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Lectures on Finite Precision Computations

Lectures on Finite Precision Computations


Part of Software, Environments and Tools

  • Date Published: February 1997
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • format: Paperback
  • isbn: 9780898713589

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About the Authors
  • 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.

    Reviews & endorsements

    '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

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    Product details

    • Date Published: February 1997
    • format: Paperback
    • isbn: 9780898713589
    • length: 250 pages
    • dimensions: 253 x 179 x 15 mm
    • weight: 0.452kg
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • Table of Contents

    Foreword Iain S. Duff
    General Presentation Notations. Part I. Computability in Finite Precision: Well-Posed Problems
    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 II. 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 III. 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 IV. 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 V. 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. PartVI. 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 VII. 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 VIII. 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
    Bibliographical Comments. Part IX. 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 V. 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 XI. More Numerical Illustrations with PRECISE: Annex: The Toolbox PRECISE for MATLAB

  • Authors

    Françoise Chaitin-Chatelin, Université de Toulouse I (Sciences Sociales)

    Valérie Frayssé, CERFACS, Toulouse

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