Numerical method

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title: "Numerical method" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["numerical-analysis"] description: "Mathematical tool to algorithmically solve equations" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Numerical_method" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical tool to algorithmically solve equations ::

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. 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 and an output data set Y, such that exists a locally lipschitz function 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

: \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 \times Y_n \rightarrow \mathbb{R}, x_n \in X_n and 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.{{cite book | last = Quarteroni, Sacco, Saleri | title = Numerical Mathematics | publisher = Springer | location = Milano | year = 2000 | page = 33 | url = http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf | access-date = 2016-09-27 | archive-url = https://web.archive.org/web/20171114040621/http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf | archive-date = 2017-11-14 | url-status = dead

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 \left { F_n \right }_{n \in \mathbb{N}} pointwise converges to F on the set S of its solutions:

: \lim F_n(x,y+t) = F(x,y,t) = 0, \quad \quad \forall (x,y,t) \in S.

When F_n=F, \forall n \in \mathbb{N} on S the method is said to be strictly consistent.

Convergence

Denote by \ell_n a sequence of admissible perturbations of x \in X for some numerical method M (i.e. x+\ell_n \in X_n \forall n \in \mathbb{N}) and with y_n(x+\ell_n) \in Y_n the value such that 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:

: \begin{align} &\forall \varepsilon 0, \exist n_0(\varepsilon) 0, \exist \delta_{\varepsilon, n_0} \text{ such that} \ &\forall n n_0, \forall \ell_n : | \ell_n | \end{align}

One can easily prove that the point-wise convergence of {y_n} _{n \in \mathbb{N}} to y implies the convergence of the associated method.

References

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