Numerical method

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 $F(x,y)=0$ be a well-posed problem, i.e. $F:X\times Y\rightarrow \mathbb {R}$ is a real or complex functional relationship, defined on the cross-product of some input data set $X$ and an output set $Y$, such that exists a locally lipschitz function $g:X\rightarrow Y$ called resolvent, which has the property that for every root $(x,y)$ of $F$, $y=g(x)$. We define numerical method for the approximation of $F(x,y)=0$, the sequence of problems
$M=\left\{M_{n}\right\}=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} }.$
We don't require the problems of which the method consists to be well-posed. Otherwise, the method is said to be stable or well-posed.

Consistence

Necessary conditions for a numerical method to effectively approximate $F(x,y)=0$ are that $x_{n}\rightarrow x$ and that $F_{n}$ behaves like $F$ when $n\rightarrow \infty$. So, a numerical method method is called consistent if and only if the sequence of functions $\left\{F_{n}\right\}_{n\in \mathbb {N} }$ pointlwise converges to $F$ on the set $S$ of its solutions:
$\lim F_{n}(x,y)=F(x,y)=0,\quad \quad \forall (x,y)\in S.$
When $F_{n}=F,\forall n\in \mathbb {N}$ on $S$ the method is said to be strictly consistent.

Convergence

Denote with $l_{n}$ a sequence of admissable perturbations of $x\in X$ for some numerical method $M$ (i.e. $x+l_{n}\in X_{n}\forall n\in \mathbb {N}$) and with $y_{n}(x+l_{n})\in Y_{n}$ the value such that $F_{n}(x+l_{n},y_{n}(x+l_{n}))=0$. A condition which the method has to satisfy to be a meaningful tool for solving the problem $F(x,y)=0$ is convergence:
${\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall l_{n}:\lVert l_{n}\rVert <\delta _{\varepsilon ,n_{0}}\Rightarrow \lVert y_{n}(x+l_{n})-y\rVert \leq \varepsilon .\end{aligned}}$
One can easily prove that the point-wise convergence of ${y_{n}}_{n\in \mathbb {N} }$ to $y$ implies the convergence of the associated method.

References

Numerical Mathematics - Quarternioni, Sacco, Saleri