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Numerical method - Wikipedia

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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.

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.^[1]

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.^[1]

Denote by ℓ n {\displaystyle \ell _{n}} a sequence of admissible perturbations of x ∈ X {\displaystyle x\in X} for some numerical method M {\displaystyle M} (i.e. x + ℓ n ∈ X n ∀ n ∈ N {\displaystyle x+\ell _{n}\in X_{n}\forall n\in \mathbb {N} } ) and with y n ( x + ℓ n ) ∈ Y n {\displaystyle y_{n}(x+\ell _{n})\in Y_{n}} the value such that F n ( x + ℓ n , y n ( x + ℓ n ) ) = 0 {\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 {\displaystyle F(x,y)=0} is convergence:

∀ ε > 0 , ∃ n 0 ( ε ) > 0 , ∃ δ ε , n 0  such that ∀ n > n 0 , ∀ ℓ n : ‖ ℓ n ‖ < δ ε , n 0 ⇒ ‖ y n ( x + ℓ n ) − y ‖ ≤ ε . {\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 { y n } n ∈ N {\displaystyle \{y_{n}\}_{n\in \mathbb {N} }} to y {\displaystyle y} implies the convergence of the associated method.^[1]

• Numerical methods for ordinary differential equations
• Numerical methods for partial differential equations

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