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Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions

Published: 05 March 2014 Publication History

Abstract

A MATLAB toolbox, IIPBF, for calculating infinite integrals involving a product of two Bessel functions Jax)Jbx), Jax)Ybx) and Yax)Ybx), for non-negative integers a,b, and a well-behaved function f(x), is described. Based on the Lucas algorithm previously developed for Jax)Jbx) only, IIPBF recasts each product as the sum of two functions whose oscillatory behavior is exploited in the three-step procedure of adaptive integration, summation, and extrapolation. The toolbox uses customised QUADPACK and IMSL functions from a MATLAB conversion of the SLATEC library. In addition, MATLAB's own quadgk function for adaptive Gauss-Kronrod quadrature results in a significant speed up compared with the original algorithm. Usage of IIPBF is described and eighteen test cases illustrate the robustness of the toolbox; five additional ones are used to compare IIPBF with the BESSELINT code for rational and exponential forms of f(x) with Jax)Jbx). Reliability for a broad range of values of ρ and τ for the three different product types as well as different orders in one case is demonstrated. An electronic appendix provides a novel derivation of formulae for five cases.

Supplementary Material

ZIP File (935.zip)
Software for IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions
a14-ratnanather-apndx.pdf (ratnanather.zip)
Supplemental movie, appendix, image and software files for, Algorithm 935: <tt>IIPBF</tt>, a <tt>MATLAB</tt> toolbox for infinite integral of products of two bessel functions

References

[1]
V. Adamchik. 1995. The evaluation of integrals of Bessel functions via G-function identities. J. Comp. Appl. Math. 64, 283--290.
[2]
D. H. Bailey and J. H. Borwein. 2012. Hand-to-hand combat with thousand-digit integrals. J. Comp. Sci. 3, 77--86. http://dx.doi.org/10.1016/j.jocs.2010.12.004.
[3]
B. Barrowes. 2009. The SLATEC library converted into MATLAB functions. http://www.mathworks.com/matlabcentral/fileexchange/14535.
[4]
R. V. Craster. 1998. Scattering by cracks beneath fluid-solid interfaces. J. Sound Vib. 209, 343--372.
[5]
A. M. J. Davis, J. H. Kim, C. Ceritoglu, and J. T. Ratnanather. 2012. A Stokesian analysis of a submerged viscous jet impinging on a planar wall. J. Fluid Mech. 712, 531--551.
[6]
A. M. J. Davis, J. H. Kim, G. M. Gunter, and J. T. Ratnanather. 2013. The Stokesian flow field of an oscillatory submerged viscous jet impinging on a planar wall. Proc. Roy. Soc. A, 469, 2157. DOI 10.1098/rspa.2013.0282.
[7]
A. M. J. Davis and H. A. Stone. 1998. Slow translation, rotation or oscillation of a disk in a rotating fluid: Effect of a plane wall or another disk. Q. J. Mech. App. Math. 51, 495--513.
[8]
B. Gebremariam, T. Duguet, and S. K. Bogner. 2010. Symbolic integration of a product of two spherical bessel functions with an additional exponential and polynomial factor. Comput. Phys. Commun. 181, 1136--1143.
[9]
M. L. Glasser. 1974. Some definite integrals of the product of two Bessel functions of the second kind: (order zero). Math. Comp. 28, 613--615.
[10]
P. Gonnet. 2012. A review of error estimation in adaptive quadrature. ACM Comput. Surv. 44, 4, 22:1--22:36.
[11]
I. S. Gradshteyn and I. M. Ryzhik. 2007. Gradshteyn and Ryzhik's Table of Integrals, Series, and Products 7th Ed. A. Jeffrey and D. Zwillinger (Eds.). Academic Press, San Diego, CA.
[12]
A. P. Hosseinbor, M. K. Chung, Y.-C. Wu, and A. L. Alexander. 2013. Bessel Fourier Orientation Reconstruction (BFOR): An analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices. Neuroimage 64, 650--670.
[13]
D. Huybrechs and S. Vandewalle. 2006. On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44, 1026--1048.
[14]
M. Ikonomou, P. Köhler, and A. F. Jacob. 1995. Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew Math. Mech. 75, 917--926.
[15]
K. Key. 2012. Is the fast Hankel transform faster than quadrature&quest; Geophysics 77, F21--F30.
[16]
M. Li, W. He, Q. Li, Z. Xu, and C. Luo. 2010. The numerical integration algorithm of dual Bessel function and its application. Int. J. Appl. Electromag. Mech. 33, 727--734.
[17]
S. K. Lucas. 1995. Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. App. Math. 64, 269--282.
[18]
S. K. Lucas and H. A. Stone. 1995. Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comp. App. Math. 64, 217--231.
[19]
R. C. McPhedran, D. H. Dawes, and T. C. Scott. 1992. On a Bessel function integral. Appl. Algebra Eng. Commun. Comput. 2, 207--216.
[20]
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. 2010. NIST Handbook of Mathematical Functions: Companion to the Digital Library of Mathematical Functions. Cambridge University Press, New York.
[21]
E. Pan, M. Bevis, F. Han, H. Zhou, and R. Zhu. 2007. Surface deformation due to loading of a layered elastic half-space: a rapid numerical kernel based on a circular loading element. Geophys. J. Int. 171, 11--24.
[22]
E. P. Petrov and P. Schwille. 2008. Translational diffusion in lipid membranes beyond the Saffman-Delbruck approximation. Biophys. J. 94, L41--L43.
[23]
R. Piessens, E. D. Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahauer. 1983. QUADPACK, A Sub-routine Package for Automatic Integration. Springer-Verlag, Berlin.
[24]
N. I. Robinson. 2002. An isotropic elastic medium containing a cylindrical borehole with a rigid plug. Int. J. Solids Struct. 39, 4889--4904.
[25]
J. Salo, H. Wei-Sallabi, and P. Vainikainen. 2006. Statistical analysis of the multiple scattering radio channel. IEEE Trans. Ant. Prop. 54, 3114--3124.
[26]
L. F. Shampine. 2008. Vectorized adaptive quadrature in MATLAB. J. Comp. App. Math. 211, 131--140.
[27]
J. D. Sherwood. 2005. Optimal probes for withdrawal of uncontaminated fluid samples. Phys. Fluids 17, 083102.
[28]
A. Sidi. 1988. A user-friendly extrapolation method for oscillatory infinite integrals. Math. Comp. 51, 249--266.
[29]
A. Sidi. 2012. A user-friendly extrapolation method for computing infinite range integrals of products of oscillatory functions. IMA J. Num. Anal. 32, 602--631.
[30]
N. P. Singh and T. Mogi. 2005. Electromagnetic response of a large circular loop source on a layered earth: A new computation method. Pure and Appl. Geophysics 162, 181--200.
[31]
D. M. Tartakovsky, J. D. Moulton, and V. A. Zlotnik. 2000. Kinematic structure of minipermeameter flow. Water Resources Res. 36, 2433--2442.
[32]
J. Van Deun and R. Cools. 2006a. Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions. ACM Trans. Math. Soft. 32, 580--596.
[33]
J. Van Deun and R. Cools. 2006b. A Matlab implementation of an algorithm for computing integrals of products of Bessel functions. Lecture Notes in Computer Science, vol. 4151, 284--295.
[34]
J. Van Deun and R. Cools. 2006c. A stable recurrence for the incomplete gamma function with imaginary second argument. Numer. Math. 104, 445--456.
[35]
J. Van Deun and R. Cools. 2008. Integrating products of Bessel functions with an additional exponential or rational factor. Comp. Phys. Comm. 178, 578--590.
[36]
Y. S. Xu, J. Lin, H. M. Li, and F. M. Wu. 2003. Analysis for the potential function of the digital microstructure image of porous media. Commun. Theor. Phys. 40, 393--394.

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        cover image ACM Transactions on Mathematical Software
        ACM Transactions on Mathematical Software  Volume 40, Issue 2
        February 2014
        161 pages
        ISSN:0098-3500
        EISSN:1557-7295
        DOI:10.1145/2594412
        Issue’s Table of Contents
        Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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        Publication History

        Published: 05 March 2014
        Accepted: 01 June 2013
        Revised: 01 September 2012
        Received: 01 February 2011
        Published in TOMS Volume 40, Issue 2

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        Author Tags

        1. Bessel function
        2. adaptive quadrature
        3. extrapolation
        4. quadrature
        5. summation

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