Numerical method

{{Short description|Mathematical tool to algorithmically solve equations}} {{more footnotes needed|date=September 2016}}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== Let <math>F(x,y)=0</math> be a well-posed problem, i.e. <math>F:X \times Y \rightarrow \mathbb{R}</math> is a real or complex functional relationship, defined on the Cartesian product of an input data set <math>X</math> and an output data set <math>Y</math>, such that exists a locally lipschitz function <math>g:X \rightarrow Y</math> called resolvent, which has the property that for every root <math>(x,y)</math> of <math>F</math>, <math>y=g(x)</math>. We define '''numerical method''' for the approximation of <math>F(x,y)=0</math>, the sequence of problems

: <math>\left \{ M_n \right \}_{n \in \mathbb{N}} = \left \{ F_n(x_n,y_n)=0 \right \}_{n \in \mathbb{N}},</math>

with <math>F_n:X_n \times Y_n \rightarrow \mathbb{R}</math>, <math>x_n \in X_n</math> and <math>y_n \in Y_n</math> for every <math>n \in \mathbb{N}</math>. 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''.<ref name="quartsaccsal">{{cite book | last = Quarteroni, Sacco, Saleri | title = Numerical Mathematics | publisher = Springer | location = Milano | year = 2000 | page = 33 | url = http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf | access-date = 2016-09-27 | archive-url = https://web.archive.org/web/20171114040621/http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf | archive-date = 2017-11-14 | url-status = dead }}</ref>

==Consistency== Necessary conditions for a numerical method to effectively approximate <math>F(x,y)=0</math> are that <math>x_n \rightarrow x</math> and that <math>F_n</math> behaves like <math>F</math> when <math>n \rightarrow \infty</math>. So, a numerical method is called ''consistent'' if and only if the sequence of functions <math>\left \{ F_n \right \}_{n \in \mathbb{N}}</math> pointwise converges to <math>F</math> on the set <math>S</math> of its solutions:

: <math> \lim F_n(x,y+t) = F(x,y,t) = 0, \quad \quad \forall (x,y,t) \in S. </math>

When <math>F_n=F, \forall n \in \mathbb{N}</math> on <math>S</math> the method is said to be ''strictly consistent''.<ref name="quartsaccsal" />

==Convergence== Denote by <math>\ell_n</math> a sequence of ''admissible perturbations'' of <math>x \in X</math> for some numerical method <math>M</math> (i.e. <math>x+\ell_n \in X_n \forall n \in \mathbb{N}</math>) and with <math>y_n(x+\ell_n) \in Y_n</math> the value such that <math>F_n(x+\ell_n,y_n(x+\ell_n)) = 0</math>. A condition which the method has to satisfy to be a meaningful tool for solving the problem <math>F(x,y)=0</math> is ''convergence'':

: <math> \begin{align} &\forall \varepsilon > 0, \exist n_0(\varepsilon) > 0, \exist \delta_{\varepsilon, n_0} \text{ such that} \\ &\forall n > n_0, \forall \ell_n : \| \ell_n \| < \delta_{\varepsilon,n_0} \Rightarrow \| y_n(x+\ell_n) - y \| \leq \varepsilon. \end{align} </math>

One can easily prove that the point-wise convergence of <math> \{y_n\} _{n \in \mathbb{N}}</math> to <math>y</math> implies the convergence of the associated method.<ref name="quartsaccsal" />

==References== {{Reflist}}

Category:Numerical analysis