Numerical method

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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.

Mathematical definition[edit]

Let be a well-posed problem, i.e. is a real or complex functional relationship, defined on the cross-product of an input data set and an output data set , such that exists a locally lipschitz function called resolvent, which has the property that for every root of , . We define numerical method for the approximation of , the sequence of problems

with , and for every . The problems of which the method consists need not be well-posed. If they are, the method is said to be stable or well-posed.[1]

Consistency[edit]

Necessary conditions for a numerical method to effectively approximate are that and that behaves like when . So, a numerical method is called consistent if and only if the sequence of functions pointwise converges to on the set of its solutions:

When on the method is said to be strictly consistent.[1]

Convergence[edit]

Denote by a sequence of admissible perturbations of for some numerical method (i.e. ) and with the value such that . A condition which the method has to satisfy to be a meaningful tool for solving the problem is convergence:

One can easily prove that the point-wise convergence of to implies the convergence of the associated method.[1]

References[edit]

  1. ^ a b c Quarteroni, Sacco, Saleri (2000). Numerical Mathematics (PDF). Milano: Springer. p. 33.CS1 maint: multiple names: authors list (link)