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

HTML articles powered by AMS MathViewer

- by Avram Sidi PDF
- Math. Comp.
**78**(2009), 1593-1612 Request permission

## 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

- Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR**0167642** - 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** - 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 - Peter R. Johnston,
*Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals*, Internat. J. Numer. Methods Engrg.**45**(1999), no. 10, 1333–1348. MR**1699754**, DOI 10.1002/(SICI)1097-0207(19990810)45:10<1333::AID-NME632>3.3.CO;2-H - Peter R. Johnston,
*Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals*, Internat. J. Numer. Methods Engrg.**47**(2000), no. 10, 1709–1730. MR**1750249**, DOI 10.1002/(SICI)1097-0207(20000410)47:10<1709::AID-NME852>3.0.CO;2-V - N. M. Korobov,
*Teoretiko-chislovye metody v priblizhennom analize*, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR**0157483** - 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 - G. Monegato and I. H. Sloan,
*Numerical solution of the generalized airfoil equation for an airfoil with a flap*, SIAM J. Numer. Anal.**34**(1997), no. 6, 2288–2305. MR**1480381**, DOI 10.1137/S0036142995295054 - Masatake Mori,
*An IMT-type double exponential formula for numerical integration*, Publ. Res. Inst. Math. Sci.**14**(1978), no. 3, 713–729. MR**527197**, DOI 10.2977/prims/1195188835 - T. W. Sag and G. Szekeres,
*Numerical evaluation of high-dimensional integrals*, Math. Comp.**18**(1964), 245–253. MR**165689**, DOI 10.1090/S0025-5718-1964-0165689-X - Avram Sidi,
*A new variable transformation for numerical integration*, Numerical integration, IV (Oberwolfach, 1992) Internat. Ser. Numer. Math., vol. 112, Birkhäuser, Basel, 1993, pp. 359–373. MR**1248416** - Avram Sidi,
*Extension of a class of periodizing variable transformations for numerical integration*, Math. Comp.**75**(2006), no. 253, 327–343. MR**2176402**, DOI 10.1090/S0025-5718-05-01773-4 - Avram Sidi,
*A novel class of symmetric and nonsymmetric periodizing variable transformations for numerical integration*, J. Sci. Comput.**31**(2007), no. 3, 391–417. MR**2320555**, DOI 10.1007/s10915-006-9110-z - 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,
*Further extension of a class of periodizing variable transformations for numerical integration*, J. Comput. Appl. Math.**221**(2008), no. 1, 132–149. MR**2458757**, DOI 10.1016/j.cam.2007.10.009

## Additional Information

**Avram Sidi**- Affiliation: Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel
- Email: asidi@cs.technion.ac.il
- Received by editor(s): March 3, 2008
- Received by editor(s) in revised form: July 28, 2008
- Published electronically: January 22, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**78**(2009), 1593-1612 - MSC (2000): Primary 40A25, 41A60, 65B15, 65D30, 65D32
- DOI: https://doi.org/10.1090/S0025-5718-09-02203-0
- MathSciNet review: 2501065