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Numerical Methods

Germund Dahlquist and Åke Björck
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This book gives wonderfully clear coverage of most topics in numerical analysis. One of its best features is a wealth of examples, illustrating both the normal cases and the pitfalls. The present book is a Dover 2003 unaltered reprint of the 1974 Prentice-Hall edition.

The book is pitched as a graduate-level text, although I think this is due to its size and broad coverage and not because it is so difficult. The prerequisites are fairly low, and it does not assume a previous course in numerical analysis.

Despite its age, the book is still up-to-date as far as techniques and coverage are concerned. The biggest weakness today is that there is very little use of computers; most modern courses would focus on using existing computer packages, such as Maple, Mathematica, or MATLAB, but none of these existed when the present book was written. The book does often ask for FORTRAN or Algol programs to work particular problems, and gives a few examples. The numerical methods themselves are still perfectly good, but the way you use them day-to-day has completely changed. This is particularly conspicuous in the examples, which are all geared toward hand or calculator work and find an answer to 5 or 6 digits. There’s also only a moderate amount of discussion of round-off error, and the discussion of machine arithmetic is very general (necessarily so, because the IEEE-754 floating point standard had not come out yet and every computer did floating point differently). Most modern books would also deal with mathematical modeling, but that is absent from the present book; all the problems are presented in isolation. By present-day standards the book is weak on numerical linear algebra, a subject that has become much more important with the rise of computers.

This is a “proofs” book and all theorems are proved. The exercises, which are numerous and well-chosen, nearly always ask the answers to particular problems and not for proofs. All exercises are answered in the back of the book, although often only the final numerical answer is given.

The authors began a much-updated two-volume revision of this book, titled Numerical Methods in Scientific Computing. Co-author Dahlquist died during the preparation of the first volume, and that is the only volume that has been published (in 2008). That volume covers about the first half of the present work, although without linear algebra.

Bottom line: Probably not a good choice for a course text, due to the lack of computers, but still very valuable as a reference.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.


Preface. Conventions.
1. Some General Principles of Numerical Calculation.
2. How to Obtain and Estimate Accuracy in Numerical Calculations.
3. Numerical Uses of Series.
4. Approximation of Functions.
5. Numerical Linear Algebra.
6. Nonlinear Equations.
7. Finite Differences with Applications to Numerical Integration, Differentiation and Interpolation.
8. Differential Equations.
9. Fourier Methods.
10. Optimization.
11. The Monte Carlo Method and Simulation.
12. Solutions to Problems.
13. Bibliography and Published Algorithms.
  Appendix Tables. Index.