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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Convergence properties of a class of product formulas for weakly singular integral equations
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by Giuliana Criscuolo, Giuseppe Mastroianni and Giovanni Monegato PDF
Math. Comp. 55 (1990), 213-230 Request permission


We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type \[ \int _{ - 1}^1 {K(x,y)f(x) dx,} \quad - 1 \leq y \leq 1,\] where the kernel $K(x,y)$ is weakly singular and the function $f(x)$ has singularities only at the endpoints $\pm 1$. In particular, when $K(x,y) = \log |x - y|$, $K(x,y) = |x - y{|^v}$, $v > - 1$, and $f(x)$ has algebraic singularities of the form ${(1 \pm x)^\sigma }$, $\sigma > - 1$, we prove that the uniform rate of convergence of the rules is $O({m^{ - 2 - 2\sigma }}{\log ^2}m)$ in the case of the first kernel, and $O({m^{ - 2 - 2\sigma - 2v}}\log m)$ if $v \leq 0$, or $O({m^{ - 2 - 2\sigma }}\log m)$ if $v > 0$, for the second, where m is the number of points in the quadrature rule.
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Math. Comp. 55 (1990), 213-230
  • MSC: Primary 65R20; Secondary 45L05
  • DOI:
  • MathSciNet review: 1023045