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{align} &\forall \varepsilon > 0, \exist n_0(\varepsilon) > 0, \exist \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{align} $$

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.