J. N. Lyness

The representation of lattice quadrature rules as multiple sums

We provide a classification of lattice rules. Applying elementary group theory, we assign to each s-dimensional lattice rule a rank m and a set of positive integer invariants n/sub 1/ n/sub 2/....,n/sub s/. The number ..nu..(Q) of abscissas required by the rule is the product n/sub 1/n/sub 2/...n/sub s/, and the rule may be expressed in a canonical form with m independent summations. Under this classification an N-point number-theoretic rule in the sense of Korobov and Conroy is a rank m = 1 rule having invariants N, 1,1,...,1, and the product trapezoidal rule using n/sup s/ points in rank m = s rule having invariants n,n...,n. Besides providing a canonical form, we give some of the properties of copy rules and of projections into lower dimensions.

Numerical integration : recent developments, software and applications

Numerical integration rules numerical integration error analysis numerical integration applications numerical integration algorithms and software. Appendix: Final program.

Algorithm 662: A Fortran software package for the numerical inversion of the Laplace transform based on Weeks' method

Here we present a brief documentation of the software package WEEKS written in FORTRAN 77. The mathematical background and general information about its performance are described in the accompanying paper on pages 163-170 of this issue. Further information of a theoretical nature may be found in [3]. Some of the design characteristics of this package are described in detail in a preliminary report [2]. However, there are minor differences between the implementation described there and that described here.

Some generalizations of the Euler-Knopp transformation

SummaryThe purpose of this paper is to construct a generalization of the Euler-Knopp transformation. Using this, one may recover previously known transformations, derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well-known functions.

The Euler-Maclaurin expansion and finite-part integrals

Abstract. In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and

The Euler Maclaurin expansion for the Cauchy Principal Value integral

\overline{\overline{G}}(p)

Quadrature Rules for Regions Having Regular Hexagonal Symmetry

, the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer,

A search program for finding optimal integration lattices

\overline{\overline{G}}(p)

On the remainder term in theN-dimensional Euler Maclaurin expansion

is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.

Notes on integration and integer sublattices

SummaryThe classical Euler Maclaurin Summation Formula expresses the difference between a definite integral over [0, 1] and its approximation using the trapezoidal rule with step lengthh=1/m as an asymptotic expansion in powers ofh together with a remainder term. Many variants of this exist some of which form the basis of extrapolation methods such as Romberg Integration. in this paper a variant in which the integral is a Cauchy Principal Value integral is derived. The corresponding variant of the Fourier Coefficient Asymptotic Expansion is also derived. The possible role of the former in numerical quadrature is discussed.