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. ${\displaystyle F:X\times Y\rightarrow \mathbb {R} }$ 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 ${\displaystyle g:X\rightarrow Y}$ 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

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

with ${\displaystyle F_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R} }$, ${\displaystyle x_{n}\in X_{n}}$ and ${\displaystyle y_{n}\in Y_{n}}$ 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.^[1]

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

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

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

Convergence

Denote by $\ell_n$ a sequence of admissable perturbations of $x\in X$ for some numerical method $M$ (i.e. ${\displaystyle x+\ell _{n}\in X_{n}\forall n\in \mathbb {N} }$) 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 $F(x,y)=0$ is convergence:

${\displaystyle {\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\leq \varepsilon .\end{aligned}}}$

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

References

1. Quarteroni, Sacco, Saleri (2000). Numerical Mathematics (PDF). Milano: Springer. p. 33.