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Mathematics of Computation
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
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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0375750-X
PII: S 0025-5718(1975)0375750-X
Keywords: Integrals, improper integrals, numerical analysis, numerical integration, numerical quadrature, quadrature, singularities, Riemann integral
Article copyright: © Copyright 1975 American Mathematical Society



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