New Euler-Maclaurin expansions and their application to quadrature over the $s$-dimensional simplex
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- by Elise de Doncker PDF
- Math. Comp. 33 (1979), 1003-1018 Request permission
Abstract:
The $\mu$-panel offset trapezodial rule for noninteger values of $\mu$, is introduced in a one-dimensional context. An asymptotic series describing the error functional is derived. The values of $\mu$ for which this is an even Euler-Maclaurin expansion are determined, together with the conditions under which it terminates after a finite number of terms. This leads to a new variant of one-dimensional Romberg integration. The theory is then extended to quadrature over the s-dimensional simplex, the basic rules being obtained by an iterated use of one-dimensional rules. The application to Romberg integration is discussed, and it is shown how Romberg integration over the simplex has properties analogous to those for standard one-dimensional Romberg integration and Romberg integration over the hypercube. Using extrapolation, quadrature rules for the s-simplex can be generated, and a set of formulas can be obtained which are the optimum so far discovered in the sense of requiring fewest function values to obtain a specific polynomial degree.References
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Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 1003-1018
- MSC: Primary 65B15; Secondary 65D32
- DOI: https://doi.org/10.1090/S0025-5718-1979-0528053-8
- MathSciNet review: 528053