Asymptotic expansions for product integration

Authors:
Frank de Hoog and Richard Weiss

Journal:
Math. Comp. **27** (1973), 295-306

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1973-0329207-0

MathSciNet review:
0329207

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A generalized Euler-Maclaurin sum formula is established for product integration based on piecewise Lagrangian interpolation. The integrands considered may have algebraic or logarithmic singularities. The results are used to obtain accurate convergence rates of numerical methods for Fredholm and Volterra integral equations with singular kernels.

**[1]**K. E. Atkinson, "The numerical solution of Fredholm integral equations of the second kind,"*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 337-348. MR**36**#7358. MR**0224314 (36:7358)****[2]**C. T. H. Baker & G. S. Hodgson, "Asymptotic expansions for integration formulas in one or more dimensions,"*SIAM J. Numer. Anal.*, v. 8, 1971, pp. 473-480. MR**44**#2339. MR**0285115 (44:2339)****[3]**P. Linz, "Numerical methods for Volterra integral equations with singular kernels,"*SIAM J. Numer. Anal.*, v. 6, 1969, pp. 365-374. MR**41**#4850. MR**0260222 (41:4850)****[4]**J. N. Lyness & B. W. Ninham, "Numerical quadrature and asymptotic expansions,"*Math. Comp.*, v. 21, 1967, pp. 162-178. MR**37**#1081. MR**0225488 (37:1081)****[5]**B. W. Ninham & J. N. Lyness, "Further asymptotic expansions for the error functional,"*Math. Comp.*, v. 23, 1969, pp. 71-83. MR**39**#3682. MR**0242351 (39:3682)****[6]**E. T. Whittaker & G. N. Watson,*A Course of Modern Analysis*, Cambridge Univ. Press, New York, 1958. MR**1424469 (97k:01072)**

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/S0025-5718-1973-0329207-0

Keywords:
Product integration,
asymptotic expansion,
Euler-Maclaurin sum formula,
integral equation

Article copyright:
© Copyright 1973
American Mathematical Society