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. $F:X \times Y \rightarrow \mathbb$ 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

: $\left \_ = \left \_,$

with $F_n:X_n \times Y_n \rightarrow \mathbb$, $x_n \in X_n$ and $y_n \in Y_n$ for every $n \in \mathbb$. 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 $\left \_$ 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 $F_n=F, \forall n \in \mathbb$ 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. $x+\ell_n \in X_n \forall n \in \mathbb$) 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'':

: $\begin &\forall \varepsilon > 0, \exist n_0(\varepsilon) > 0, \exist \delta_ \text \\ &\forall n > n_0, \forall \ell_n : \, \ell_n \, < \delta_ \Rightarrow \, y_n(x+\ell_n) - y \, \leq \varepsilon. \end$

One can easily prove that the point-wise convergence of $\ _$ to $y$ implies the convergence of the associated method is function.

References

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Numerical analysis