On the Structure of Fully Symmetric Multidimensional Quadrature Rules

Keast, P.; Lyness, J. N.

doi:10.1137/0716002

Cited by 29 publications

(27 citation statements)

References 3 publications

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“…There has, however, been a considerable effort [19][20][21] to find minimal integration rules for special regions such as the n-cube (C,), the n-sphere (S,), the surface of the n-sphere ( U n ) , and the n-simplex (T,) [an integration rule over region R, of degree d with N points integrates all monomials, over R, of degree d exists with less than N points]. We will use the special points given by Stroud [19], Lebedev [22], and McLaren [23] for integration over the surface of a three-dimensional sphere (ZJ3).…”

Section: Integration In More Dimensionsmentioning

confidence: 99%

Three‐dimensional numerical integration for electronic structure calculations

Boerrigter¹,

Velde²,

Baerends³

1988

Int J of Quantum Chemistry

775

388

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Two three-dimensional numerical integration schemes are presented for molecular integrands such as matrix elements of one-electron operators occurring in the Fock operator and expectation values of oneelectron operators describing molecular properties. The schemes are based on a judicious partitioning of space so that product-Gauss integration rules can be used in each region. Convergence with the number of integration points is such that very high accuracy (8-10 digits) may be obtained with a modest number of points. The use of point group symmetry to reduce the required number of points is discussed. Examples are given for overlap, nuclear potential, and electric field gradient integrals.

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Section: Integration In More Dimensionsmentioning

confidence: 99%

Three‐dimensional numerical integration for electronic structure calculations

Boerrigter¹,

Velde²,

Baerends³

1988

Int J of Quantum Chemistry

775

388

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“…These fully symmetric integration rules have received a considerable amount of attention in the numerical analysis literature (see e.g. Lyness, 1965;McNamee and Stenger, 1967;Rabinowitz and Richter, 1969;Mantel and Rabinowitz, 1977;Keast and Lyness, 1979;Genz, 1986). Here we give the basic definitions of fully symmetric integration rules and some of their important properties.…”

Section: Definitions and Notationmentioning

confidence: 99%

Positive embedded integration in Bayesian analysis

Δελλαπόρτας

Wright²

1991

Stat Comput

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This paper describes how embedded sequences of positive interpolatory integration rules (PIIRs) obtained from Gauss-Hermite product rules can be applied in Bayesian analysis. These embedded sequences are very promising for two major reasons. First, they provide a rich class of spatially distributed rules which are particularly useful in high dimensions. Second, they provide a way of producing more efficient integration strategies by enabling approximations to be updated sequentially through the addition of new nodes at each step rather than through changing to a completely new set of nodes. Moreover, as points are added successive rules change naturally from spatially distributed non-product rules to product rules. This feature is particularly attractive when the rules are used for the evaluation of marginal posterior densities. We illustrate the use of embedded sequences of PIIRs in two examples. These illustrate how embedded sequences can be applied to improve the efficiency of the adaptive integration strategy currently in use.

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“…For definitions of the concepts fully symmetric sets, regions and rules, see [1] or [6]. A representative of a fully symmetric set is called a generator.…”

Section: I[f] = I_ F~)dx__ Zwf~_) Q[f]mentioning

confidence: 99%