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.
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
We define numerical method for the approximation of
, the sequence of problems with
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.
[1] Necessary conditions for a numerical method to effectively approximate
behaves like
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
the method is said to be strictly consistent.
[1] Denote by
ℓ
a sequence of admissible perturbations of
for some numerical method
ℓ
ℓ
ℓ
ℓ
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
implies the convergence of the associated method.