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Mathematics of Computation

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

 

 

Adaptive integration and improper integrals


Author: Seymour Haber
Journal: Math. Comp. 29 (1975), 806-809
MSC: Primary 65D30
DOI: https://doi.org/10.1090/S0025-5718-1975-0375750-X
MathSciNet review: 0375750
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Abstract: Let R be the class of all functions that are properly Riemann-integrable on [0, 1], and let IR be the class of all functions that are properly Riemann-integrable on [a, 1] for all $ a > 0$ and for which

$\displaystyle \mathop {\lim }\limits_{a \to {0^+}} \int_a^1 {f(x)\;dx} $

exists and is finite. There are computational schemes that produce a convergent sequence of approximations to the integral of any function in R; the trapezoid rule is one. In this paper, it is shown that there is no computational scheme that uses only evaluations of the integrand, that is similarly effective for IR.

References [Enhancements On Off] (What's this?)

  • [1] J. R. RICE, A Metalgorithm for Adaptive Quadrature, CSDTR89, Purdue University, March, 1973.
  • [2] 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

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1975-0375750-X
Keywords: Integrals, improper integrals, numerical analysis, numerical integration, numerical quadrature, quadrature, singularities, Riemann integral
Article copyright: © Copyright 1975 American Mathematical Society