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 <math>F(x,y)=0</math> be a well-posed problem, i.e. <math>F:X \times Y \rightarrow \mathbb{R}</math> is a real or complex functional relationship, defined on the Cartesian product of an input data set <math>X</math> and an output data set <math>Y</math>, such that exists a locally lipschitz function <math>g:X \rightarrow Y</math> called resolvent, which has the property that for every root <math>(x,y)</math> of <math>F</math>, <math>y=g(x)</math>. We define numerical method for the approximation of <math>F(x,y)=0</math>, the sequence of problems
: <math>\left \{ M_n \right \}_{n \in \mathbb{N = \left \{ F_n(x_n,y_n)=0 \right \}_{n \in \mathbb{N,</math>
with <math>F_n:X_n \times Y_n \rightarrow \mathbb{R}</math>, <math>x_n \in X_n</math> and <math>y_n \in Y_n</math> for every <math>n \in \mathbb{N}</math>. 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 <math>F(x,y)=0</math> are that <math>x_n \rightarrow x</math> and that <math>F_n</math> behaves like <math>F</math> when <math>n \rightarrow \infty</math>. So, a numerical method is called consistent if and only if the sequence of functions <math>\left \{ F_n \right \}_{n \in \mathbb{N</math> pointwise converges to <math>F</math> on the set <math>S</math> of its solutions:
: <math>
\lim F_n(x,y+t) = F(x,y,t) = 0, \quad \quad \forall (x,y,t) \in S.
</math>
When <math>F_n=F, \forall n \in \mathbb{N}</math> on <math>S</math> the method is said to be strictly consistent.