# Numerical method

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

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 an input data set $X$  and an output data 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

$\left\{M_{n}\right\}_{n\in \mathbb {N} }=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} },$

with $F_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R}$ , $x_{n}\in X_{n}$  and $y_{n}\in Y_{n}$  for every $n\in \mathbb {N}$ . 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.

## Consistency

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 is called consistent if and only if the sequence of functions $\left\{F_{n}\right\}_{n\in \mathbb {N} }$  pointwise converges to $F$  on the set $S$  of its solutions:

$\lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.$

When $F_{n}=F,\forall n\in \mathbb {N}$  on $S$  the method is said to be strictly consistent.

## Convergence

Denote by $\ell _{n}$  a sequence of admissible perturbations of $x\in X$  for some numerical method $M$  (i.e. $x+\ell _{n}\in X_{n}\forall n\in \mathbb {N}$ ) and with $y_{n}(x+\ell _{n})\in Y_{n}$  the value such that $F_{n}(x+\ell _{n},y_{n}(x+\ell _{n}))=0$ . A condition which the method has to satisfy to be a meaningful tool for solving the problem $F(x,y)=0$  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 \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\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 is function.