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Lectures on Finite Precision ComputationsLectures on Finite Precision Computations

Françoise Chaitin-Chatelin and Valérie Frayssé



Software, Environments, and Tools 1


Chaitin-Chatelin and Frayssé provide a rigorous basis for error analysis and asses the quality and reliability of computations. ... Problems and algorithm derivations, toolboxes for computer experimentation, are given in a clear succinct form. ... --D. E. Bentil, CHOICE, March 1997, Vol. 34,No. 7


Devoted to the assessment of the quality of numerical results produced by computers, this book addresses the question: How does finite precision affect the convergence of numerical methods on the computer when convergence has been proven in exact arithmetic?

Finite precision computations are at the heart of the daily activities of many engineers and researchers in all branches of applied mathematics. Written in an informal style, the book combines techniques from engineering and mathematics to describe the rigorous and novel theory of computability in finite precision. In the challenging cases of nonlinear problems, theoretical analysis is supplemented by software tools to explore the stability on the computer.

Roundoff errors are often considered negatively, as a severe limitation on the purity of exact computations. The authors show how the necessarily finite precision of the computer arithmetic can be turned into an asset to describe physical phenomena.

Special Features

  • Discusses the influence of nonnormality on the reliability of algorithms and methods in relation with Physics and Technology.
  • Shows rounding errors to be treatable by classical analysis due to the framework of backward error analysis.
  • Presents a unified theory of convergence and stability by means of elementary mathematics.
  • Illustrates how to take advantage of modern programming environments to do experimental investigation about the stability of a problem or about the reliability of an algorithm or a numerical method.
  • Contains, for the first time in a book, a unified survey of normwise/componentwise error analysis for linear algebra (linear systems, least-squares and eigenproblems) and roots of polynomials.


Audience

This book has a wide appeal: from the beginning numeratician who wants to understand issues of backward and forward analysis and the influence of norms and the various measures for determining and analyzing computing errors, to the experienced engineer who wants to assess the quality of his computer simulations. Software developers, numerical analysts, engineers, and other aplied mathematicians as well as physicists will also benefit from this book. A background of basic analysis and linear algebra and some experience in scientific computing are suggested.

Contents

Foreword by Iain S. Duff; Preface; General Presentation Notations. Part One: Computability in Finite Precision. Well-Posed Problems; Approximations; Convergence in Exact Arithmetic; Computability in Finite Precision; Gaussian Elimination; Forward Error Analysis; The Influence of Singularities; Numerical Stability in Exact Arithmetic; Computability in Finite Precision for Iterative and Approximate Methods; The Limit of Numerical Stability in Finite Precision; Arithmetically Robust Convergence; The Computed Logistic; Bibliographical Comments. Part Two: Measures of Stability for Regular Problems. Choice of Data and Class of Perturbations; Choice of Norms: Scaling; Conditioning of Regular Problems; Simple Roots of Polynomials; Factorizations of a Complex Matrix; Solving Linear Systems; Functions of a Square Matrix; Concluding Remarks; Bibliographical Comments. Part Three: Computation in the Neighbourhood of a Singularity. Singular Problems Which are Well-Posed; Condition Numbers of Hölder-Singularities; Computability of Ill-Posed Problems; Singularities of z ----> A - zI; Distances to Singularity; Unfolding of Singularity; Spectral Portraits; Bibliographical Comments. Part Four: Arithmetic Quality of Reliable Algorithms. Forward and Backward Analyses; Backward Error; Quality of Reliable Software; Formulae for Backward Errors; Influence of the Class of Perturbations; Iterative Refinement for Backward Stability; Robust Reliability and Arithmetic Quality; Bibliographical Comments. Part Five: Numerical Stability in Finite Precision. Iterative and Approximate Methods; Numerical Convergence of Iterative Solvers; Stopping Criteria in Finite Precision; Robust Convergence; The Computed Logistic Revisited; Care of Use; Bibliographical Comments. Part Six: Software Tools for Round-Off Error Analysis in Algorithms. A Historical Perspective; The Assessment of the Quality of the Numerical Software; Backward Error Analysis in Libraries; Sensitivity Analysis; Interval Analysis; Probabilisitc Models; Computer Algebra; Bibliographical Comments. Part Seven: The Toolbox PRECISE for Computer Experimentation. What is PRECISE?; Module for Backward Error Analysis; Sample Size; Backward Analysis with PRECISE; Dangerous Border and Unfolding of a Singularity; Summary of Module 1; Bibliographical Comments. Part Eight: Experiments with PRECISE. Format of the Examples; Backward Error Analysis for Linear Systems; Computer Unfolding of Singularity; Dangerous Border and Distance to Singularity; Roots of Polynomials; Eigenvalue Problems; Conclusion; Bibliographical Comments. Part Nine: Robustness to Nonnormality. Nonnormality and Spectral Instability; Nonnormality in Physics and Technology; Convergence of Numerical Methods in Exact Arithmetic; Influence on Numerical Software; Bibliographical Comments. Part Ten. Qualitative Computing. Sensitivity and Pseudosolutions for F (x) = y; Pseudospectra of Matrices; Pseudozeroes of Polynomials; Divergence Portrait for the Complex Logistic Iteration; Qualitative Computation of a Jordan Form; Beyond Linear Perturbation Theory; Bibliographical Comments. Part Eleven: More Numerical Illustrations with PRECISE. Annex: The Toolbox PRECISE for MATLAB; Index; Bibliography.

1996 / xvi + 235 pages / Softcover / ISBN-13: 978-0-898713-58-9 / ISBN-10: 0-89871-358-7 /
List Price $79.50 /SIAM Member Price $55.65 / Order Code SE01
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