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Mathematics of Computation

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An error estimate for Stenger’s quadrature formula

Authors: S. Beighton and B. Noble
Journal: Math. Comp. 38 (1982), 539-545
MSC: Primary 65D30
MathSciNet review: 645669
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Abstract: The basis of this paper is the quadrature formula \[ \int _{ - 1}^1 {f(x) dx \approx \log q\sum \limits _{m = - M}^M {\frac {{2{q^m}}}{{{{(1 + {q^m})}^2}}}f\left ( {\frac {{{q^m} - 1}}{{{q^m} + 1}}} \right )} ,} \] where $q = \exp (2h)$, h being a chosen step length. This formula has been derived from the Trapezoidal Rule formula by F. Stenger. An explicit form of the error is given for the case where the integrand has a factor of the form ${(1 - x)^\alpha }{(1 + x)^\beta },\alpha ,\beta > - 1$. Application is made to the evaluation of Cauchy principal value integrals with endpoint singularities and an appropriate error form is derived.

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Article copyright: © Copyright 1982 American Mathematical Society