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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



New Euler-Maclaurin expansions and their application to quadrature over the $ s$-dimensional simplex

Author: Elise de Doncker
Journal: Math. Comp. 33 (1979), 1003-1018
MSC: Primary 65B15; Secondary 65D32
MathSciNet review: 528053
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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.

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Article copyright: © Copyright 1979 American Mathematical Society