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
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:
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:
- Quarteroni, Sacco, Saleri (2000). Numerical Mathematics (PDF). Milano: Springer. p. 33.CS1 maint: multiple names: authors list (link)