Abstract

We provide two Matlab programs to compute integrals of the form

Program summary

Program title: BESSELINTR, BESSELINTC

Catalogue identifier: AEAH_v1_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAH_v1_0.html

Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland

Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html

No. of lines in distributed program, including test data, etc.: 1601

No. of bytes in distributed program, including test data, etc.: 13 161

Distribution format: tar.gz

Programming language: Matlab (version 6.5), Octave (version 2.1.69)

Computer: All supporting Matlab or Octave

Operating system: All supporting Matlab or Octave

RAM: For k Bessel functions our program needs approximately (500+140k) double precision variables

Classification: 4.11

Nature of problem: The problem consists in integrating an arbitrary product of Bessel functions with an additional rational or exponential factor over a semi-infinite interval. Difficulties arise from the irregular oscillatory behaviour and the possible slow decay of the integrand, which prevents truncation at a finite point.

Solution method: The interval of integration is split into a finite and infinite part. The integral over the finite part is computed using Gauss–Legendre quadrature. The integrand on the infinite part is approximated using asymptotic expansions and this approximation is integrated exactly with the aid of the upper incomplete gamma function. In the case where a rational factor is present, this factor is first expanded in a Taylor series around infinity.

Restrictions: Some (and eventually all) numerical accuracy is lost when one or more of the parameters r,c,a_i or v_i grow very large, or when r becomes small.

Running time: Less than 0.02 s for a simple problem (two Bessel functions, small parameters), a few seconds for a more complex problem (more than six Bessel functions, large parameters), in Matlab 7.4 (R2007a) on a 2.4 GHz AMD Opteron Processor 250.

References:

[1] J. Van Deun, R. Cools, Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions, ACM Trans. Math. Software 32 (4) (2006) 580–596.

Keywords: Quadrature; Bessel functions

PACS classification codes: 02.30.Gp; 02.60.Jh

Article Outline

1. Introduction

2. The algorithm

2.1. Description

2.2. Error analysis of infinite part

2.2.1. Original error analysis

2.2.2. Error analysis for I⁽¹⁾

2.2.3. Error analysis for I⁽²⁾

2.3. Cost function and fudge parameters

3. Implementation issues

4. Limitations

5. Experiments

5.1. Analytic expressions

5.2. Test program

Appendix A. Test run output

References

This paper and its associated computer program are available via the Computer Physics Communications homepage on Science Direct (http://www.sciencedirect.com/science/journal/00104655).