poisson equation and natural cubic spline interpolation. Thomas' algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant (either by rows or columns) or symmetric positive definite; for a more precise characterization of stability of Thomas' algorithm, see Higham Theorem 9.12. If stability is required in the general case, Gaussian elimination with partial pivoting (GEPP) is recommended instead. | https://en.wikipedia.org/wiki/Thomas_algorithm |
In numerical mathematics, Beam and Warming scheme or Beam–Warming implicit scheme introduced in 1978 by Richard M. Beam and R. F. Warming, is a second order accurate implicit scheme, mainly used for solving non-linear hyperbolic equations. It is not used much nowadays. | https://en.wikipedia.org/wiki/Beam_and_Warming_scheme |
In numerical mathematics, a non-compact stencil is a type of discretization method, where any node surrounding the node of interest may be used in the calculation. Its computational time grows with an increase of layers of nodes used. Non-compact stencils may be compared to Compact stencils. | https://en.wikipedia.org/wiki/Non-compact_stencil |
In numerical mathematics, artificial precision is a source of error that occurs when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input. For example, a person enters their birthday as the date 1984-01-01 but it is stored in a database as 1984-01-01T00:00:00Z which introduces the artificial precision of the hour, minute, and second they were born, and may even affect the date, depending on the user's actual place of birth. This is also an example of false precision, which is artificial precision specifically of numerical quantities or measures. | https://en.wikipedia.org/wiki/Artificial_precision |
In numerical mathematics, hierarchical matrices (H-matrices) are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension n {\displaystyle n} can be represented efficiently in O ( n ) {\displaystyle O(n)} units of storage by storing only its non-zero entries, a non-sparse matrix would require O ( n 2 ) {\displaystyle O(n^{2})} units of storage, and using this type of matrices for large problems would therefore be prohibitively expensive in terms of storage and computing time. Hierarchical matrices provide an approximation requiring only O ( n k log ( n ) ) {\displaystyle O(nk\,\log(n))} units of storage, where k {\displaystyle k} is a parameter controlling the accuracy of the approximation. | https://en.wikipedia.org/wiki/Hierarchical_matrix |
In typical applications, e.g., when discretizing integral equations, preconditioning the resulting systems of linear equations, or solving elliptic partial differential equations, a rank proportional to log ( 1 / ϵ ) γ {\displaystyle \log(1/\epsilon )^{\gamma }} with a small constant γ {\displaystyle \gamma } is sufficient to ensure an accuracy of ϵ {\displaystyle \epsilon } . Compared to many other data-sparse representations of non-sparse matrices, hierarchical matrices offer a major advantage: the results of matrix arithmetic operations like matrix multiplication, factorization or inversion can be approximated in O ( n k α log ( n ) β ) {\displaystyle O(nk^{\alpha }\,\log(n)^{\beta })} operations, where α , β ∈ { 1 , 2 , 3 } . {\displaystyle \alpha ,\beta \in \{1,2,3\}.} | https://en.wikipedia.org/wiki/Hierarchical_matrix |
In numerical mathematics, interval propagation or interval constraint propagation is the problem of contracting interval domains associated to variables of R without removing any value that is consistent with a set of constraints (i.e., equations or inequalities). It can be used to propagate uncertainties in the situation where errors are represented by intervals. Interval propagation considers an estimation problem as a constraint satisfaction problem. | https://en.wikipedia.org/wiki/Interval_propagation |
In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems.Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. They are also used for the solution of linear equations for linear least-squares problems and also for systems of linear inequalities, such as those arising in linear programming. They have also been developed for solving nonlinear systems of equations.Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. | https://en.wikipedia.org/wiki/Relaxation_(iterative_method) |
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation, it resembles repeated application of a local smoothing filter to the solution vector. These are not to be confused with relaxation methods in mathematical optimization, which approximate a difficult problem by a simpler problem whose "relaxed" solution provides information about the solution of the original problem. | https://en.wikipedia.org/wiki/Relaxation_(iterative_method) |
In numerical mathematics, the Uzawa iteration is an algorithm for solving saddle point problems. It is named after Hirofumi Uzawa and was originally introduced in the context of concave programming. | https://en.wikipedia.org/wiki/Uzawa_iteration |
In numerical mathematics, the boundary knot method (BKM) is proposed as an alternative boundary-type meshfree distance function collocation scheme. Recent decades have witnessed a research boom on the meshfree numerical PDE techniques since the construction of a mesh in the standard finite element method and boundary element method is not trivial especially for moving boundary, and higher-dimensional problems. The boundary knot method is different from the other methods based on the fundamental solutions, such as boundary element method, method of fundamental solutions and singular boundary method in that the former does not require special techniques to cure the singularity. The BKM is truly meshfree, spectral convergent (numerical observations), symmetric (self-adjoint PDEs), integration-free, and easy to learn and implement. The method has successfully been tested to the Helmholtz, diffusion, convection-diffusion, and Possion equations with very irregular 2D and 3D domains. | https://en.wikipedia.org/wiki/Boundary_knot_method |
In numerical mathematics, the constant strain triangle element, also known as the CST element or T3 element, is a type of element used in finite element analysis which is used to provide an approximate solution in a 2D domain to the exact solution of a given differential equation. The name of this element reflects how the partial derivatives of this element's shape function are linear functions. When applied to plane stress and plane strain problems, this means that the approximate solution obtained for the stress and strain fields are constant throughout the element's domain. The element provides an approximation for the exact solution of a partial differential equation which is parametrized barycentric coordinate system (mathematics) | https://en.wikipedia.org/wiki/Constant_strain_triangle_element |
In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless). Some core properties are required to prove the convergence of a GDM. These core properties enable complete proofs of convergence of the GDM for elliptic and parabolic problems, linear or non-linear. | https://en.wikipedia.org/wiki/Gradient_discretization_method |
For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM (the quantities C D {\displaystyle C_{D}} , S D {\displaystyle S_{D}} and W D {\displaystyle W_{D}} , see below). For non-linear problems, the proofs are based on compactness techniques and do not require any non-physical strong regularity assumption on the solution or the model data. Non-linear models for which such convergence proof of the GDM have been carried out comprise: the Stefan problem which is modelling a melting material, two-phase flows in porous media, the Richards equation of underground water flow, the fully non-linear Leray—Lions equations.Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to conforming Finite Elements, Mixed Finite Elements, nonconforming Finite Elements, and, in the case of more recent schemes, the Discontinuous Galerkin method, Hybrid Mixed Mimetic method, the Nodal Mimetic Finite Difference method, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes | https://en.wikipedia.org/wiki/Gradient_discretization_method |
In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to solve certain partial differential equations whose fundamental solution is explicitly known. The RMM is a strong-form collocation method with merits being meshless, integration-free, easy-to-implement, and high stability. Until now this method has been successfully applied to some typical problems, such as potential, acoustics, water wave, and inverse problems of bounded and unbounded domains. | https://en.wikipedia.org/wiki/Regularized_meshless_method |
In numerical methods for stochastic differential equations, the Markov chain approximation method (MCAM) belongs to the several numerical (schemes) approaches used in stochastic control theory. Regrettably the simple adaptation of the deterministic schemes for matching up to stochastic models such as the Runge–Kutta method does not work at all. It is a powerful and widely usable set of ideas, due to the current infancy of stochastic control it might be even said 'insights.' | https://en.wikipedia.org/wiki/Markov_chain_approximation_method |
for numerical and other approximations problems in stochastic processes. They represent counterparts from deterministic control theory such as optimal control theory.The basic idea of the MCAM is to approximate the original controlled process by a chosen controlled markov process on a finite state space. In case of need, one must as well approximate the cost function for one that matches up the Markov chain chosen to approximate the original stochastic process. | https://en.wikipedia.org/wiki/Markov_chain_approximation_method |
In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten. | https://en.wikipedia.org/wiki/Total_variation_diminishing |
In numerical models and mathematical models, there are two different approaches to describe the motion of matter: Eulerian and Lagrangian. In geology, both approaches are commonly used to model fluid flow like mantle convection, where an Eulerian grid is used for computation and Lagrangian markers are used to visualize the motion. Recently, there have been models that try to describe different parts using different approaches to combine the advantages of these two approaches. This combined approach is called the arbitrary Lagrangian-Eulerian approach. | https://en.wikipedia.org/wiki/Numerical_modeling_(geology) |
In numerical optimization, meta-optimization is the use of one optimization method to tune another optimization method. Meta-optimization is reported to have been used as early as in the late 1970s by Mercer and Sampson for finding optimal parameter settings of a genetic algorithm. Meta-optimization and related concepts are also known in the literature as meta-evolution, super-optimization, automated parameter calibration, hyper-heuristics, etc. | https://en.wikipedia.org/wiki/Meta-optimization |
In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information. It does so by gradually improving an approximation to the Hessian matrix of the loss function, obtained only from gradient evaluations (or approximate gradient evaluations) via a generalized secant method.Since the updates of the BFGS curvature matrix do not require matrix inversion, its computational complexity is only O ( n 2 ) {\displaystyle {\mathcal {O}}(n^{2})} , compared to O ( n 3 ) {\displaystyle {\mathcal {O}}(n^{3})} in Newton's method. Also in common use is L-BFGS, which is a limited-memory version of BFGS that is particularly suited to problems with very large numbers of variables (e.g., >1000). The BFGS-B variant handles simple box constraints.The algorithm is named after Charles George Broyden, Roger Fletcher, Donald Goldfarb and David Shanno. | https://en.wikipedia.org/wiki/BFGS_method |
In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function f ( x ) {\displaystyle \displaystyle f(x)} f ( x ) = ‖ A x − b ‖ 2 , {\displaystyle \displaystyle f(x)=\|Ax-b\|^{2},} the minimum of f {\displaystyle f} is obtained when the gradient is 0: ∇ x f = 2 A T ( A x − b ) = 0 {\displaystyle \nabla _{x}f=2A^{T}(Ax-b)=0} .Whereas linear conjugate gradient seeks a solution to the linear equation A T A x = A T b {\displaystyle \displaystyle A^{T}Ax=A^{T}b} , the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient ∇ x f {\displaystyle \nabla _{x}f} alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum and the second derivative is non-singular there. Given a function f ( x ) {\displaystyle \displaystyle f(x)} of N {\displaystyle N} variables to minimize, its gradient ∇ x f {\displaystyle \nabla _{x}f} indicates the direction of maximum increase. | https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient |
The algorithm stops when it finds the minimum, determined when no progress is made after a direction reset (i.e. in the steepest descent direction), or when some tolerance criterion is reached. Within a linear approximation, the parameters α {\displaystyle \displaystyle \alpha } and β {\displaystyle \displaystyle \beta } are the same as in the linear conjugate gradient method but have been obtained with line searches. The conjugate gradient method can follow narrow (ill-conditioned) valleys, where the steepest descent method slows down and follows a criss-cross pattern. | https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient |
Four of the best known formulas for β n {\displaystyle \displaystyle \beta _{n}} are named after their developers: Fletcher–Reeves: β n F R = Δ x n T Δ x n Δ x n − 1 T Δ x n − 1 . {\displaystyle \beta _{n}^{FR}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.} Polak–Ribière: β n P R = Δ x n T ( Δ x n − Δ x n − 1 ) Δ x n − 1 T Δ x n − 1 . | https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient |
{\displaystyle \beta _{n}^{PR}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.} Hestenes-Stiefel: β n H S = Δ x n T ( Δ x n − Δ x n − 1 ) − s n − 1 T ( Δ x n − Δ x n − 1 ) . {\displaystyle \beta _{n}^{HS}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{-s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.} | https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient |
Dai–Yuan: β n D Y = Δ x n T Δ x n − s n − 1 T ( Δ x n − Δ x n − 1 ) . {\displaystyle \beta _{n}^{DY}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{-s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.} | https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient |
.These formulas are equivalent for a quadratic function, but for nonlinear optimization the preferred formula is a matter of heuristics or taste. A popular choice is β = max { 0 , β P R } {\displaystyle \displaystyle \beta =\max\{0,\beta ^{PR}\}} , which provides a direction reset automatically.Algorithms based on Newton's method potentially converge much faster. There, both step direction and length are computed from the gradient as the solution of a linear system of equations, with the coefficient matrix being the exact Hessian matrix (for Newton's method proper) or an estimate thereof (in the quasi-Newton methods, where the observed change in the gradient during the iterations is used to update the Hessian estimate). | https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient |
For high-dimensional problems, the exact computation of the Hessian is usually prohibitively expensive, and even its storage can be problematic, requiring O ( N 2 ) {\displaystyle O(N^{2})} memory (but see the limited-memory L-BFGS quasi-Newton method). The conjugate gradient method can also be derived using optimal control theory. In this accelerated optimization theory, the conjugate gradient method falls out as a nonlinear optimal feedback controller, u = k ( x , x ˙ ) := − γ a ∇ x f ( x ) − γ b x ˙ {\displaystyle u=k(x,{\dot {x}}):=-\gamma _{a}\nabla _{x}f(x)-\gamma _{b}{\dot {x}}} for the double integrator system, x ¨ = u {\displaystyle {\ddot {x}}=u} The quantities γ a > 0 {\displaystyle \gamma _{a}>0} and γ b > 0 {\displaystyle \gamma _{b}>0} are variable feedback gains. | https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient |
In numerical order and with their modern French and English Braille equivalents, the letters are: Not quite half of the letters retained their French Braille values. | https://en.wikipedia.org/wiki/American_Braille |
In numerical order by decade, the letters are: For the purposes of accommodating a foreign alphabet, the letters ì, ä, ò may be added: There are also numerous contractions and abbreviations in French braille. | https://en.wikipedia.org/wiki/French_Braille |
In numerical physics the method is used to find solutions of the unidimensional Schrödinger equation for arbitrary potentials. An example of which is solving the radial equation for a spherically symmetric potential. In this example, after separating the variables and analytically solving the angular equation, we are left with the following equation of the radial function R ( r ) {\displaystyle R(r)}: d d r ( r 2 d R d r ) − 2 m r 2 ℏ 2 ( V ( r ) − E ) R ( r ) = l ( l + 1 ) R ( r ) . {\displaystyle {\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)-{\frac {2mr^{2}}{\hbar ^{2}}}(V(r)-E)R(r)=l(l+1)R(r).} | https://en.wikipedia.org/wiki/Numerov's_method |
{\displaystyle V_{\text{eff}}(r)=V(r)+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}=V(r)+{\frac {L^{2}}{2mr^{2}}},\quad L^{2}=l(l+1)\hbar ^{2}.} This equation we can proceed to solve the same way we would have solved the one-dimensional Schrödinger equation. We can rewrite the equation a little bit differently and thus see the possible application of Numerov's method more clearly: d 2 u d r 2 = − 2 m ℏ 2 ( E − V eff ( r ) ) u ( r ) , {\displaystyle {\frac {d^{2}u}{dr^{2}}}=-{\frac {2m}{\hbar ^{2}}}(E-V_{\text{eff}}(r))u(r),} g ( r ) = 2 m ℏ 2 ( E − V eff ( r ) ) , {\displaystyle g(r)={\frac {2m}{\hbar ^{2}}}(E-V_{\text{eff}}(r)),} s ( r ) = 0. {\displaystyle s(r)=0.} | https://en.wikipedia.org/wiki/Numerov's_method |
In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters to the model vary about a mean. For instance, in a system modeled as a function of two variables z = f ( x , y ) . {\displaystyle z\,=\,f(x,y).} Error analysis deals with the propagation of the numerical errors in x {\displaystyle x} and y {\displaystyle y} (around mean values x ¯ {\displaystyle {\bar {x}}} and y ¯ {\displaystyle {\bar {y}}} ) to error in z {\displaystyle z} (around a mean z ¯ {\displaystyle {\bar {z}}} ).In numerical analysis, error analysis comprises both forward error analysis and backward error analysis. | https://en.wikipedia.org/wiki/Error_analysis_(mathematics) |
In numerical solution of differential equations, WENO (weighted essentially non-oscillatory) methods are classes of high-resolution schemes. WENO are used in the numerical solution of hyperbolic partial differential equations. These methods were developed from ENO methods (essentially non-oscillatory). | https://en.wikipedia.org/wiki/WENO_methods |
The first WENO scheme was developed by Liu, Osher and Chan in 1994. In 1996, Guang-Sh and Chi-Wang Shu developed a new WENO scheme called WENO-JS. Nowadays, there are many WENO methods. | https://en.wikipedia.org/wiki/WENO_methods |
In numerical weather prediction applications, data assimilation is most widely known as a method for combining observations of meteorological variables such as temperature and atmospheric pressure with prior forecasts in order to initialize numerical forecast models. | https://en.wikipedia.org/wiki/Data_assimilation |
In numerically challenging applications, sophisticated preconditioners are used, which may lead to variable preconditioning, changing between iterations. Even if the preconditioner is symmetric positive-definite on every iteration, the fact that it may change makes the arguments above invalid, and in practical tests leads to a significant slow down of the convergence of the algorithm presented above. Using the Polak–Ribière formula β k := r k + 1 T ( z k + 1 − z k ) r k T z k {\displaystyle \beta _{k}:={\frac {\mathbf {r} _{k+1}^{\mathsf {T}}\left(\mathbf {z} _{k+1}-\mathbf {z} _{k}\right)}{\mathbf {r} _{k}^{\mathsf {T}}\mathbf {z} _{k}}}} instead of the Fletcher–Reeves formula β k := r k + 1 T z k + 1 r k T z k {\displaystyle \beta _{k}:={\frac {\mathbf {r} _{k+1}^{\mathsf {T}}\mathbf {z} _{k+1}}{\mathbf {r} _{k}^{\mathsf {T}}\mathbf {z} _{k}}}} may dramatically improve the convergence in this case. This version of the preconditioned conjugate gradient method can be called flexible, as it allows for variable preconditioning. | https://en.wikipedia.org/wiki/Conjugate_Gradient_method |
The flexible version is also shown to be robust even if the preconditioner is not symmetric positive definite (SPD). The implementation of the flexible version requires storing an extra vector. For a fixed SPD preconditioner, r k + 1 T z k = 0 , {\displaystyle \mathbf {r} _{k+1}^{\mathsf {T}}\mathbf {z} _{k}=0,} so both formulas for βk are equivalent in exact arithmetic, i.e., without the round-off error. The mathematical explanation of the better convergence behavior of the method with the Polak–Ribière formula is that the method is locally optimal in this case, in particular, it does not converge slower than the locally optimal steepest descent method. | https://en.wikipedia.org/wiki/Conjugate_Gradient_method |
In numerology, 11:11 is considered to be a significant moment in time for an event to occur. It is seen as an example of synchronicity, as well as a favorable sign or a suggestion towards the presence of spiritual influence. It is additionally thought that the repetition of numbers in the sequence adds "intensity" to them and increases the numerological effect.Critics highlight the lack of substantial evidence for this assertion, and they gesture towards confirmation bias and post-hoc analysis as a scientific explanation for any claims related to the significance or importance of 11:11 and other such sequences. Through observations made in the study of statistics, specifically chaos theory and the law of truly large numbers, skeptics explain these anecdotal observations as a coincidence and an inevitability, rather than as any particular indication towards significance. | https://en.wikipedia.org/wiki/11:11_(numerology) |
In numerous bird species, a breeding pair receives support in raising its young from other "helper" birds, including help with the feeding of its fledglings. Some will even go as far as protecting an unrelated bird's young from predators. | https://en.wikipedia.org/wiki/Altruism_in_animals |
In numerous books, movies, video games, etc., the hotline between Washington and Moscow is represented by a red phone, although the real hotline has never been a telephone line. A hotline telephone was depicted in the film Fail-Safe as the "Red 1 / Ultimate 1 Touch phone", and also in Stanley Kubrick's film Dr. Strangelove, both from 1964 and both loosely based on Peter George's Cold War thriller novel Red Alert from 1958. A more realistic depiction of the Hotline was Tom Clancy's novel The Sum of All Fears from 1991 and its 2002 film adaptation, in which a text-based computer communications system was depicted, resembling the actual Hotline equipment from the 1980s and 1990s. In the 1990 HBO film By Dawn's Early Light, the White House Situation Room equipment that receives the (translated) hotline message, apparently relayed by the Pentagon-NMCC MOLINK team, is depicted as a teleprinter (and not as a fax machine, the technology already in use at the NMCC itself by that year). | https://en.wikipedia.org/wiki/Moscow–Washington_hotline |
A telephone is used in the intro cinematic of the video game Command & Conquer: Red Alert 2. The call is placed by the US president to the Kremlin in the wake of a global Soviet invasion. In the 2005 episode of the british sci-fi show Doctor Who "World War Three" the Slitheen await a phone call to plunge the planet into Nuclear armageddon on an actual red telephone, directly pastiching the cold war fears related to the hotline. | https://en.wikipedia.org/wiki/Moscow–Washington_hotline |
In numerous cases, Hollywood studios adapting literary works into film added a happy ending which did not appear in the original. Mary Shelley's 1818 novel, Frankenstein, ended with the deaths of Victor Frankenstein and Elizabeth Lavenza. In the 1931 film adaption they survive and marry. C. S. Forester's 1935 novel The African Queen has a British couple, stranded in Africa during the First World War, hatch a plot to sink a German gunboat; they make an enormous, dedicated struggle, with boundless effort and sacrifice, but at the last moment their quest ends with failure and futility. | https://en.wikipedia.org/wiki/Happy_ending |
In the 1951 film adaptation they succeed, and get to see the German boat sink (just in time to save them from being hanged by the Germans). Truman Capote's 1958 novella Breakfast at Tiffany's ended with the main character, Holly Golightly, going her own solitary way and disappearing from the male protagonist's life. | https://en.wikipedia.org/wiki/Happy_ending |
In the 1961 film made on its base she finally accepts the love he offers her and the film ends with their warmly embracing, oblivious of a pouring rain. Hans Christian Andersen's fairy tale The Little Mermaid ends with the protagonist mermaid making a noble sacrifice, resigned to seeing her beloved prince marrying another woman. She is, however, unexpectedly rewarded for the sacrifice by the chance to earn herself an immortal soul by further good deeds in air-spirit form. | https://en.wikipedia.org/wiki/Happy_ending |
As written by Andersen, acquiring an immortal soul had been her main objective from the start, with the prince chiefly a means to this end, and thus Andersen may have meant this as a "spiritual happy ending". However, the 1989 Disney adaptation paid much less attention to the spiritual aspect and focused on the love interest, and they replaced Andersen's ending and presented the American public with a less subtle one and more conventional ending: the mermaid does get to happily marry her prince. Disney later added a sequel, obviously impossible for the Andersen original, focused on the child born of that marriage. | https://en.wikipedia.org/wiki/Happy_ending |
Herman Wouk's novel Marjorie Morningstar ends with the formerly vibrant protagonist giving up her dreams of an artistic career, marrying a mediocre middle-class man approved by her parents and becoming totally reconciled to the commonplace life of a suburban housewife and mother. In her review for Slate Magazine, Alana Newhouse wrote that "most female readers cry when they reach the end of this book, and for good reason. Marjorie Morningstar, as they came to know her, has become another woman entirely"; Newhouse expressed the opinion that an adaptation to a film or a stage play which would keep the book's ending "would not run for a week". | https://en.wikipedia.org/wiki/Happy_ending |
But the makers of the film version did change the ending, letting Marjorie end up in the loving arms of a talented, sensitive and warm-hearted playwright – whom she unwisely rejected in the book, and who in the film version can be expected to encourage and support her in launching her own artistic career. George Orwell's Nineteen Eighty-Four has a particularly harsh ending, with the protagonists Winston and Julia being totally broken by the totalitarian state against which they tried to rebel, their subversive "criminal thoughts" driven out of their minds and being forced to betray each other and destroy their love for each other. The 1956 adaptation had two alternate endings made. | https://en.wikipedia.org/wiki/Happy_ending |
One, faithful to the Orwell original, ended with a rehabilitated and brainwashed Winston fervently joining the crowd cheering "Long live Big Brother!". The alternate ending had Winston rebelling against his brainwashing and starting to shout, "Down with Big Brother", whereupon he is shot down. Julia runs to his aid and suffers the same fate. Clearly, the theme of two individuals waging a foredoomed rebellion against an all-powerful oppressive state effectively precluded the option of so radically changing the ending as to let them live "happily ever after". The maximum realistic change was to create a "moral happy end", letting Winston and Julia keep their integrity and dignity and die as martyrs. | https://en.wikipedia.org/wiki/Happy_ending |
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