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
be a well-posed problem, i.e.
is a real or complex functional relationship, defined on the cross-product of an input data set
and an output data set
, such that exists a locally lipschitz function
called resolvent, which has the property that for every root
of
,
. We define numerical method for the approximation of
, 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} },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a48d2e4b6af62707e361ceb02633c6fa86c185a0)
with
,
and
for every
. 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
are that
and that
behaves like
when
. So, a numerical method is called consistent if and only if the sequence of functions
pointwise converges to
on the set
of its solutions:
![{\displaystyle \lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3cfb36e901d5de5f0d59ec08406430595354b7)
When
on
the method is said to be strictly consistent.[1]
Convergence
Denote by
a sequence of admissible perturbations of
for some numerical method
(i.e.
) and with
the value such that
. A condition which the method has to satisfy to be a meaningful tool for solving the problem
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dea0e99df283bc0ad41e30da628607e9e7cb8474)
One can easily prove that the point-wise convergence of
to
implies the convergence of the associated method is function.[1]