In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, 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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
functional relationship, defined on the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of problems
:
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''.
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:
:
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 . A condition which the method has to satisfy to be a meaningful tool for solving the problem is ''convergence'':
:
One can easily prove that the point-wise convergence of to implies the convergence of the associated method.[
]
See also
* Numerical methods for ordinary differential equations
* Numerical methods for partial differential equations
References
{{Authority control
Numerical analysis