id stringlengths 14 14 | text stringlengths 9 3.55k | source stringlengths 1 250 |
|---|---|---|
c_ytgvdzhjwbd0 | 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 un... | Quasi-Monte Carlo method |
c_5s58zx3am2rr | 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 ... | Quasi-Monte Carlo method |
c_pg1mu88qe4f9 | 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 evalu... | Quasi-Monte Carlo method |
c_jijaahybykj2 | 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... | Secant Method |
c_bm0dy4uie501 | 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 val... | Shooting method |
c_wweeufryrz0m | 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... | Singular boundary method |
c_8o72q736kern | 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 other... | Singular boundary method |
c_evq2uo8aqpxz | 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... | Split-step method |
c_0nbcs9d797b1 | 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 mechanis... | Split-step method |
c_x2tiyx3cpdze | 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... | Geometrical theory of diffraction |
c_xjzrm6962lch | 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 Willia... | Transfinite interpolation |
c_cvx8woq12bnt | 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 Ala... | Von Neumann stability analysis |
c_euwzm8x8at0t | 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 r... | Determination of the day of the week |
c_gpjschta32w3 | 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... | Determination of the day of the week |
c_xjln06f02nsa | 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 ... | Determination of the day of the week |
c_a0xknb31wtm8 | 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 ran... | Pseudo code |
c_cg70tfbyujp8 | 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'... | Gauss' law for magnetism |
c_95zv62piftx9 | 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. | Minkowski difference |
c_3cgv6wx3yyv5 | 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. | Numeric control |
c_i2kfkccw5u0g | 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... | Numeric control |
c_t8pjah953300 | 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... | Simpson's Rule |
c_q0s7lbqzlhb9 | 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. | Simpson's Rule |
c_e13sdrmmm844 | 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... | Simpson's Rule |
c_kd7iquky5uby | 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 b... | MISER algorithm |
c_dvr1enavo1vr | 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 ... | MISER algorithm |
c_jf4xi33y8zxv | 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} . Thi... | MISER algorithm |
c_j6v028vt9a3c | 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... | MISER algorithm |
c_uhxws85equxi | 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. | Givens rotation |
c_yq5w3ncygjtj | 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 ′ . {\displaysty... | Jacobi rotation |
c_1fm1xehirge2 | {\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 &\d... | Jacobi rotation |
c_x5mo5f84iy8d | 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 &\vd... | Jacobi rotation |
c_88bq0wpl2a11 | 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 K... | Jacobi rotation |
c_zewmuer9zd31 | {\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 ... | Jacobi rotation |
c_pk3klz7lt90t | 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 {\displa... | Jacobi rotation |
c_7lo1gr8u0p9s | 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. | Incomplete LU factorization |
c_f539k55b0wov | 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. | Bartels–Stewart algorithm |
c_6ql1q15bvnb4 | 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... | Bartels–Stewart algorithm |
c_16vpd8clsd2k | 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 th... | Cuthill–McKee algorithm |
c_z8usyn3qtbgd | 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}... | Cuthill–McKee algorithm |
c_r5hf6k8ptkhm | 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 Jacob... | Gauss–Seidel method |
c_thbrdmn6lrrc | 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 t... | Jacobi eigenvalue algorithm |
c_4r0l5pdama78 | 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. | Jacobi method |
c_ues0fg0iqkvj | 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. | Jacobi method |
c_hzkixonyhw2h | 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 perfo... | QR algorithm |
c_nnrccd51iw8y | 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 | Rayleigh–Ritz method |
c_4mhsf0garo5o | 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, factor... | Alternating direction implicit method |
c_p95wqj2jqglw | 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 tha... | Biconjugate gradient stabilized method |
c_altzttq1wm1w | 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 dif... | Derivation of the conjugate gradient method |
c_yt017im86efm | 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. | Successive over-relaxation |
c_1gupy9mboa4g | 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... | Successive over-relaxation |
c_7vmkoyz54nmu | 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... | Thomas algorithm |
c_i6gvkuh49g7k | 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 mat... | Thomas algorithm |
c_h5tzfabmnaf5 | 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. | Beam and Warming scheme |
c_dt8204l8lz5w | 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. | Non-compact stencil |
c_6coermr6r35a | 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 1984-01-01T... | Artificial precision |
c_95rqsb7964ye | 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 wou... | Hierarchical matrix |
c_o3xtlmepebem | 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 suff... | Hierarchical matrix |
c_jmkago9z494i | 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 situat... | Interval propagation |
c_a4rgchsg8b21 | 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 l... | Relaxation (iterative method) |
c_l33zlrvjz1qb | 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 loc... | Relaxation (iterative method) |
c_3xaawp450zxs | 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. | Uzawa iteration |
c_hfwspbvfq24p | 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 e... | Boundary knot method |
c_tgyynwk1a3i0 | 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 ... | Constant strain triangle element |
c_r3f2l162zyso | 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 o... | Gradient discretization method |
c_0b3uo1mq19qh | 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 ... | Gradient discretization method |
c_6ddt13sgpiw6 | 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 collo... | Regularized meshless method |
c_7e8wz26a430r | 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 me... | Markov chain approximation method |
c_5gdc2j0u8cf9 | 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 nee... | Markov chain approximation method |
c_zwqk62v25pky | 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. | Total variation diminishing |
c_pzzc8ukz63ai | 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... | Numerical modeling (geology) |
c_v6xwbcthnkle | 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 concept... | Meta-optimization |
c_p2mgj89i002n | 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... | BFGS method |
c_arq2qmqxm1rf | 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 obt... | Nonlinear conjugate gradient |
c_wk6xv5eik1kl | 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... | Nonlinear conjugate gradient |
c_wve9mx9vtvdm | 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 (... | Nonlinear conjugate gradient |
c_hquo97uurswj | {\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}(\Del... | Nonlinear conjugate gradient |
c_vly63g7nl06j | 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})}}.} | Nonlinear conjugate gradient |
c_m0husbcbcgmb | .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 Newt... | Nonlinear conjugate gradient |
c_b84epvqaacb8 | 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 co... | Nonlinear conjugate gradient |
c_hqfs6pz86bj5 | 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. | American Braille |
c_y6t71uw6yces | 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. | French Braille |
c_19tem2zlbe8y | 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 l... | Numerov's method |
c_i0x15pjdgcq3 | {\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 th... | Numerov's method |
c_ncvu5tn2jsxi | 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 pr... | Error analysis (mathematics) |
c_8sx5f7t1g1m1 | 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). | WENO methods |
c_686o2rusywgj | 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. | WENO methods |
c_s00v4wjo6x4h | 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. | Data assimilation |
c_bw3mj356oo7j | 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 le... | Conjugate Gradient method |
c_xpdonyx2jeib | 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 bot... | Conjugate Gradient method |
c_c490hwcruiru | 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 t... | 11:11 (numerology) |
c_zxkid9gs42yu | 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. | Altruism in animals |
c_g4i4e86pkglf | 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. Strange... | Moscow–Washington hotline |
c_enl20rrnui60 | 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 int... | Moscow–Washington hotline |
c_j2w7aiw99o3e | 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... | Happy ending |
c_c1xoelg6500z | 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. | Happy ending |
c_1tbn81e6vqp2 | 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 anot... | Happy ending |
c_8yv8lqwojtt7 | 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 in... | Happy ending |
c_daw51p6onjl0 | 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... | Happy ending |
c_m5snppx16y02 | 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... | Happy ending |
c_iptfgci0qtt2 | 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... | Happy ending |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.