Numerical Integration

1. Front Matter

   Pages i-xii

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2. Numerical Integration Rules

   1. A Survey of Methods for Constructing Cubature Formulae

      • Ronald Cools

      Pages 1-24

   2. Linear Representations of Finite Groups and The Ideal Theoretical Construction of G-Invariant Cubature Formulas

      • Karin Gatermann

      Pages 25-35

   3. On the Number of Nodes of Odd Degree Cubature Formulae for Integrals with Jacobi Weights on a Simplex

      • H. Berens, H. J. Schmid

      Pages 37-44

   4. On Quadrature Formulae Near Gaussian Quadrature

      • K.-J. Förster

      Pages 45-54

   5. Numerical Integration in High Dimensions — the Lattice Rule Approach

      • Ian H. Sloan

      Pages 55-69

   6. Existence Theorems for Efficient Lattice Rules

      • Harald Niederreiter

      Pages 71-80

   7. Sinc Quadratures for Cauchy Principal Value Integrals

      • Bernard Bialecki

      Pages 81-92

   8. Interpolatory Product Integration in the Presence of Singularities: L _p Theory

      • Philip Rabinowitz, William E. Smith

      Pages 93-109

   9. The Numerical Evaluation of Definite Integrals Affected by Singularities Near the Interval of Integration

      • D. B. Hunter

      Pages 111-120

   10. Application of Computer Algebra Software to the Derivation of Numerical Integration Rules for Singular and Hypersingular Integrals

       • Nikolaos I. Ioakimidis

       Pages 121-131

3. Numerical Integration Error Analysis

   1. Remainder Estimates for Analytic Functions

      • Walter Gautschi

      Pages 133-145

   2. Error Bounds Based on Approximation Theory

      • H. Brass

      Pages 147-163

   3. One Sided L1-Approximation and Bounds for Peano Kernels

      • Knut Petras

      Pages 165-174

   4. An Algebraic Study of the Levin Transformation in Numerical Integration

      • Ricolindo Cariño, Ian Robinson, Elise de Doncker

      Pages 175-186

4. Numerical Integration Applications

   1. Developments in Solving Integral Equations Numerically

      • Günther Hämmerlin

      Pages 187-201

   2. Numerical Integration of Singular and Hypersingular Integrals in Boundary Element Methods

      • Christoph Schwab, Wolfgang L. Wendland

      Pages 203-218

   3. On Handling Singularities in Finite Elements

      • J. N. Lyness

      Pages 219-233

   4. A Robust Numerical Integration Method for 3-D Boundary Element Analysis and Its Error Analysis using Complex Function Theory

      • Ken Hayami

      Pages 235-248

   5. On the Numerical Calculation of Multidimensional Integrals Appearing in the Theory of Underwater Acoustics

      • Jarle Berntsen

      Pages 249-265

This volume contains refereed papers and extended abstracts of papers presented at the NATO Advanced Research Workshop entitled 'Numerical Integration: Recent Develop­ ments, Software and Applications', held at the University of Bergen, Bergen, Norway, June 17-21,1991. The Workshop was attended by thirty-eight scientists. A total of eight NATO countries were represented. Eleven invited lectures and twenty-three contributed lectures were presented, of which twenty-five appear in full in this volume, together with three extended abstracts and one note. The main focus of the workshop was to survey recent progress in the theory of methods for the calculation of integrals and show how the theoretical results have been used in software development and in practical applications. The papers in this volume fall into four broad categories: numerical integration rules, numerical integration error analysis, numerical integration applications and numerical integration algorithms and software. It is five years since the last workshop of this nature was held, at Dalhousie University in Halifax, Canada, in 1986. Recent theoretical developments have mostly occurred in the area of integration rule construction. For polynomial integrating rules, invariant theory and ideal theory have been used to provide lower bounds on the numbers of points for different types of multidimensional rules, and to help in structuring the nonlinear systems which must be solved to determine the points and weights for the rules. Many new optimal or near optimal rules have been found for a variety of integration regions using these techniques.

• Numerical integration
• STATISTICA
• algorithms
• cubature
• finite element method

• Department of Informatics, University of Bergen, Bergen, Norway

  Terje O. Espelid

• School of Electrical Engineering and Computer Science, Washington State University, Pullman, USA

  Alan Genz

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