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Methods of Numerical Integration

Philip J. Davis and Philip Rabinowitz
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This book is an annotated catalog or encyclopedia of techniques in numerical integration, with the annotations giving particular attention to the types of function for which each method is best suited. It deals with integration only in the sense of evaluating integrals, and does not discuss numerical solution of differential equations. It concentrates on one-dimensional integrals, with one chapter on multiple integrals, and the treatment of Monte Carlo (sampling) methods is relatively weak. The chapter on error analysis is especially good, particularly since practitioners tend to take results on faith without any error analysis at all.

Like all good numerical analysis books, this one emphasizes that thinking should precede calculating: “Whenever possible, a problem should be analyzed and put into a proper form before it is run on a computer.’ (p. 5) Scattered throughout the book are examples of transformations that make an integral more tractable numerically. The book emphasizes the need to characterize the function before picking an integration method. The characterizations include not only an analysis of the singularities, but also broad behavioral characteristics such as whether the function is periodic, rapidly oscillatory, or has some special property such as satisfying a differential equation.

The book is an unaltered reprint of the 1984 second edition, but is still very up-to-date. For example, it covers all the integration methods used in Mathematica except for Duffy’s coordinates, and it is much more thorough (in its special subject of integration) than most numerical analysis texts. This is partly because it omits nearly all the proofs; the reader is referred to texts or original papers that contain the proof. The bibliography is enormous.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  • Preface to First Edition
  • Preface to Second Edition
  • Chapter 1. Introduction
    1. Why Numerical Integration?
    2. Formal Differentiation and Integration on Computers
    3. Numerical Integration and Its Appeal in Mathematics
    4. Limitations of Numerical Integration
    5. The Riemann Integral
    6. Improper Integrals
    7. The Riemann Integral in Higher Dimensions
    8. More General Integrals
    9. The Smoothness of Functions and ApproximateIntegration
    10. Weight Functions
    11. Some Useful Formulas
    12. Orthogonal Polynomials
    13. Short Guide to the Orthogonal Polynomials
    14. Some Sets of Polynomials Orthogonal over Figures in the Complex Plane
    15. Extrapolation and Speed-Up
    16. Numerical Integration and the Numerical Solution of Integral Equations
  • Chapter 2. Approximate Integration Over a Finite Interval
    1. Primitive Rules
    2. Simpson's Rule
    3. Nonequally Spaced Abscissas
    4. Compound Rules
    5. Integration Formulas of Interpolatory Type
    6. Integration Formulas of Open Type
    7. Integration Rules of Gauss Type
    8. Integration Rules Using Derivative Data
    9. Integration of Periodic Functions
    10. Integration of Rapidly Oscillatory Functions
    11. Contour Integrals
    12. Improper Integrals (Finite Interval)
    13. Indefinite Integration
  • Chapter 3. Approximate Integration Over Infinite Intervals
    1. Change of Variable
    2. Proceeding to the Limit
    3. Truncation of the Infinite Interval
    4. Primitive Rules for the Infinite Interval
    5. Formulas of Interpolatory Type
    6. Gaussian Formulas for the Infinite Interval
    7. Convergence of Formulas of Gauss Type for Singly and Doubly Infinite Intervals
    8. Oscillatory Integrands
    9. The Fourier Transform
    10. The Laplace Transform and Its Numerical Inversion
  • Chapter 4. Error Analysis
    1. Types of Errors
    2. Roundoff Error for a Fixed Integration Rule
    3. Truncation Error
    4. Special Devices
    5. Error Estimates through Differences
    6. Error Estimates through the Theory of Analytic Functions
    7. Application of Functional Analysis to Numerical Integration
    8. Errors for Integrands with Low Continuity
    9. Practical Error Estimation
  • Chapter 5. Approximate Integration in Two or More Dimensions
    1. Introduction
    2. Some Elementary Multiple Integrals over Standard Regions
    3. Change of Order of Integration
    4. Change of Variables
    5. Decomposition into Elementary Regions
    6. Cartesian Products and Product Rules
    7. Rules Exact for Monomials
    8. Compound Rules
    9. Multiple Integration by Sampling
    10. The Present State of the Art
  • Chapter 6. Automatic Integration
    1. The Goals of Automatic Integration
    2. Some Automatic Integrators
    3. Romberg Integration
    4. Automatic Integration Using Tschebyscheff Polynomials
    5. Automatic Integration in Several Variables
    6. Concluding Remarks
  • Appendix 1: On the Practical Evaluation of Integrals (Milton Abramowitz)
  • Appendix 2: FORTRAN Programs
  • Appendix 3: Bibliography of ALGOL, FORTRAN, and PL/I Procedures
  • Appendix 4: Bibliography of Tables
  • Appendix 5: Bibliography of Books and Articles
  • Index