Numerical Integration: Recent Developments, Software and Applications

This volume contains refereed papers and extended abstracts of papers presented at the NATO Advanced Research Workshop entitled 'Numerical Integration: Recent Developments, Software and Applications', held at Dalhousie University, Halifax, Canada, August 11-15, 1986. The Workshop was attended by thirty-six scientists from eleven NATO countries. Thirteen invited lectures and twenty-two contributed lectures were presented, of which twenty-five appear in full in this volume, together with extended abstracts of the remaining ten. It is more than ten years since the last workshop of this nature was held, in Los Alamos in 1975. Many developments have occurred in quadrature in the intervening years, and it seemed an opportune time to bring together again researchers in this area. The development of QUADPACK by Piessens, de Doncker, Uberhuber and Kahaner has changed the focus of research in the area of one dimensional quadrature from the construction of new rules to an emphasis on reliable robust software. There has been a dramatic growth in interest in the testing and evaluation of software, stimulated by the work of Lyness and Kaganove, Einarsson, and Piessens. The earlier research of Patterson into Kronrod extensions of Gauss rules, followed by the work of Monegato, and Piessens and Branders, has greatly increased interest in Gauss-based formulas for one-dimensional integration.

Table of Contents:
Theoretical Aspects of One-Dimensional Quadrature.- The convergence of noninterpolatory product integration rules.- Some quadrature rules for finite trigonometric and related integrals.- Modified Clenshaw-Curtis integration and applications to numerical computation of integral transforms.- Fast generation of quadrature rules with some special properties.- Quadrature methods for the determination of zeros of transcendental functions - a review.- Computation of the index of an analytic function.- Approximate computation of strongly oscillatory integrals with compound precision.- Bounds and approximations for the zeros of classical orthogonal polynomials. Their use in generating quadrature rules..- Indefinite integration formulas based on the sinc expansion.- On positive quadrature rules.- Quadrature rules with end-point corrections - comments on a paper by Garloff, Solak and Szydelko.- Comparison of Gauss-Hermite and midpoint quadrature with application to the Voigt function.- Theoretical Aspects of Multiple Quadrature.- On sequences of imbedded integration rules.- Asymptotic expansions and their applications in numerical integration.- Construction of three-dimensional cubature formulae with points on regular and semi-regular polytopes.- Construction of sequences of embedded cubature formulae for circular symmetric planar regions.- On the construction of higher degree three dimensional embedded integration rules.- Fully symmetric integration rules for the unit four-cube.- On the construction of cubature formulae with few nodes using Groebner bases.- Quasi-random sequences for optimization and numerical integration.- Non-fortuitous, non-product, non-fully symmetric cubature structures.- Algorithms, Software and Applications.- A parallelization of adaptive integration methods.- The numerical evaluation of multiple integrals on parallel computers.- Vectorisation of one dimensional quadrature codes.- Performance of self-validating adaptive quadrature.- Using weight functions in self-validating quadrature.- On the construction of a practical Ermakov-Zolotukhin multiple integrator.- Interactive numerical quadrature.- The design of a generic quadrature library in Ada.- The integration of the multivariate normal density function for the triangular method.- Numerical integration in scalar wave scattering, with application to acoustic scattering by fish.- Software Classification and Testing.- Testing multiple integrators.- A package for testing multiple integration subroutines.- A test of some well known one dimensional general purpose automatic quadrature routines.- Development of useful quadrature software, with particular emphasis on microcomputers.- Appendix I: Final Program.- Appendix II: Discussion.- List of Participants.- List of Contributors.