Numerical analysis: Difference between revisions

Revision as of 22:12, 21 April 2025

Latest revision as of 12:01, 23 June 2025

(2 intermediate revisions by 2 users not shown)

Line 166:

{{Main|Numerical integration}}

Numerical integration, in some instances also known as numerical [[Numericalquadrature integration(mathematics)|quadrature]], asks for the value of a definite [[integral]].<ref>{{cite book |last1=Davis |first1=P.J. |last2=Rabinowitz |first2=P. |title=Methods of numerical integration |publisher=Courier Corporation |date=2007 |isbn=978-0-486-45339-2 |url={{GBurl|gGCKdqka0HAC|pg=PR5}}}}</ref> Popular methods use one of the [[Newton–Cotes formulas]] (like the midpoint rule or [[Simpson's rule]]) or [[Gaussian quadrature]].<ref>{{MathWorld|author=Weisstein, Eric W. |title=Gaussian Quadrature |id=GaussianQuadrature}}</ref> These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use [[Monte Carlo method|Monte Carlo]] or [[quasi-Monte Carlo method]]s (see [[Monte Carlo integration]]<ref>{{cite book |first=John |last=Geweke |chapter=15. Monte carlo simulation and numerical integration |chapter-url=https://www.sciencedirect.com/science/article/pii/S1574002196010179) |doi=10.1016/S1574-0021(96)01017-9|title=Handbook of Computational Economics |publisher=Elsevier |volume=1 |date=1996 |isbn=9780444898579 |pages=731–800 }}</ref>), or, in modestly large dimensions, the method of [[sparse grid]]s.

===Differential equations===

Line 176:

==Software==

{{Main|List of numerical-analysis software|Comparison of numerical-analysis software|List of programming languages by type#Numerical analysis|l3=Numerical analysis programming languages|List of numerical libraries}}

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The [[Netlib]] repository contains various collections of software routines for numerical problems, mostly in [[Fortran]] and [[C (programming language)|C]]. Commercial products implementing many different numerical algorithms include the [[IMSL Numerical Libraries|IMSL]] and [[Numerical Algorithms Group|NAG]] libraries; a [[free software|free-software]] alternative is the [[GNU Scientific Library]].