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 be a well-posed problem, i.e. is a real or complex functional relationship, defined on the Cartesian 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.

Consistency

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.

Convergence

Denote by a sequence of admissible perturbations of for some numerical method 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.