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Elements of Numerical Analysis

Radhey S. Gupta
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

This is a strongly methods-oriented introduction to numerical analysis at the advanced undergraduate or beginning graduate level. It is notable in at least two respects. It has a very wide range of topics for an introductory text, and it includes a level of detail in explanations and worked examples that is very unusual.

The topics that are treated range from very elementary to quite advanced. On the elementary side, for example, is basic information about binary numbers and computer representation of floating point numbers. The author explicitly goes through an example where he carries out the conversion of a decimal to its binary representation. From the advanced side is a short final chapter on free and moving boundary value problems (which is still an active research area). In between, virtually every major area of numerical analysis is treated. Besides the more or less standard introductory topics, the author includes chapters on the numerical solution of partial differential equations, integral equations and finite element methods. The second edition of the book has added a short chapter on Fourier series, the Fourier transform and the Fast Fourier Transform.

The author makes few if any references to actual computer implementations of any of the methods. Nor is there mention of any of commonly used software such as MATLAB or Mathematica. While there are disadvantages to this, it does at least focus attention on the methods and not the software. Throughout the book, the author is consistently intent on working through all the details. In the chapter on linear equations, for example, the he explicitly carries out all the steps of a reduction of a 4x4 matrix to LU form using Crout’s method. Later in the book, treating the computation of eigenvalues of symmetric matrices, he reduces a 4x4 matrix to tridiagonal form using Givens’ method — step by step over more than three pages. These are not exceptions; this is how virtually all the examples are presented. Where proofs are provided, they have a similar level of detail.

The methods-based approach the author uses has some drawbacks. In particular, it’s just too easy to lose track of the bigger picture in the proliferation of details. Little — and certainly insufficient — guidance is offered the reader about which methods work best for which problems. Often enough there are significantly better methods than the ones the author presents. It is disappointing, for example, to see little mention of matrix factorization and none at all of singular value decomposition in the chapters on linear algebra.

Having said that, I find that I genuinely like the book. The hands-on feeling it conveys is oddly appealing and offers a pleasing contrast to numerical analysis books that concentrate intensively on the use of software.

Each chapter has a modest number of exercises, and solutions of many are included in an appendix. Because the treatment of each topic is so well supported with examples, this book would work pretty well for self-study. However, I’d recommend that students use it in conjunction with another book (perhaps something like Numerical Recipes) that provides a broader context and more guidance for selection of appropriate methods.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1. Errors in computation
2. Linear equations and eigenvalue problem
3. Nonlinear equations
4. Interpolation
5. Numerical differentiation
6. Numerical integration
7. Ordinary differential equations
8. Splines and their applications
9. Method of least squares and Chebyshev approximation
10. Eigenvalues of symmetric matrices
11. Partial differential equations
12. Finite element method
13. Integral equations
14. Difference equations
15. Fourier series, discrete Fourier transform and fast Fourier transform
16. Free and moving boundary problems: a brief introduction