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Stencil (numerical analysis) — the geometric arrangements of grid points affected by a basic step of the algorithm Compact stencil — stencil which only uses a few grid points, usually only the immediate and diagonal neighbours Higher-order compact finite difference scheme; Non-compact stencil — any stencil that is not compact
An exception occurs in numerical smoothing and differentiation where an analytical expression is required. If the matrix X T X is well-conditioned and positive definite, implying that it has full rank, the normal equations can be solved directly by using the Cholesky decomposition R T R, where R is an upper triangular matrix, giving:
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:
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones.
An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
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.
Numerical methods for ordinary differential equations approximate solutions to initial value problems of the form ′ = (,), =.. The result is approximations for the value of () at discrete times : = +, where is the time step (sometimes referred to as ) and is an integer.
P. Padé approximant; Padé table; Pairwise summation; Parareal; Partial differential algebraic equation; Particle method; Peano kernel theorem; Piecewise linear continuation