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.
Let be a well-posed problem, i.e. is a real or complex functional relationship, defined on the cross-product of an input data set and an output data set , such that exists a locally lipschitz function called resolvent, which has the property that for every root of , . We define numerical method for the approximation of , the sequence of problems
with , and 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.
Necessary conditions for a numerical method to effectively approximate are that and that behaves like when . So, a numerical method is called consistent if and only if the sequence of functions pointwise converges to on the set of its solutions:
When on the method is said to be strictly consistent.
Denote by a sequence of admissible perturbations of for some numerical method (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 is convergence:
One can easily prove that the point-wise convergence of to implies the convergence of the associated method.