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 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\}_{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.^[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 $\left\{F_{n}\right\}_{n\in \mathbb {N} }$ 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.^[1]

Convergence

Denote by $\ell_n$ a sequence of admissable 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.^[1]

References

1 2 3 Quarteroni, Sacco, Saleri (2000). Numerical Mathematics (PDF). Milano: Springer. p. 33.

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