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 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. 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  (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 is function. See also
 Numerical methods for ordinary differential equations
 Numerical methods for partial differential equations

References

Numerical analysis