Experimental Mathematics

A comparison of three high-precision quadrature schemes

David H. Bailey, Karthik Jeyabalan, and Xiaoye S. Li

Full-text: Open access

Abstract

The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high precision a large class of integrals typical of those encountered in experimental mathematics, relying on a minimum of a priori information regarding the function to be integrated. Such a facility is useful, for example, to permit the experimental identification of definite integrals based on their numerical values. The performance and accuracy of these three quadrature schemes are compared using a suite of 15 integrals, ranging from continuous, well-behaved functions on finite intervals to functions with infinite derivatives and blow-up singularities at endpoints, as well as several integrals on an infinite interval. In results using 412-digit arithmetic, we achieve at least 400-digit accuracy, using two of the programs, for all problems except one highly oscillatory function on an infinite interval. Similar results were obtained using 1,012-digit arithmetic.

Article information

Source
Experiment. Math., Volume 14, Issue 3 (2005), 317-329.

Dates
First available in Project Euclid: 3 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.em/1128371757

Mathematical Reviews number (MathSciNet)
MR2172710

Zentralblatt MATH identifier
1082.65028

Subjects
Primary: 65D30: Numerical integration

Keywords
Numerical quadrature numerical integration arbitrary precision

Citation

Bailey, David H.; Jeyabalan, Karthik; Li, Xiaoye S. A comparison of three high-precision quadrature schemes. Experiment. Math. 14 (2005), no. 3, 317--329. https://projecteuclid.org/euclid.em/1128371757


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