Rates of convergence of Gaussian quadrature for singular integrands

Authors:
D. S. Lubinsky and P. Rabinowitz

Journal:
Math. Comp. **43** (1984), 219-242

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1984-0744932-2

MathSciNet review:
744932

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Abstract: The authors obtain the rates of convergence (or divergence) of Gaussian quadrature on 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 bounded smooth weight function on , the error in *n*-point Gaussian quadrature of 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]**M. M. Chawla and M. K. Jain,*Asymptotic error estimates for the Gauss quadrature formula*, Math. Comp.**22**(1968), 91–97. MR**0223094**, https://doi.org/10.1090/S0025-5718-1968-0223094-5**[2]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR**0448814****[3]**Philip J. Davis and Philip Rabinowitz,*Ignoring the singularity in approximate integration*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 367–383. MR**0195256****[4]**M. E. A. el-Tom,*On ignoring the singularity in approximate integration*, SIAM J. Numer. Anal.**8**(1971), 412–424. MR**0293852**, https://doi.org/10.1137/0708039**[5]**Alan Feldstein and Richard K. Miller,*Error bounds for compound quadrature of weakly singular integrals*, Math. Comp.**25**(1971), 505–520. MR**0297127**, https://doi.org/10.1090/S0025-5718-1971-0297127-4**[6]**G. Freud,*Orthogonal Polynomials*, Pergamon Press, New York, 1966.**[7]**Walter Gautschi,*Numerical quadrature in the presence of a singularity*, SIAM J. Numer. Anal.**4**(1967), 357–362. MR**0218014**, https://doi.org/10.1137/0704031**[8]**D. S. Lubinsky and Avram Sidi,*Convergence of product integration rules for functions with interior and endpoint singularities over bounded and unbounded intervals*, Math. Comp.**46**(1986), no. 173, 229–245. MR**815845**, https://doi.org/10.1090/S0025-5718-1986-0815845-4**[9]**R. K. Miller,*On ignoring the singularity in numerical quadrature*, Math. Comp.**25**(1971), 521–532. MR**0301901**, https://doi.org/10.1090/S0025-5718-1971-0301901-5**[10]**Charles F. Osgood and Oved Shisha,*Numerical quadrature of improper integrals and the dominated integral*, J. Approximation Theory**20**(1977), no. 1, 139–152. MR**0448823****[11]**Philip Rabinowitz,*Gaussian integration in the presence of a singularity*, SIAM J. Numer. Anal.**4**(1967), 191–201. MR**0213016**, https://doi.org/10.1137/0704018**[12]**Philip Rabinowitz,*Error bounds in Gaussian integration of functions of low-order continuity*, Math. Comp.**22**(1968), 431–434. MR**0226861**, https://doi.org/10.1090/S0025-5718-1968-0226861-7**[13]**Philip Rabinowitz,*Ignoring the singularity in numerical integration*, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 361–368. MR**0656727****[14]**P. Rabinowitz, "Gaussian integration of functions with branch point singularities,"*Internat. J. Comput. Math.*, v. 2, 1970, pp. 297-306.**[15]**Philip Rabinowitz and Ian H. Sloan,*Product integration in the presence of a singularity*, SIAM J. Numer. Anal.**21**(1984), no. 1, 149–166. MR**731219**, https://doi.org/10.1137/0721010**[16]**Theodore J. Rivlin,*An introduction to the approximation of functions*, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1969. MR**0249885****[17]**Vladimir G. Sprindžuk,*Metric theory of Diophantine approximations*, V. H. Winston & Sons, Washington, D.C.; A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979. Translated from the Russian and edited by Richard A. Silverman; With a foreword by Donald J. Newman; Scripta Series in Mathematics. MR**548467****[18]**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/S0025-5718-1984-0744932-2

Article copyright:
© Copyright 1984
American Mathematical Society