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 be a well-posed problem, i.e. \( {\displaystyle F:X\times Y\rightarrow \mathbb {R} }\) is a real or complex functional relationship, defined on the cross-product of an input data set X and an output data set Y, such that exists a locally lipschitz function \( {\displaystyle g:X\rightarrow Y} \) called resolvent, which has the property that for every root (x,y) of F, y=g(x). We define numerical method for the approximation of F(x,y)=0, the sequence of problems
\( {\displaystyle \left\{M_{n}\right\}_{n\in \mathbb {N} }=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} },} \)
with F \( {\displaystyle F_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R} } \), \( {\displaystyle x_{n}\in X_{n}} \) and \( {\displaystyle y_{n}\in Y_{n}} \) for every \( n\in \mathbb {N} \) . 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.[1]
Consistency
Necessary conditions for a numerical method to effectively approximate F(x,y)=0 are that \( x_{n}\rightarrow x \) and that \( F_{n} \) behaves like F when \( n\rightarrow \infty \) . So, a numerical method is called consistent if and only if the sequence of functions \( {\displaystyle \left\{F_{n}\right\}_{n\in \mathbb {N} }} \) pointwise converges to F on the set S of its solutions:
\( {\displaystyle \lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.} \)
When \( {\displaystyle F_{n}=F,\forall n\in \mathbb {N} } \) on S the method is said to be strictly consistent.[1]
Convergence
Denote by \( \ell_n \) a sequence of admissible perturbations of \( x\in X \) for some numerical method M (i.e. \( {\displaystyle x+\ell _{n}\in X_{n}\forall n\in \mathbb {N} } \)) and with \( {\displaystyle y_{n}(x+\ell _{n})\in Y_{n}} \) the value such that \( {\displaystyle F_{n}(x+\ell _{n},y_{n}(x+\ell _{n}))=0} \). A condition which the method has to satisfy to be a meaningful tool for solving the problem F(x,y)=0 is convergence:
\( {\displaystyle {\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\leq \varepsilon .\end{aligned}}} \)
One can easily prove that the point-wise convergence of \( {\displaystyle \{y_{n}\}_{n\in \mathbb {N} }} \) implies the convergence of the associated method is function.[1]
References
Quarteroni, Sacco, Saleri (2000). Numerical Mathematics (PDF). Milano: Springer. p. 33. Archived from the original (PDF) on 2017-11-14. Retrieved 2016-09-27.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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