Accuracy and stability of numerical algorithms.

*(English)*Zbl 0847.65010
Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xxviii, 688 p. (1996).

This text may become the new ‘Bible’ about accuracy and stability for the solution of systems of linear equations. It covers 688 pages carefully collected, investigated, and written, dedicated to two pioneers in this field, Alan M. Turing and James H. Wilkinson.

After two introductory chapters about ‘Finite precision computation’ and ‘Floating point arithmetic’ (FPA) there are twenty-one chapters covering the range from the basics like inner products to ‘The fast Fourier transform and applications’. The last three chapters deal with ‘Automatic error analysis’, ‘Software issues in FPA’, and ‘Test matrices’. Five appendices, 1134(!) references, and two indices are added. The area of eigenvalue and singular value computations is omitted as well as Toeplitz systems and parallel algorithms.

Each chapter starts with a number of quotations which are an art gallery of numerical facts, names, and sometimes curiosities. Then typically the chapters contain sections about solution methods, perturbation theory, error analysis, historical perspectives, notes and references, and problems. One finds a lot of theorems, most of them with proofs, as well as facts about software development with special emphasis on LAPACK. The error bounds and condition estimator in LAPACK are explained. New results like error analyses for iterative refinement, Gauss-Jordan elimination, and Newton interpolating polynomial are integrated with more or less well-known existing results. There are a lot of mostly new exercises, from classroom-suited to research problems. The ‘solutions to problems’-appendix covers fifty pages.

Summarizing one will find that this book is a very suitable and comprehensive reference for research in numerical linear algebra, software usage and development, and for numerical linear algebra courses.

After two introductory chapters about ‘Finite precision computation’ and ‘Floating point arithmetic’ (FPA) there are twenty-one chapters covering the range from the basics like inner products to ‘The fast Fourier transform and applications’. The last three chapters deal with ‘Automatic error analysis’, ‘Software issues in FPA’, and ‘Test matrices’. Five appendices, 1134(!) references, and two indices are added. The area of eigenvalue and singular value computations is omitted as well as Toeplitz systems and parallel algorithms.

Each chapter starts with a number of quotations which are an art gallery of numerical facts, names, and sometimes curiosities. Then typically the chapters contain sections about solution methods, perturbation theory, error analysis, historical perspectives, notes and references, and problems. One finds a lot of theorems, most of them with proofs, as well as facts about software development with special emphasis on LAPACK. The error bounds and condition estimator in LAPACK are explained. New results like error analyses for iterative refinement, Gauss-Jordan elimination, and Newton interpolating polynomial are integrated with more or less well-known existing results. There are a lot of mostly new exercises, from classroom-suited to research problems. The ‘solutions to problems’-appendix covers fifty pages.

Summarizing one will find that this book is a very suitable and comprehensive reference for research in numerical linear algebra, software usage and development, and for numerical linear algebra courses.

Reviewer: N.Köckler (Paderborn)

##### MSC:

65Fxx | Numerical linear algebra |

15-04 | Software, source code, etc. for problems pertaining to linear algebra |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65G50 | Roundoff error |