# Numerical method

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{{Short description|Mathematical tool to algorithmically solve equations}}
{{more footnotes needed|date=September 2016}}In [numerical analysis](/source/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](/source/programming_language) is called a numerical algorithm.

==Mathematical definition==
Let <math>F(x,y)=0</math> be a [well-posed problem](/source/well-posed_problem_(numerical_analysis)), i.e. <math>F:X \times Y \rightarrow \mathbb{R}</math> is a [real](/source/Real_number) or [complex](/source/Complex_Numbers) functional relationship, defined on the [Cartesian product](/source/Cartesian_product) of an input data set <math>X</math> and an output data set <math>Y</math>, such that exists a [locally lipschitz](/source/Lipschitz_continuity) function <math>g:X \rightarrow Y</math> called [resolvent](/source/Resolvent_(direct_problem)), 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](/source/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" />

==See also==
* [Numerical methods for ordinary differential equations](/source/Numerical_methods_for_ordinary_differential_equations)
* [Numerical methods for partial differential equations](/source/Numerical_methods_for_partial_differential_equations)

==References==
{{Reflist}}

{{Authority control}}

Category:Numerical analysis

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Adapted from the Wikipedia article [Numerical method](https://en.wikipedia.org/wiki/Numerical_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Numerical_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.