Numerical method

Template:Short description Template:More footnotes neededIn 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. $F:X\times Y\to ℝ$ 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 $g:X\to Y$ 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

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

with $F_n:X_n\times Y_n\to ℝ$, $x_n\in X_n$ 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 $x_n\to x$ and that $F_n$ behaves like $F$ when $n\to \mathrm{\infty }$. 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:

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

When $F_n=F,\mathrm{\forall }n\in \mathbb{N}$ 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. $x+{\ell }_n\in X_n\mathrm{\forall }n\in \mathbb{N}$) 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:

$\begin{array}{cc}& \mathrm{\forall }\epsilon >0,\mathrm{\exists }{n}_0(\epsilon )>0,\mathrm{\exists }{\delta }_{\epsilon ,n_0}\text{ such that}\\ & \mathrm{\forall }n>n_0,\mathrm{\forall }{\ell }_n:\Vert {\ell }_n\Vert <{\delta }_{\epsilon ,n_0}\Rightarrow \Vert y_n(x+{\ell }_n)-y\Vert \le \epsilon .\end{array}$

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.

See also

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