New EulerMaclaurin expansions and their application to quadrature over the dimensional simplex
Author:
Elise de Doncker
Journal:
Math. Comp. 33 (1979), 10031018
MSC:
Primary 65B15; Secondary 65D32
MathSciNet review:
528053
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The panel offset trapezodial rule for noninteger values of , is introduced in a onedimensional context. An asymptotic series describing the error functional is derived. The values of for which this is an even EulerMaclaurin expansion are determined, together with the conditions under which it terminates after a finite number of terms. This leads to a new variant of onedimensional Romberg integration. The theory is then extended to quadrature over the sdimensional simplex, the basic rules being obtained by an iterated use of onedimensional 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 onedimensional Romberg integration and Romberg integration over the hypercube. Using extrapolation, quadrature rules for the ssimplex 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.
 [1]
Christopher
T. H. Baker and Graham
S. Hodgson, Asymptotic expansions for integration formulas in one
or more dimensions, SIAM J. Numer. Anal. 8 (1971),
473–480. MR 0285115
(44 #2339)
 [2]
Axel
Grundmann and H.
M. Möller, Invariant integration formulas for the
𝑛simplex by combinatorial methods, SIAM J. Numer. Anal.
15 (1978), no. 2, 282–290. MR 488881
(81e:41045), http://dx.doi.org/10.1137/0715019
 [3]
J.
N. Lyness, Quadrature over a simplex. I. A representation for the
integrand function, SIAM J. Numer. Anal. 15 (1978),
no. 1, 122–133. MR 0468118
(57 #7957)
 [4]
J.
N. Lyness and K.
K. Puri, The EulerMaclaurin expansion for the
simplex, Math. Comp. 27 (1973), 273–293. MR 0375752
(51 #11942), http://dx.doi.org/10.1090/S00255718197303757521
 [5]
A.
H. Stroud, Approximate calculation of multiple integrals,
PrenticeHall, Inc., Englewood Cliffs, N.J., 1971. PrenticeHall Series in
Automatic Computation. MR 0327006
(48 #5348)
 [1]
 C. T. H. BAKER & G. S. HODGSON, "Asymptotic expansions for integration formulas in one or more dimensions," SIAM J. Numer. Anal., v. 8, 1971, pp. 473480. MR 0285115 (44:2339)
 [2]
 A. GRUNDMANN & H. M. MÖLLER, "Invariant integration formulas for the nsimplex by combinatorial methods," SIAM J. Numer. Anal., v. IS, 1978, pp. 282290. MR 488881 (81e:41045)
 [3]
 J. N. LYNESS, "Quadrature over a simplex: Part 1. A representation of the integrand function," SIAM J. Numer. Anal., v. 15, 1978, pp. 122133. MR 0468118 (57:7957)
 [4]
 J. N. LYNESS & K. K. PURI, "The EulerMaclaurin expansion for the simplex," Math. Comp., v. 27, 1973, pp. 273293. MR 0375752 (51:11942)
 [5]
 A. H. STROUD, Approximate Calculation of Multiple Integrals, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0327006 (48:5348)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65B15,
65D32
Retrieve articles in all journals
with MSC:
65B15,
65D32
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905280538
PII:
S 00255718(1979)05280538
Article copyright:
© Copyright 1979
American Mathematical Society
