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wiki_12400_chunk_0 | Multigrid methods | In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation. They have also been widely used for more-complicated non-symmetric and nonlinear systems of e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12401_chunk_0 | Numerical method | In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12402_chunk_0 | Gaussian integration | In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl F... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12403_chunk_0 | Gaussian integration | This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on . The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint sin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12404_chunk_0 | Superconvergence | In numerical analysis, a superconvergent or supraconvergent method is one which converges faster than generally expected (superconvergence or supraconvergence). For example, in the Finite Element Method approximation to Poisson's equation in two dimensions, using piecewise linear elements, the average error in the grad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12405_chunk_0 | Mesh refinement | In numerical analysis, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated. When solutions are calculated numerically, they are often limited to pre-determined quan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12406_chunk_0 | Mesh refinement | Adaptive mesh refinement provides such a dynamic programming environment for adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas of multi-dimensional graphs which need precision while leaving the other regions of the multi-dimensional graphs at lower ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12407_chunk_0 | Incomplete Cholesky factorization | In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method. The Cholesky factorization of a positive ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12408_chunk_0 | Incomplete Cholesky factorization | An incomplete Cholesky factorization is given by a sparse lower triangular matrix K that is in some sense close to L. The corresponding preconditioner is KK*. One popular way to find such a matrix K is to use the algorithm for finding the exact Cholesky decomposition in which K has the same sparsity pattern as A (any e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12409_chunk_0 | Tetrahedral angle | In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have wide a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12410_chunk_0 | Discretization error | In numerical analysis, computational physics, and simulation, discretization error is the error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a lattice. Discretization error can usually be reduced by using a more finely... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12411_chunk_0 | Matrix decomposition | In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12412_chunk_0 | Finite Difference Method | In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the sol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12413_chunk_0 | Fixed point iteration | In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f {\displaystyle f} defined on the real numbers with real values and given a point x 0 {\displaystyle x_{0}} in the domain of f {\displaystyle f} , the fixed-point iteration is which giv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12414_chunk_0 | Unstable fixed point | In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. Specifically, given a function f {\displaystyle f} with the same domain and codomain, a point x 0 {\displaystyle x_{0}} in the domain of f {\displaystyle f} , the fixed-point iteration is which gives rise to the sequence x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12415_chunk_0 | Five-point stencil | In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12416_chunk_0 | Nine-point stencil | In numerical analysis, given a square grid in two dimensions, the nine-point stencil of a point in the grid is a stencil made up of the point itself together with its eight "neighbors". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiation. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12417_chunk_0 | Random-restart hill climbing | In numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by making an incremental change to the solution. If the change produces a b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12418_chunk_0 | Random-restart hill climbing | It is easy to find an initial solution that visits all the cities but will likely be very poor compared to the optimal solution. The algorithm starts with such a solution and makes small improvements to it, such as switching the order in which two cities are visited. Eventually, a much shorter route is likely to be obt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12419_chunk_0 | Random-restart hill climbing | Hill climbing finds optimal solutions for convex problems – for other problems it will find only local optima (solutions that cannot be improved upon by any neighboring configurations), which are not necessarily the best possible solution (the global optimum) out of all possible solutions (the search space). Examples o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12420_chunk_0 | Random-restart hill climbing | The relative simplicity of the algorithm makes it a popular first choice amongst optimizing algorithms. It is used widely in artificial intelligence, for reaching a goal state from a starting node. Different choices for next nodes and starting nodes are used in related algorithms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12421_chunk_0 | Random-restart hill climbing | Although more advanced algorithms such as simulated annealing or tabu search may give better results, in some situations hill climbing works just as well. Hill climbing can often produce a better result than other algorithms when the amount of time available to perform a search is limited, such as with real-time system... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12422_chunk_0 | Interpolative decomposition | In numerical analysis, interpolative decomposition (ID) factors a matrix as the product of two matrices, one of which contains selected columns from the original matrix, and the other of which has a subset of columns consisting of the identity matrix and all its values are no greater than 2 in absolute value. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12423_chunk_0 | Inverse iteration | In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method. It appears to have origin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12424_chunk_0 | Inverse iteration | The inverse power iteration algorithm starts with an approximation μ {\displaystyle \mu } for the eigenvalue corresponding to the desired eigenvector and a vector b 0 {\displaystyle b_{0}} , either a randomly selected vector or an approximation to the eigenvector. The method is described by the iteration where C k {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12425_chunk_0 | Inverse iteration | Since eigenvectors are defined up to multiplication by constant, the choice of C k {\displaystyle C_{k}} can be arbitrary in theory; practical aspects of the choice of C k {\displaystyle C_{k}} are discussed below. At every iteration, the vector b k {\displaystyle b_{k}} is multiplied by the matrix ( A − μ I ) − 1 {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12426_chunk_0 | Inverse iteration | {\displaystyle (A-\mu I)^{-1}.} The closer the approximation μ {\displaystyle \mu } to the eigenvalue is chosen, the faster the algorithm converges; however, incorrect choice of μ {\displaystyle \mu } can lead to slow convergence or to the convergence to an eigenvector other than the one desired. In practice, the metho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12427_chunk_0 | Inverse quadratic interpolation | In numerical analysis, inverse quadratic interpolation is a root-finding algorithm, meaning that it is an algorithm for solving equations of the form f(x) = 0. The idea is to use quadratic interpolation to approximate the inverse of f. This algorithm is rarely used on its own, but it is important because it forms part ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12428_chunk_0 | Leapfrog method | In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form or equivalently of the form particularly in the case of a dynamical system of classical mechanics. The method is known by different names in different disciplines. In particular, it is similar to the v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12429_chunk_0 | Leapfrog method | Leapfrog integration is equivalent to updating positions x ( t ) {\displaystyle x(t)} and velocities v ( t ) = x ˙ ( t ) {\displaystyle v(t)={\dot {x}}(t)} at different interleaved time points, staggered in such a way that they "leapfrog" over each other. Leapfrog integration is a second-order method, in contrast to Eu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12430_chunk_0 | Bandwidth (matrix theory) | In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the banded property corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12431_chunk_0 | Bandwidth (matrix theory) | These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12432_chunk_0 | Mortar methods | In numerical analysis, mortar methods are discretization methods for partial differential equations, which use separate finite element discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciousl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12433_chunk_0 | Multi-time-step integration | In numerical analysis, multi-time-step integration, also referred to as multiple-step or asynchronous time integration, is a numerical time-integration method that uses different time-steps or time-integrators for different parts of the problem. There are different approaches to multi-time-step integration. They are ba... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12434_chunk_0 | Multi-time-step integration | However, this is not the case in a strong algorithm. In the past few years a number of research articles have addressed the development of strong multi-time-step algorithms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12435_chunk_0 | Multi-time-step integration | In either case, strong or weak, the numerical accuracy and stability needs to be carefully studied. Other approaches to multi-time-step integration in the context of operator splitting methods have also been developed; i.e., multi-rate GARK method and multi-step methods for molecular dynamics simulations. == References... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12436_chunk_0 | Multivariate interpolation | In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points ( x i , y i , z i , … ) {\displaystyle (x_{i}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12437_chunk_0 | Nested dissection | In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff.Nested dissection consists of the following steps: Form a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12438_chunk_0 | Nested dissection | Perform Cholesky decomposition (a variant of Gaussian elimination for symmetric matrices), ordering the elimination of the variables by the recursive structure of the partition: each of the two subgraphs formed by removing the separator is eliminated first, and then the separator vertices are eliminated.As a consequenc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12439_chunk_0 | Adaptive numerical differentiation | In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12440_chunk_0 | Matrix eigenvalue problem | In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12441_chunk_0 | Order of accuracy | In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider u {\displaystyle u} , the exact solution to a differential equation in an appropriate normed space ( V , | | | | ) {\displaystyle (V,||\ ||)} . Consider a n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12442_chunk_0 | Order of accuracy | Using the big O notation an n {\displaystyle n} th-order accurate numerical method is notated as | | u − u h | | = O ( h n ) {\displaystyle ||u-u_{h}||=O(h^{n})} This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12443_chunk_0 | Pairwise summation | In numerical analysis, pairwise summation, also called cascade summation, is a technique to sum a sequence of finite-precision floating-point numbers that substantially reduces the accumulated round-off error compared to naively accumulating the sum in sequence. Although there are other techniques such as Kahan summati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12444_chunk_0 | Pairwise summation | Its worst-case roundoff errors grow asymptotically as at most O(ε log n), where ε is the machine precision (assuming a fixed condition number, as discussed below). In comparison, the naive technique of accumulating the sum in sequence (adding each xi one at a time for i = 1, ..., n) has roundoff errors that grow at wor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12445_chunk_0 | Interpolating polynomial | In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset.Given a set of n + 1 data points ( x 0 , y 0 ) , … , ( x n , y n ) {\displaystyle (x_{0},y_{0}),\ldots ,(x_{n},y_{n})} , with no tw... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12446_chunk_0 | Predictor-corrector method | In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps: The initial, "prediction" step, starts from a function fitted to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12447_chunk_0 | Stochastic tunneling | In numerical analysis, stochastic tunneling (STUN) is an approach to global optimization based on the Monte Carlo method-sampling of the function to be objective minimized in which the function is nonlinearly transformed to allow for easier tunneling among regions containing function minima. Easier tunneling allows for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12448_chunk_0 | Cash–Karp method | More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12449_chunk_0 | Clenshaw algorithm | In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. The method was published by Charles William Clenshaw in 1955. It is a generalization of Horner's method for evaluating a linear combination of monomials. It gen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12450_chunk_0 | Crank-Nicolson method | In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12451_chunk_0 | Dormand–Prince method | In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations (ODE). The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12452_chunk_0 | Dormand–Prince method | The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the "First Same As Last" (FSAL) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince chose the coefficients of their method to minimize the error of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12453_chunk_0 | FTCS scheme | In numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations. It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation. Whe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12454_chunk_0 | ITP Method | In numerical analysis, the ITP method, short for Interpolate Truncate and Project, is the first root-finding algorithm that achieves the superlinear convergence of the secant method while retaining the optimal worst-case performance of the bisection method. It is also the first method with guaranteed average performanc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12455_chunk_0 | ITP Method | As a bracketing strategy, in each iteration the ITP queries the value of the function on one point and discards the part of the interval between two points where the function value shares the same sign. The queried point is calculated with three steps: it interpolates finding the regula falsi estimate, then it perturbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12456_chunk_0 | Kahan summation | In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-precision floating-point numbers, compared to the obvious approach. This is done by keeping a separate running compensation (a variabl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12457_chunk_0 | Lagrange form | Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration, Shamir's secret sharing s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12458_chunk_0 | Lax–Richtmyer theorem | In numerical analysis, the Lax equivalence theorem is a fundamental theorem in the analysis of finite difference methods for the numerical solution of partial differential equations. It states that for a consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12459_chunk_0 | Lax–Richtmyer theorem | Stability in this context means that a matrix norm of the matrix used in the iteration is at most unity, called (practical) Lax–Richtmyer stability. Often a von Neumann stability analysis is substituted for convenience, although von Neumann stability only implies Lax–Richtmyer stability in certain cases. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12460_chunk_0 | Lax–Richtmyer theorem | This theorem is due to Peter Lax. It is sometimes called the Lax–Richtmyer theorem, after Peter Lax and Robert D. Richtmyer. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12461_chunk_0 | Quadrature formula | In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration (also called quadrature) based on evaluating the integrand at equally spaced points. They are named after Isaac Newton and Roger Cotes. Newton–C... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12462_chunk_0 | Peano kernel theorem | In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12463_chunk_0 | Schur complement method | In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domain decomposition method, also called iterative substructuring. A finite element problem is split into non-overlapping subdomains, and the unknowns in the interiors of the subdomains ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12464_chunk_0 | Shanks transformation | In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12465_chunk_0 | Durand–Kerner method | In numerical analysis, the Weierstrass method or Durand–Kerner method, discovered by Karl Weierstrass in 1891 and rediscovered independently by Durand in 1960 and Kerner in 1966, is a root-finding algorithm for solving polynomial equations. In other words, the method can be used to solve numerically the equation f(x) =... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12466_chunk_0 | Balancing domain decomposition method | In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method. In each iteration, it combines the solution of local problems on non-overlapping subdomains wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12467_chunk_0 | Balancing domain decomposition method | The dual counterpart to BDD is FETI, which enforces the equality of the solution between the subdomain by Lagrange multipliers. The base versions of BDD and FETI are not mathematically equivalent, though a special version of FETI designed to be robust for hard problems has the same eigenvalues and thus essentially the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12468_chunk_0 | Ill-conditioned matrix | In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Ve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12469_chunk_0 | Ill-conditioned matrix | The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of the matrix. More generally, condition numbers can be defined for non-linear functions in several varia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12470_chunk_0 | Ill-conditioned matrix | A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. In non-mathematical terms, an ill-conditioned problem is one where, for a small change in the inputs (the independent variables) there is a large change in the answer or depe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12471_chunk_0 | Ill-conditioned matrix | The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called backward stability; in general, a backward stable algorithm can be expected to accurately solve well-c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12472_chunk_0 | Ill-conditioned matrix | Numerical analysis textbooks give formulas for the condition numbers of problems and identify known backward stable algorithms. As a rule of thumb, if the condition number κ ( A ) = 10 k {\displaystyle \kappa (A)=10^{k}} , then you may lose up to k {\displaystyle k} digits of accuracy on top of what would be lost to th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12473_chunk_0 | Interval FEM | In numerical analysis, the interval finite element method (interval FEM) is a finite element method that uses interval parameters. Interval FEM can be applied in situations where it is not possible to get reliable probabilistic characteristics of the structure. This is important in concrete structures, wood structures,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12474_chunk_0 | Interval FEM | The goal of the Interval Finite Element is to find upper and lower bounds of different characteristics of the model (e.g. stress, displacements, yield surface etc.) and use these results in the design process. This is so called worst case design, which is closely related to the limit state design. Worst case design req... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12475_chunk_0 | Local linearization method | In numerical analysis, the local linearization (LL) method is a general strategy for designing numerical integrators for differential equations based on a local (piecewise) linearization of the given equation on consecutive time intervals. The numerical integrators are then iteratively defined as the solution of the re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12476_chunk_0 | Minimum degree algorithm | In numerical analysis, the minimum degree algorithm is an algorithm used to permute the rows and columns of a symmetric sparse matrix before applying the Cholesky decomposition, to reduce the number of non-zeros in the Cholesky factor. This results in reduced storage requirements and means that the Cholesky factor can ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12477_chunk_0 | Minimum degree algorithm | Given a linear system A x = b {\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} } where A is an n × n {\displaystyle n\times n} real symmetric sparse square matrix. The Cholesky factor L will typically suffer 'fill in', that is have more non-zeros than the upper triangle of A. We seek a permutation matrix P, so that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12478_chunk_0 | Minimum degree algorithm | {\displaystyle \left(\mathbf {P} ^{T}\mathbf {A} \mathbf {P} \right)\left(\mathbf {P} ^{T}\mathbf {x} \right)=\mathbf {P} ^{T}\mathbf {b} .} The problem of finding the best ordering is an NP-complete problem and is thus intractable, so heuristic methods are used instead. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12479_chunk_0 | Minimum degree algorithm | The minimum degree algorithm is derived from a method first proposed by Markowitz in 1959 for non-symmetric linear programming problems, which is loosely described as follows. At each step in Gaussian elimination row and column permutations are performed so as to minimize the number of off diagonal non-zeros in the piv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12480_chunk_0 | Minimum degree algorithm | The graph referred to is the graph with n vertices, with vertices i and j connected by an edge when a i j ≠ 0 {\displaystyle a_{ij}\neq 0} , and the degree is the degree of the vertices. A crucial aspect of such algorithms is a tie breaking strategy when there is a choice of renumbering resulting in the same degree. A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12481_chunk_0 | Mixed finite element method | In numerical analysis, the mixed finite element method, is a type of finite element method in which extra fields to be solved are introduced during the posing a partial differential equation problem. Somewhat related is the hybrid finite element method. The extra fields are constrained by using Lagrange multiplier fiel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12482_chunk_0 | Mixed finite element method | The mixed finite element method is efficient for some problems that would be numerically ill-posed if discretized by using the irreducible finite element method; one example of such problems is to compute the stress and strain fields in an almost incompressible elastic body. In mixed methods, the Lagrange multiplier fi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12483_chunk_0 | Mixed finite element method | In sparse direct solvers, pivoting may be needed, where ultimately the resulting matrix has 2x2 blocks on the diagonal, rather than a working towards a completely pure LLH Cholesky decomposition for positive definite symmetric or Hermitian systems. Pivoting may result in unpredictable memory usage increases. For iterat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12484_chunk_0 | Mixed finite element method | In hybrid methods, the Lagrange fields are for jumps of fields between elements, living on the boundary of the elements, weakly enforcing continuity; continuity from fields in the elements does not need to be enforced through shared degrees of freedom between elements anymore. Both mixing and hybridization can be appli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12485_chunk_0 | Linear convergence | In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence ( x n ) {\displaystyle (x_{n})} that converges to x ∗ {\displaystyle x^{*}} is said to have order of convergence q ≥ 1 {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12486_chunk_0 | Linear convergence | The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology, in this case, is different from the terminology for iterative methods. Series accelerat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12487_chunk_0 | Quasi-Monte Carlo method | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12488_chunk_0 | 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 evalu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12489_chunk_0 | Secant 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12490_chunk_0 | Singular boundary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12491_chunk_0 | 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 other... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12492_chunk_0 | Split-step 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12493_chunk_0 | 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 mechanis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12494_chunk_0 | Transfinite interpolation | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12495_chunk_0 | Von Neumann stability analysis | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12496_chunk_0 | Pseudo code | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12497_chunk_0 | Gauss' law for magnetism | 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'... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12498_chunk_0 | Minkowski difference | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12499_chunk_0 | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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