Numerical method explained

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 x Y → 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 g:X → 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} = \left \{ F _{n(x} _{n,y} _{n)=0} \right \} _{n} ,

with

F _{n:X} _{n} x Y _{n} → R , x _{n} \in X _{n} and y _{n} \in Y _{n} for every n \in 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} → x and that F _{n} behaves like F when n → inf ty . So, a numerical method is called consistent if and only if the sequence of functions \left \{ F _{n} \right \} _{n} pointwise converges to F on the set S of its solutions:\lim F _{n(x,y+t)} = F(x,y,t) = 0 , \forall (x,y,t) \in S.

When

F _{n=F,} \forall n \in 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 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}} suchthat \\ &\forall n > n _{0,} \forall \ell _{n} : \| \ell _{n} \| < \delta _{\varepsilon,n _{0}} ⇒ \| y _{n(x+\ell} _{n)} - y \| \leq \varepsilon. \end{align}

One can easily prove that the point-wise convergence of

\{y _{n\}} _{n} to y implies the convergence of the associated method is function.

See also

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

Notes and References

1. Book: Quarteroni, Sacco, Saleri . Numerical Mathematics. Springer. Milano. 2000. 33. 2016-09-27. https://web.archive.org/web/20171114040621/http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf. 2017-11-14. dead.