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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 & M. K. Jain, "Error estimates for the Gauss quadrature formula,"*Math. Comp.*, v. 22, 1980, pp. 91-97. MR**0223094 (36:6143)****[2]**P. J. Davis & P. Rabinowitz,*Methods of Numerical Integration*, Academic Press, New York, 1975. MR**0448814 (56:7119)****[3]**P. J. Davis & P. Rabinowitz, "Ignoring the singularity in approximate integration,"*SIAM J. Numer. Anal.*, v. 2, 1965, pp. 367-383. MR**0195256 (33:3459)****[4]**M. A. El-Tom, "On ignoring the singularity in approximate integration,"*SIAM J. Numer. Anal.*, v. 8, 1971, pp. 412-424. MR**0293852 (45:2928)****[5]**A. Feldstein & R. K. Miller, "Error bounds for compound quadrature of weakly singular integrals,"*Math. Comp.*, v. 25, 1971, pp. 505-520. MR**0297127 (45:6185)****[6]**G. Freud,*Orthogonal Polynomials*, Pergamon Press, New York, 1966.**[7]**W. Gautschi, "Numerical quadrature in the presence of a singularity,"*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 357-362. MR**0218014 (36:1103)****[8]**D. S. Lubinsky & A. Sidi,*Convergence of Product Integration Rules for Functions with Interior and Endpoint Singularities over Bounded and Unbounded Intervals*, Technion Computer Science Preprint No. 215, Technion, Haifa, 1981. MR**815845 (87j:41072)****[9]**R. K. Miller, "On ignoring the singularity in numerical quadrature."*Math. Comp.*, v. 25, 1971, pp. 521-532. MR**0301901 (46:1056)****[10]**C. F. Osgood & O. Shisha, "Numerical quadrature of improper integrals and the dominated integral,"*J. Approx. Theory*, v. 20, 1977, pp. 139-152. MR**0448823 (56:7128)****[11]**P. Rabinowitz, "Gaussian integration in the presence of a singularity,"*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 197-201. MR**0213016 (35:3881)****[12]**P. Rabinowitz, "Error in Gaussian integration of functions of low order continuity,"*Math. Comp.*, v. 22, 1968, pp. 431-434. MR**0226861 (37:2447)****[13]**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)****[14]**P. Rabinowitz, "Gaussian integration of functions with branch point singularities,"*Internat. J. Comput. Math.*, v. 2, 1970, pp. 297-306.**[15]**P. Rabinowitz & I. H. Sloan, "Product integration in the presence of a singularity,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 149-166. MR**731219 (85c:65023)****[16]**T. J. Rivlin,*An Introduction to the Approximation of Functions*, Blaisdell, Waltham, Mass., 1969. MR**0249885 (40:3126)****[17]**V. G. Sprindzuk, (transl. R. A. Silverman),*Metric Theory of Diophantine Approximations*, Winston-Wiley, Washington, D.C., 1969. MR**548467 (80k:10048)****[18]**G. Szegö,*Orthogonal Polynomials*, rev. ed., Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, R.I., 1959.

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

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

Article copyright:
© Copyright 1984
American Mathematical Society