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Innumerical analysis, anumerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in aprogramming language is called a numerical algorithm.

Let${\displaystyle F(x,y)=0}$ be awell-posed problem, i.e.${\displaystyle F:X\times Y\to \mathbb{R}}$ is areal orcomplex functional relationship, defined on theCartesian product of an input data set${\displaystyle X}$ and an output data set${\displaystyle Y}$, such that exists alocally lipschitz function${\displaystyle g:X\to Y}$ calledresolvent, which has the property that for every root${\displaystyle (x,y)}$ of${\displaystyle F}$,${\displaystyle y=g(x)}$. We definenumerical method for the approximation of${\displaystyle F(x,y)=0}$, thesequence of problems

${\displaystyle {\left\{M_n\right\}}_{n\in \mathbb{N}}={\left\{F_n(x_n,y_n)=0\right\}}_{n\in \mathbb{N}},}$

with${\displaystyle F_n:X_n\times Y_n\to \mathbb{R}}$,${\displaystyle x_n\in X_n}$ and${\displaystyle y_n\in Y_n}$ for every${\displaystyle n\in \mathbb{N}}$. The problems of which the method consists need not be well-posed. If they are, the method is said to bestable orwell-posed.^[1]

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

${\displaystyle lim{F}_n(x,y+t)=F(x,y,t)=0,\mathrm{\forall }(x,y,t)\in S.}$

When${\displaystyle F_n=F,\mathrm{\forall }n\in \mathbb{N}}$ on${\displaystyle S}$ the method is said to bestrictly consistent.^[1]

Denote by${\displaystyle {\ell }_n}$ a sequence ofadmissible perturbations of${\displaystyle x\in X}$ for some numerical method${\displaystyle M}$ (i.e.${\displaystyle x+{\ell }_n\in X_n\mathrm{\forall }n\in \mathbb{N}}$) and with${\displaystyle y_n(x+{\ell }_n)\in Y_n}$ the value such that${\displaystyle F_n(x+{\ell }_n,y_n(x+{\ell }_n))=0}$. A condition which the method has to satisfy to be a meaningful tool for solving the problem${\displaystyle F(x,y)=0}$ isconvergence:

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

• Numerical methods for ordinary differential equations
• Numerical methods for partial differential equations

• Numerical analysis

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