Numerical method

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.

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. ==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 \| 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. ==See also== Numerical methods for ordinary differential equations Numerical methods for partial differential equations ==References==

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