A comparison of some numerical integration programs

Casaletto, J.; Pickett, Mary S.; Rice, John R.

doi:10.1145/1198442.1198448

Cited by 28 publications

(10 citation statements)

References 2 publications

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“…Whereas other authors [Casaletto et al 1969;Hillstrom 1970;Kahaner 1971;Malcolm and Simpson 1975;Robinson 1979;Krommer andÜberhuber 1998;Favati et al 1991a] have focused on comparing different algorithms as a whole, using sets of functions chosen to best represent typical integrands, we will focus here only on · P. GONNET specific error estimators and on integrands chosen such that they specifically should or should not cause the error estimator to fail. For these test functions we will not consider the usual metrics of efficiency, i.e.…”

Section: Methodsmentioning

confidence: 99%

A review of error estimation in adaptive quadrature

Gonnet

2012

ACM Comput. Surv.

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The most critical component of any adaptive numerical quadrature routine is the estimation of the integration error. Since the publication of the first algorithms in the 1960s, many error estimation schemes have been presented, evaluated and discussed. This paper presents a review of existing error estimation techniques and discusses their differences and their common features. Some common shortcomings of these algorithms are discussed and a new general error estimation technique is presented. 12:right 13: push interval [a k , m] with integral I left and error ε left onto H 14: push interval [m, b k ] with integral I right and error ε right onto H 15: end while 16: return I

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Section: Methodsmentioning

confidence: 99%

A review of error estimation in adaptive quadrature

Gonnet

2012

ACM Comput. Surv.

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“…For m = 1 two extensive sets of test integrals which appear in the numerical analysis literature have been used to make a comparative computation. The first set is due to Casaletto et al (1969) and contains 50 functions ranging from polynomials up to degree 20 through functions with discontinuities; see also Evans (1993, Kahaner (1971) which includes in addition some harder examples (Evans, 1993, point Gaussian formula, the (7-15)-point Gauss-Kronrod method implemented in gkint and GL.integrate.1D given in Kuonen (2001, app. A, Table A.1) which uses the Gauss-Legendre (GL) rule with M points.…”

Section: Comparisonmentioning

confidence: 99%

Numerical Integration inS-PLUSorR: A Survey

Kuonen¹

2003

J. Stat. Soft.

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This paper reviews current quadrature methods for approximate calculation of integrals within S-Plus or R. Starting with the general framework, Gaussian quadrature will be discussed first, followed by adaptive rules and Monte Carlo methods. Finally, a comparison of the methods presented is given. The aim of this survey paper is to help readers, not expert in computing, to apply numerical integration methods and to realize that numerical analysis is an art, not a science.Calculations were made on a Sun SPARC Ultra 60 workstation with 1Gb RAM using S-Plus or R. S-Plus is a value-added version of S (Becker et al., 1988;Chambers, 1998) sold by Insightful Corporation. The S language is often the vehicle of choice for research in statistical methodology and R (Ihaka and Gentleman, 1996; www.R-project.org) provides a free software route to participation in that activity. Good reference books for S-Plus and R are Ripley (2000, 2002).The tools available in S-Plus to measure resources differ between versions. The principal resource considered in this paper is CPU time. It is clear that more resources, like memory usage, should be considered but we found it sufficient to consider CPU time

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“…The sample of functions presented by has the additional advantage of having been considered at various times in the literature. Other samples for possible consideration are given by O'Hara and Smith [7], and by Casaletto, Pickett and Rice [2].…”

Section: Test Cases and Comparisonsmentioning

confidence: 99%

Algorithm 36 SNIFF: Efficient self-tuning algorithm for numerical integration

Garribba

Quartapelle

Reina³

1978

Computing

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--ZusammenfassungAlgorithm 36. SNIFF: Efficient Serf-Tuning Algorithm for Numerical Integration. A new adaptive algorithm for the integration of analytic functions is presented. The algorithm processes the integration interval by generating local subintervals whose length is controlled through a feedback loop. Control is performed by means of a relation derived on an analytical basis and valid for an arbitrary integration rule: two different estimates of an integral are used to compute the interval length necessary to obtain an integral estimate with accuracy within the assigned error bounds. The implied method for local generation of subintervals and an effective assumption of error partition among subintervals give rise to an adaptive algorithm provided with an accurate and very efficient integration process. The particular algorithm obtained by choosing the 6-point Gaug-Legendre integration rule is considered and extensive comparisons are made with other outstanding integration algorithms.Algorithmus 36. Giinstiger selbsteichender Algorithmus fiir numerische Integration. Hier wird ein neuer selbsteichender Algorithmus ffir die Integration der analytischen Funktionen dargestellt. Der Algorithmus wirkt auf das Integrationsintervall indem er 6rtliche Unterintervalle hervorruft, deren L~inge dutch ein neues System fiberwacht wird. Die Uberwachung ergibt sich aus einer Formel, die yon einer analytischen Basis abgeleitet wird und fiir jede beliebige Integrationsregel gilt: mit zwei verschiedenen Anniiherungen eines Integrals wird die Intervallfinge berechnet, die notwendig ist, um eine Integralannfiherung mit der innerhalb zugelassener Fehlergrenzen notwendigen Genauigkeit zu erreichen. Die daraus entstehende Methode ffir die Erzeugung der Unterintervalle und eine wirksame Auswahl der Fehlerverteilung unter die Unterintervalle bringen einen selbstanpassenden Algorithmus hervor, der fiber einen genauen und h6chst giinstigen Integrationsprozel3 verfiigt. Darauf wird der besondere Algorithmus betrachtet, der sich mit Hilfe der 6-Punkte-Gaug-LegendreIntegrationsregel ergibt. AuBerdem wird er mit den besten unter den anderen Integrationsalgorithmen verglichen.

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A comparison of some numerical integration programs

Cited by 28 publications

References 2 publications

A review of error estimation in adaptive quadrature

A review of error estimation in adaptive quadrature

Numerical Integration inS-PLUSorR: A Survey

Algorithm 36 SNIFF: Efficient self-tuning algorithm for numerical integration

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