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. 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 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 with, x_n \in X_n and y_n \in Y_n for every. 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 and that F_n behaves like F when. So, a numerical method is called consistent if and only if the sequence of functions pointwise converges to F on the set S of its solutions: When 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. ) and with the value such that. A condition which the method has to satisfy to be a meaningful tool for solving the problem F(x,y)=0 is convergence: One can easily prove that the point-wise convergence of to y implies the convergence of the associated method is function.