Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Variable transformations and Gauss-Legendre quadrature for integrals with endpoint singularities

Author: Avram Sidi
Journal: Math. Comp. 78 (2009), 1593-1612
MSC (2000): Primary 40A25, 41A60, 65B15, 65D30, 65D32
Published electronically: January 22, 2009
MathSciNet review: 2501065
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Gauss-Legendre quadrature formulas have excellent convergence properties when applied to integrals $ \int^1_0f(x) dx$ with $ f\in C^\infty[0,1]$. However, their performance deteriorates when the integrands $ f(x)$ are in $ C^\infty(0,1)$ but are singular at $ x=0$ and/or $ x=1$. One way of improving the performance of Gauss-Legendre quadrature in such cases is by combining it with a suitable variable transformation such that the transformed integrand has weaker singularities than those of $ f(x)$. Thus, if $ x=\psi(t)$ is a variable transformation that maps $ [0,1]$ onto itself, we apply Gauss-Legendre quadrature to the transformed integral $ \int^1_{0}f(\psi(t))\psi'(t) dt$, whose singularities at $ t=0$ and/or $ t=1$ are weaker than those of $ f(x)$ at $ x=0$ and/or $ x=1$. In this work, we first define a new class of variable transformations we denote $ \widetilde{\mathcal{S}}_{p,q}$, where $ p$ and $ q$ are two positive parameters that characterize it. We also give a simple and easily computable representative of this class. Next, by invoking some recent results by the author concerning asymptotic expansions of Gauss-Legendre quadrature approximations as the number of abscissas tends to infinity, we present a thorough study of convergence of the combined approximation procedure, with variable transformations from $ \widetilde{\mathcal{S}}_{p,q}$. We show how optimal results can be obtained by adjusting the parameters $ p$ and $ q$ of the variable transformation in an appropriate fashion. We also give numerical examples that confirm the theoretical results.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I.A. Stegun.
    Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
    Number 55 in Nat. Bur. Standards Appl. Math. Series. US Government Printing Office, Washington, D.C., 1964. MR 0167642 (29:4914)
  • 2. P.J. Davis and P. Rabinowitz.
    Methods of Numerical Integration.
    Academic Press, New York, second edition, 1984. MR 760629 (86d:65004)
  • 3. M. Iri, S. Moriguti, and Y. Takasawa.
    On a certain quadrature formula.
    Kokyuroku of Res. Inst. for Math. Sci. Kyoto Univ., 91:82-118, 1970.
    In Japanese. English translation in J. Comp. Appl. Math., 17:3-20, 1987. MR 884257 (88j:65057)
  • 4. P.R. Johnston.
    Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals.
    Intern. J. Numer. Methods Engrg., 45:1333-1348, 1999. MR 1699754 (2000c:74101)
  • 5. P.R. Johnston.
    Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals.
    Intern. J. Numer. Methods Engrg., 47:1709-1730, 2000. MR 1750249
  • 6. N.M. Korobov.
    Number-Theoretic Methods of Approximate Analysis.
    GIFL, Moscow, 1963.
    In Russian. MR 0157483 (28:716)
  • 7. G. Monegato and L. Scuderi.
    Numerical integration of functions with boundary singularities.
    J. Comp. Appl. Math., 112:201-214, 1999. MR 1728460
  • 8. G. Monegato and I.H. Sloan.
    Numerical solution of the generalized airfoil equation for an airfoil with a flap.
    SIAM J. Numer. Anal., 34:2288-2305, 1997. MR 1480381 (98f:45009)
  • 9. M. Mori.
    An IMT-type double exponential formula for numerical integration.
    Publ. Res. Inst. Math. Sci. Kyoto Univ., 14:713-729, 1978. MR 527197 (81c:65012)
  • 10. T.W. Sag and G. Szekeres.
    Numerical evaluation of high-dimensional integrals.
    Math. Comp., 18:245-253, 1964. MR 0165689 (29:2969)
  • 11. A. Sidi.
    A new variable transformation for numerical integration.
    In H. Brass and G. Hämmerlin, editors, Numerical Integration IV, number 112 in ISNM, pages 359-373, Basel, 1993. Birkhäuser. MR 1248416 (94k:65032)
  • 12. A. Sidi.
    Extension of a class of periodizing variable transformations for numerical integration.
    Math. Comp., 75:327-343, 2006. MR 2176402 (2006g:41066)
  • 13. A. Sidi.
    A novel class of symmetric and nonsymmetric periodizing variable transformations for numerical integration.
    J. Sci. Comput., 31:391-417, 2007. MR 2320555 (2008f:65046)
  • 14. A. Sidi.
    Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities.
    Math. Comp., 78:241-253, 2009. MR 2448705
  • 15. A. Sidi.
    Further extension of a class of periodizing variable transformations for numerical integration.
    J. Comp. Appl. Math., 221:132-149, 2008. MR 2458757

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 40A25, 41A60, 65B15, 65D30, 65D32

Retrieve articles in all journals with MSC (2000): 40A25, 41A60, 65B15, 65D30, 65D32

Additional Information

Avram Sidi
Affiliation: Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel

Keywords: Variable transformations, Gauss--Legendre quadrature, singular integrals, endpoint singularities, asymptotic expansions, Euler--Maclaurin expansions.
Received by editor(s): March 3, 2008
Received by editor(s) in revised form: July 28, 2008
Published electronically: January 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society