Abstract
Garloff, Solak and Szydelko [1] have proposed a modified trapezoidal rule quadrature with correction terms involving two points outside the range of integration. The rule is exact for cubics. We show:
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1)
There is an arbitrary factor in their formula but it would require a complex value to make the quadrature exact for quintics.
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2)
An analogous quadrature can be based on the midpoint rule.
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3)
By combining the two rules the quadrature is made exact for quintics. Furthermore the procedure is progressive so that the number of divisions can be doubled without wasting function evaluations.
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4)
The method can be used for integrands which are not real beyond an endpoint such as (x 1/2 and (xlnx which vanish at the endpoints by defining (f (endpoint-(x) as -(f (endpoint+(x).
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References
J. Garloff, W. Solak and Z. Szydelko, New integration formulas which use nodes outside the integration interval, J. Franklin Institute, 321 (1986), pp. 115–126.
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© 1987 D. Reidel Publishing Company
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Squire, W. (1987). Quadrature Rules with End-Point Corrections: Comments on a Paper by Garloff, Solak and Szydelko. In: Keast, P., Fairweather, G. (eds) Numerical Integration. NATO ASI Series, vol 203. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3889-2_11
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DOI: https://doi.org/10.1007/978-94-009-3889-2_11
Publisher Name: Springer, Dordrecht
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