Rates of convergence of Gauss, Lobatto, and Radau integration rules for singular integrands

Author:
Philip Rabinowitz

Journal:
Math. Comp. **47** (1986), 625-638

MSC:
Primary 65D30; Secondary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856707-6

MathSciNet review:
856707

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Abstract: Rates of convergence (or divergence) are obtained in the application of Gauss, Lobatto, and Radau integration rules to functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a generalized Jacobi weight function on , the error in applying an *n*-point rule to is , if and if , provided we avoid the singularity. If we ignore the singularity *y*, the error is for almost all choices of *y*. These assertions are sharp with respect to order.

**[1]**G. Freud,*Orthogonal Polynomials*, Pergamon Press, New York, 1966.**[2]**D. S. Lubinsky & P. Rabinowitz, "Rates of convergence of Gaussian quadrature for singular integrands,"*Math. Comp.*, v. 43, 1984, pp. 219-242. MR**744932 (86b:65018)****[3]**P. Nevai, "Mean convergence of Lagrange interpolation. III,"*Trans. Amer. Math. Soc.*, v. 282, 1984, pp. 669-698. MR**732113 (85c:41009)****[4]**P. Rabinowitz, "Ignoring the singularity in numerical integration," in*Topics in Numerical Analysis III*(J. J. H. Miller, ed.), Academic Press, London, 1977, pp. 361-368. MR**0656727 (58:31750)****[5]**P. Rabinowitz, "Numerical integration in the presence of an interior singularity,"*J. Comput. Appl. Math.*(To appear.) MR**884259 (88e:65022)****[6]**G. Szegö,*Orthogonal Polynomials*, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959.

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0856707-6

Article copyright:
© Copyright 1986
American Mathematical Society