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
be a
well-posed problem, i.e.
is a
real or complex functional relationship, defined on the
Cartesian 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
\left\{Mn\right\}n=\left\{Fn(xn,yn)=0\right\}n,
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:
\limFn(x,y+t)=F(x,y,t)=0, \forall(x,y,t)\inS.
When
on
the method is said to be
strictly consistent.
Convergence
Denote by
a sequence of
admissible perturbations of
for some numerical method
(i.e.
) and with
the value such that
Fn(x+\elln,yn(x+\elln))=0
. A condition which the method has to satisfy to be a meaningful tool for solving the problem
is
convergence:
\begin{align}
&\forall\varepsilon>0,\existn0(\varepsilon)>0,\exist
suchthat\\
&\foralln>n0,\forall\elln:\|\elln\|<
⇒ \|yn(x+\elln)-y\|\leq\varepsilon.
\end{align}
One can easily prove that the point-wise convergence of
to
implies the convergence of the associated method.
See also
Notes and References
- 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.
