numerical analysis

 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 ${\displaystyle �(�,�)=0}$ be a well-posed problem, i.e. ${\displaystyle �:�\times �\to \mathbb{�}}$ is a real or complex functional relationship, defined on the cross-product of an input data set ${\displaystyle �}$ and an output data set ${\displaystyle �}$, such that exists a locally lipschitz function ${\displaystyle �:�\to �}$ called resolvent, which has the property that for every root ${\displaystyle (�,�)}$ of ${\displaystyle �}$, ${\displaystyle �=�(�)}$. We define numerical method for the approximation of ${\displaystyle �(�,�)=0}$, the sequence of problems

${\displaystyle {\left\{{�}_{�}\right\}}_{�\in \mathbb{�}}={\left\{{�}_{�}({�}_{�},{�}_{�})=0\right\}}_{�\in \mathbb{�}},}$

with ${\displaystyle {�}_{�}:{�}_{�}\times {�}_{�}\to \mathbb{�}}$, ${\displaystyle {�}_{�}\in {�}_{�}}$ and ${\displaystyle {�}_{�}\in {�}_{�}}$ for every ${\displaystyle �\in \mathbb{�}}$. 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 ${\displaystyle �(�,�)=0}$ are that ${\displaystyle {�}_{�}\to �}$ and that ${\displaystyle {�}_{�}}$ behaves like ${\displaystyle �}$ when ${\displaystyle �\to \mathrm{\infty }}$. So, a numerical method is called consistent if and only if the sequence of functions ${\displaystyle {\left\{{�}_{�}\right\}}_{�\in \mathbb{�}}}$ pointwise converges to ${\displaystyle �}$ on the set ${\displaystyle �}$ of its solutions:

${\displaystyle lim{�}_{�}(�,�+�)=�(�,�,�)=0,\mathrm{\forall }(�,�,�)\in �.}$

When ${\displaystyle {�}_{�}=�,\mathrm{\forall }�\in \mathbb{�}}$ on ${\displaystyle �}$ the method is said to be strictly consistent.^[1]

Convergence[edit]

Denote by ${\displaystyle {\ell }_{�}}$ a sequence of admissible perturbations of ${\displaystyle �\in �}$ for some numerical method ${\displaystyle �}$ (i.e. ${\displaystyle �+{\ell }_{�}\in {�}_{�}\mathrm{\forall }�\in \mathbb{�}}$) and with ${\displaystyle {�}_{�}(�+{\ell }_{�})\in {�}_{�}}$ the value such that ${\displaystyle {�}_{�}(�+{\ell }_{�},{�}_{�}(�+{\ell }_{�}))=0}$. A condition which the method has to satisfy to be a meaningful tool for solving the problem ${\displaystyle �(�,�)=0}$ is convergence:

One can easily prove that the point-wise convergence of ${\displaystyle \{{�}_{�}{\}}_{�\in \mathbb{�}}}$ to ${\displaystyle �}$ implies the convergence of the associated method is function.^[1]