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

Given a well-posed problem $F(x,y)=0$ (i.e. a functional relationship $F$ such that exists a locally lipschitz resolvent $g$ such that $F(x,y)=0\Rightarrow y=g(x)$), a numerical method is a sequence of direct problems
$M_{n}=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} }.$

For the method to approximate the well-posed problem it's necessary that $x_{n}\rightarrow x$ and that $F_{n}$ behaves like $F$ when $n\rightarrow \infty$. A numerical method is stable when every element of the sequence (every problem $F_{n}$) is well-posed. The method is said to be consistent if and only if the approximated problems $F_{n}$ progressively behave like $F$ on the exact solutions, or in other words that
${\text{if }}(x,y):F(x,y)=0\Rightarrow F_{n}(x,y)\rightarrow 0{\text{ when }}n\rightarrow \infty .$
The method is instead strictly consistent when $F_{n}(x,y)=0,\forall n\in \mathbb {N}$.
A sufficient condition for the method to define meaningful approximations of the solution is that the approximations $y_{n}(x+l_{n})$ of the resolvent by the numerical method on n-th step, evaluated on some perturbed data $x+l_{n}$ are arbitrary close to the exact solution evaluated on the exact data. In formulae:
${\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall l_{n}:\lVert l_{n}\rVert <\delta _{\varepsilon ,n_{0}}\Rightarrow \lVert y-y_{n}(x+l_{n})\rVert \leq \varepsilon .\end{aligned}}$

References

Numerical Mathematics - Quarternioni, Sacco, Saleri