Lectures on finite precision computations.

*(English)*Zbl 0846.65020
Software - Environments - Tools, 1. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xiv, 235 p. (1996).

This book deals with the very important nowadays issue of finite precision computations and reliability of computer results. It is shown that the stability and convergence in exact arithmetic can be ruined in a computer implementation because of the finite precision available. This should be expected only in the neighbourhood of singularities. There are a lot of examples showing what difficulties can arise in numerical computation.

An important feature of the book is the unified treatment of normwise and componentwise error analysis for the solution of linear systems, eigenvalues, least squares, and roots of polynomials. A new toolbox PRECISE (under MATLAB) is presented which can help the user to perform computations near a singularity, thus, showing the limits of finite precision computation to a given problem. The book will definitely help to disseminate the knowledge of forward and backward error analysis among specialists in other fields, too.

Here is the brief contents of the book: General presentation, Computability in finite precision, Measures of stability for regular problems, Computation in the neighbourhood of a singularity, Arithmetic quality of reliable algorithms, Numerical stability in finite precision, Software tools for roundoff error analysis in algorithms, The toolbox PRECISE for computer experimentation, Experiments with PRECISE, Robustness to nonnormality, Qualitative computing, More numerical illustrations with PRECISE.

An important feature of the book is the unified treatment of normwise and componentwise error analysis for the solution of linear systems, eigenvalues, least squares, and roots of polynomials. A new toolbox PRECISE (under MATLAB) is presented which can help the user to perform computations near a singularity, thus, showing the limits of finite precision computation to a given problem. The book will definitely help to disseminate the knowledge of forward and backward error analysis among specialists in other fields, too.

Here is the brief contents of the book: General presentation, Computability in finite precision, Measures of stability for regular problems, Computation in the neighbourhood of a singularity, Arithmetic quality of reliable algorithms, Numerical stability in finite precision, Software tools for roundoff error analysis in algorithms, The toolbox PRECISE for computer experimentation, Experiments with PRECISE, Robustness to nonnormality, Qualitative computing, More numerical illustrations with PRECISE.

Reviewer: P.Y.Yalamov (Russe)

##### MSC:

65G50 | Roundoff error |

65Fxx | Numerical linear algebra |

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