×

A discussion of a new error estimate for adaptive quadrature. (English) Zbl 0677.65018

“A critical part of any automated integration routine is the way we estimate the error. When constructing an error estimate one has to compromise between two contradictory aims: reliability and economy.” These are the authors’ opening sentences in a paper which approaches this problem in a realistic way, preserving a fine balance between theoretical expectation and numerical experiment. The authors treat the problem of how to handle four numerical approximations \(B_ 1,B_ 2,A_ 1\), and \(A_ 2\) to the same integral I (perhaps over a small interval); here \(B_ 1\) and \(A_ 1\) employ two different rules (for example the Gauss- Legendre rule and the corresponding Gauss-Kronrod rule), and \(B_ 2\) and \(A_ 2\) employ the two copy versions of the respective rules.
In a contrived situation in which convergence is monotonic, the estimate \(I\cong A_ 2+\epsilon\) with \(\epsilon =(A_ 2-B_ 2)(A_ 2-A_ 1)/(B_ 2-B_ 1-A_ 2+A_ 1)\) is reasonable, and in a previous paper by D. P. Laurie [J. Comput. Anal. Math. 12-13, 425-431 (1985; Zbl 0589.65020)] conditions are given for I to lie in \([A_ 2,A_ 2+\epsilon]\). (These conditions depend on I, whose numerical value is of course not available.)
The authors discuss previous error estimates based on these conditions. These previous estimates work well in “the asymptotic region”. However, the point of adaptive quadrature is to economize in function values and this has the effect of arranging the calculation so that this asymptotic region is rarely encountered. With this in mind, the authors propose a more sophisticated error estimate; they confirm by numerical experiment their expectation that this one is more expensive and much more reliable.
The reviewer recommends this paper to any quadrature routine constructor who is about to impose a practical convergence criterion.
Reviewer: J.N.Lyness

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures

Citations:

Zbl 0589.65020

Software:

CUBTRI; QUADPACK
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Berntsen,A Test of some wellknown quadrature routines. Reports in Informatics 20, Dept. of Inf., Univ. of Bergen, Norway, 1986.
[2] P. J. Davis and P. Rabinowitz,Methods of Numerical Integration. Academic Press, 1984. · Zbl 0537.65020
[3] D. P. Laurie,Cubtri-automatic cubature over a triangle. ACM Trans. Math. Software, 8: 210–218, 1982. · Zbl 0478.65011 · doi:10.1145/355993.356001
[4] D. P. Laurie,Practical error estimation in numerical integration. Jour. of Comp. and Appl. Math., 12 & 13: 425–431, 1985. · Zbl 0589.65020 · doi:10.1016/0377-0427(85)90036-6
[5] D. P. Laurie,Sharper error estimate in adaptive quadrature. BIT, 23: 258–261, 1983. · Zbl 0539.41028 · doi:10.1007/BF02218446
[6] J. N. Lyness,When not to use an automatic quadrature routine. SIAM Rev., 25: 63–87, 1983. · Zbl 0508.65006 · doi:10.1137/1025003
[7] R. Piessens, E. de Doncker-Kapenga, C. W. Ueberhuber, and D. K. Kahaner,QU ADPACK. Springer-Verlag, 1983.
[8] T. Sørevik,Reliable and efficient adaptive quadrature algorithms. Thesis for the degree Doctor Scientiarum, Department of Informatics, University of Bergen, Norway, 1988.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.