On ignoring the singularity in numerical quadrature

Author:
R. K. Miller

Journal:
Math. Comp. **25** (1971), 521-532

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1971-0301901-5

MathSciNet review:
0301901

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies the convergence of numerical quadratures of singular integrands. The singularities are ignored in the sense that whenever a singularity occurs the integrand is redefined to be zero. Several convergence theorems are proved under the assumption that the integrand can be dominated near each singularity by a monotone, integrable function.

**[1]**P. J. Davis & P. Rabinowitz, ``Ignoring the singularity in approximate integration,''*J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.*, v. 2, 1965, pp. 367-383. MR**33**#3459. MR**0195256 (33:3459)****[2]**P. Rabinowitz, ``Gaussian integration in the presence of a singularity,''*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 191-201. MR**35**#3881. MR**0213016 (35:3881)****[3]**W. Gautschi, ``Numerical quadrature in the presence of a singularity,''*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 357-362. MR**36**#1103. MR**0218014 (36:1103)****[4]**A. Sard,*Linear Approximation*, Math. Surveys, no. 9, Amer. Math. Soc., Providence, R.I., 1963. MR**28**#1429. MR**0158203 (28:1429)**

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/S0025-5718-1971-0301901-5

Keywords:
Quadrature,
singular quadrature,
singular integrals,
ignoring the singularity,
error bounds,
weak singularity

Article copyright:
© Copyright 1971
American Mathematical Society