The modified composite Gauss type rules for singular integrals using Puiseux expansions

Wang, Tongke; Liu, Zhifang; Zhang, Zhiyue

doi:10.1090/mcom/3105

The modified composite Gauss type rules for singular integrals using Puiseux expansions
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by Tongke Wang, Zhifang Liu and Zhiyue Zhang PDF

Math. Comp. 86 (2017), 345-373 Request permission

Abstract:

This paper is devoted to designing some modified composite Gauss type rules for the integrals involving algebraic and logarithmic endpoint singularities, at which the integrands possess the Puiseux expansions in series. Firstly, the error asymptotic expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sum of the Hurwitz zeta function and its higher derivatives. Secondly, the deduced error asymptotic expansion is applied to the composite Gauss-Legendre and Gauss-Kronrod rules. By simplifying the evaluations of the Hurwitz zeta function and its derivatives, two modified composite Gaussian rules and their error estimates are obtained. The methods can also effectively deal with infinite range singular integrals, the singular and oscillatory Fourier transforms and the Cauchy principal value integrals by simple variable transformations. The advantage of the practical algorithms is that three ways can be used to increase the accuracy of the algorithms, which are decreasing the step length of the composite rule, increasing the orders of the Puiseux expansions and increasing the number of nodes of the Gaussian quadrature rules. Finally, the excellent performance of the proposed methods is demonstrated through several typical numerical examples. It is shown that the algorithms can be used to automatically evaluate a wide range of singular integrals over finite or infinite intervals.

References

Fuensanta Aroca, Giovanna Ilardi, and Lucía López de Medrano, Puiseux power series solutions for systems of equations, Internat. J. Math. 21 (2010), no. 11, 1439–1459. MR 2747737, DOI 10.1142/S0129167X10006574
D. H. Bailey and J. M. Borwein, High-precision numerical integration: progress and challenges, J. Symbolic Comput. 46 (2011), no. 7, 741–754. MR 2795208, DOI 10.1016/j.jsc.2010.08.010
Kevin A. Broughan, Vanishing of the integral of the Hurwitz zeta function, Bull. Austral. Math. Soc. 65 (2002), no. 1, 121–127. MR 1889386, DOI 10.1017/S000497270002013X
D. Calvetti, G. H. Golub, W. B. Gragg, and L. Reichel, Computation of Gauss-Kronrod quadrature rules, Math. Comp. 69 (2000), no. 231, 1035–1052. MR 1677474, DOI 10.1090/S0025-5718-00-01174-1
Bejoy K. Choudhury, The Riemann zeta-function and its derivatives, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1940, 477–499. MR 1356175, DOI 10.1098/rspa.1995.0096
D. V. Chudnovsky and G. V. Chudnovsky, On expansion of algebraic functions in power and Puiseux series. II, J. Complexity 3 (1987), no. 1, 1–25. MR 883165, DOI 10.1016/0885-064X(87)90002-1
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
Sven Ehrich, High order error constants of Gauss-Kronrod quadrature formulas, Analysis 16 (1996), no. 4, 335–345. MR 1429458, DOI 10.1524/anly.1996.16.4.335
Sven Ehrich, Stieltjes polynomials and the error of Gauss-Kronrod quadrature formulas, Applications and computation of orthogonal polynomials (Oberwolfach, 1998) Internat. Ser. Numer. Math., vol. 131, Birkhäuser, Basel, 1999, pp. 57–77. MR 1722715
Walter Gautschi, Gauss quadrature routines for two classes of logarithmic weight functions, Numer. Algorithms 55 (2010), no. 2-3, 265–277. MR 2720632, DOI 10.1007/s11075-010-9366-0
P. Gonnet, A review of error estimation in adaptive quadrature, ACM Comput. Surv. 44 (2012), 1–36.
Daan Huybrechs and Stefan Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. 44 (2006), no. 3, 1026–1048. MR 2231854, DOI 10.1137/050636814
Daan Huybrechs and Ronald Cools, On generalized Gaussian quadrature rules for singular and nearly singular integrals, SIAM J. Numer. Anal. 47 (2008/09), no. 1, 719–739. MR 2475959, DOI 10.1137/080723417
Masao Iri, Sigeiti Moriguti, and Yoshimitsu Takasawa, On a certain quadrature formula, J. Comput. Appl. Math. 17 (1987), no. 1-2, 3–20. MR 884257, DOI 10.1016/0377-0427(87)90034-3
Fredrik Johansson, Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, Numer. Algorithms 69 (2015), no. 2, 253–270. MR 3350381, DOI 10.1007/s11075-014-9893-1
S. Kanemitsu, H. Kumagai, H. M. Srivastava, and M. Yoshimoto, Some integral and asymptotic formulas associated with the Hurwitz zeta function, Appl. Math. Comput. 154 (2004), no. 3, 641–664. MR 2072810, DOI 10.1016/S0096-3003(03)00740-9
Hongchao Kang and Xinping Shao, Fast computation of singular oscillatory Fourier transforms, Abstr. Appl. Anal. , posted on (2014), Art. ID 984834, 8. MR 3240576, DOI 10.1155/2014/984834
Kiran M. Kolwankar and Anil D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6 (1996), no. 4, 505–513. MR 1425755, DOI 10.1063/1.166197
K. M. Kolwankar, Recursive local fractional derivative, http://arxiv.org/abs/1312.7675v1 (2013).
M. Kzaz, Convergence acceleration of the Gauss-Laguerre quadrature formula, Appl. Numer. Math. 29 (1999), no. 2, 201–220. MR 1666533, DOI 10.1016/S0168-9274(98)00075-0
Dirk P. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997), no. 219, 1133–1145. MR 1422788, DOI 10.1090/S0025-5718-97-00861-2
Z. F. Liu, T. K. Wang and G. H. Gao, A local fractional Taylor expansion and its computation for insufficiently smooth functions, E. Asian J. Appl. Math. 5 (2015), pp. 176–191.
D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Math. Comp. 43 (1984), no. 167, 219–242. MR 744932, DOI 10.1090/S0025-5718-1984-0744932-2
J. N. Lyness and B. W. Ninham, Numerical quadrature and asymptotic expansions, Math. Comp. 21 (1967), 162–178. MR 225488, DOI 10.1090/S0025-5718-1967-0225488-X
Hassan Majidian, Numerical approximation of highly oscillatory integrals on semi-finite intervals by steepest descent method, Numer. Algorithms 63 (2013), no. 3, 537–548. MR 3071190, DOI 10.1007/s11075-012-9639-x
John McDonald, Fractional power series solutions for systems of equations, Discrete Comput. Geom. 27 (2002), no. 4, 501–529. MR 1902675, DOI 10.1007/s00454-001-0077-0
G. V. Milovanović, Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures, Comput. Math. Appl. 36 (1998), no. 8, 19–39. MR 1653827, DOI 10.1016/S0898-1221(98)00180-1
G. Monegato and J. N. Lyness, The Euler-Maclaurin expansion and finite-part integrals, Numer. Math. 81 (1998), no. 2, 273–291. MR 1657780, DOI 10.1007/s002110050392
G. Monegato and L. Scuderi, Numerical integration of functions with boundary singularities, J. Comput. Appl. Math. 112 (1999), no. 1-2, 201–214. Numerical evaluation of integrals. MR 1728460, DOI 10.1016/S0377-0427(99)00230-7
Giovanni Monegato, An overview of the computational aspects of Kronrod quadrature rules, Numer. Algorithms 26 (2001), no. 2, 173–196. MR 1829797, DOI 10.1023/A:1016640617732
Masatake Mori, Quadrature formulas obtained by variable transformation and the DE-rule, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 119–130. MR 793948, DOI 10.1016/0377-0427(85)90011-1
Israel Navot, An extension of the Euler-Maclaurin summation formula to functions with a branch singularity, J. Math. and Phys. 40 (1961), 271–276. MR 140876
I. Navot, A further extension of the Euler-Maclaurin summation formula, J. Math. Phys. 41 (1962), 155–163.
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
Annamaria Palamara Orsi, Nested quadrature rules for Cauchy principal value integrals, J. Comput. Appl. Math. 25 (1989), no. 3, 251–266. MR 999090, DOI 10.1016/0377-0427(89)90030-7
Knut Petras, On the computation of the Gauss-Legendre quadrature formula with a given precision, J. Comput. Appl. Math. 112 (1999), no. 1-2, 253–267. Numerical evaluation of integrals. MR 1728464, DOI 10.1016/S0377-0427(99)00225-3
Adrien Poteaux and Marc Rybowicz, Good reduction of Puiseux series and applications, J. Symbolic Comput. 47 (2012), no. 1, 32–63. MR 2854846, DOI 10.1016/j.jsc.2011.08.008
Avram Sidi, Practical extrapolation methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 10, Cambridge University Press, Cambridge, 2003. Theory and applications. MR 1994507, DOI 10.1017/CBO9780511546815
Avram Sidi, Euler-Maclaurin expansions for integrals with endpoint singularities: a new perspective, Numer. Math. 98 (2004), no. 2, 371–387. MR 2092747, DOI 10.1007/s00211-004-0539-4
Avram Sidi, Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities, Math. Comp. 78 (2009), no. 265, 241–253. MR 2448705, DOI 10.1090/S0025-5718-08-02135-2
Avram Sidi, Variable transformations and Gauss-Legendre quadrature for integrals with endpoint singularities, Math. Comp. 78 (2009), no. 267, 1593–1612. MR 2501065, DOI 10.1090/S0025-5718-09-02203-0
Avram Sidi, Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, Math. Comp. 81 (2012), no. 280, 2159–2173. MR 2945150, DOI 10.1090/S0025-5718-2012-02597-X
Avram Sidi, Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities, Constr. Approx. 36 (2012), no. 3, 331–352. MR 2996435, DOI 10.1007/s00365-011-9140-0
Avram Sidi, Richardson extrapolation on some recent numerical quadrature formulas for singular and hypersingular integrals and its study of stability, J. Sci. Comput. 60 (2014), no. 1, 141–159. MR 3246095, DOI 10.1007/s10915-013-9788-7
J. F. Steffensen, Interpolation, Chelsea Publishing Co., New York, N. Y., 1950. 2d ed. MR 0036799
Paul N. Swarztrauber, On computing the points and weights for Gauss-Legendre quadrature, SIAM J. Sci. Comput. 24 (2002), no. 3, 945–954. MR 1950519, DOI 10.1137/S1064827500379690
Pierre Verlinden, Acceleration of Gauss-Legendre quadrature for an integrand with an endpoint singularity, J. Comput. Appl. Math. 77 (1997), no. 1-2, 277–287. ROLLS Symposium (Leipzig, 1996). MR 1440013, DOI 10.1016/S0377-0427(96)00131-8
Tongke Wang, Na Li, and Guanghua Gao, The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities, Int. J. Comput. Math. 92 (2015), no. 3, 579–590. MR 3298070, DOI 10.1080/00207160.2014.902447
Shuhuang Xiang and Folkmar Bornemann, On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity, SIAM J. Numer. Anal. 50 (2012), no. 5, 2581–2587. MR 3022232, DOI 10.1137/120869845