An error estimate for Stenger’s quadrature formula
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- by S. Beighton and B. Noble PDF
- Math. Comp. 38 (1982), 539-545 Request permission
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.References
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685
- Frank Stenger, Integration formulae based on the trapezoidal formula, J. Inst. Math. Appl. 12 (1973), 103–114. MR 381261, DOI 10.1093/imamat/12.1.103
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp. 38 (1982), 539-545
- MSC: Primary 65D30
- DOI: https://doi.org/10.1090/S0025-5718-1982-0645669-9
- MathSciNet review: 645669