This is the third edition of a famous work on the basics of numerical analysis. It is a well-written textbook for advanced undergraduate/beginning graduate students containing both classical methods and modern approaches to numerical mathematics. The theory is illustrated by many interesting examples, and carefully selected exercises lead the reader to a better understanding of the topics discussed. References at the end of each chapter, and a list of monographs on numerical methods at the end of the book, motivate a deeper study of explained techniques.

The book starts with a discussion of the general effects of input and round-off errors on the result of a calculation. Interpolation is considered – in particular, trigonometric interpolation and interpolation by polynomials, rational functions, and splines. Here, B-splines are also treated, including their use in the context of multi-resolution methods. After a description of methods of numerical quadrature, direct methods for solving systems of linear equations are thoroughly discussed, including the Gaussian elimination, the Choleski decomposition, the simplex method, and orthogonalisation techniques with their use in solving linear least-squares problems, and elimination methods for sparse matrices are mentioned. The next chapter is concerned with iterative methods suitable for finding zeros and minimum points of a given function, and contains a detailed discussion of Newton’s method. The authors then describe various normal forms of matrices and several methods for reducing matrices to (tri)diagonal form or Hessenberg form and explain the main algorithms for computing eigenvalues and eigenvectors, including the LR and QR algorithms. A long chapter is devoted to numerical solutions of ordinary differential equations, where the authors treat initial-value problems (one-step and multistep methods) and boundary-value problems (simple and multiple shooting methods), and sensitivity analysis and handling of discontinuities are considered. The final chapter describes iterative methods for solutions of large systems of linear equations, including a nice description of various Krylov space methods (including preconditioning techniques), such as CG, GMRES, QMR, Bi-CG, Bi-CGSTAB, and the presentation of multigrid methods.

The thrid edition contains new material and several improved passages and will be useful also for those who already use the previous editions.

Reviewer:

pkn