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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 definitionEdit

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]

ConsistencyEdit

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]

ConvergenceEdit

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]

ReferencesEdit

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