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