Lectures on Finite Precision Computations
Will be published approx. on 28. February 1997
Book
Paperback/Softback
251 pages
978-0-89871-358-9 (ISBN)
Description
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.
Round-off 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 when describing physical phenomena.
Special Features:
Discusses the influence of nonnormality on the reliability of algorithms and methods in relation to 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 of the stability of a problem or of 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.
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.
Round-off 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 when describing physical phenomena.
Special Features:
Discusses the influence of nonnormality on the reliability of algorithms and methods in relation to 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 of the stability of a problem or of 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.
Reviews / Votes
'Chaitin-Chatelin and Fraysse 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, CHOICEMore details
Series
Language
English
Place of publication
New York
United States
Target group
College/higher education
Professional and scholarly
Product notice
Paperback (trade)
Dimensions
Height: 253 mm
Width: 179 mm
Thickness: 15 mm
Weight
452 gr
ISBN-13
978-0-89871-358-9 (9780898713589)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
Foreword
Preface
Notation
Chapter 1: General Presentation. Coupling
Chaotic Computations
Computability in Finite Precision
Numerical Quality of Computations
Role of Singularities
Spectral Instability and Nonnormality
Influence on Nonnumerical Software
Qualitative Computing
Experimental Mathematics
Sense of Errors: For a Rehabilitation of Finite Precision Computations
Chapter 2: 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
Chapter 3: 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. Chapter 4: Computation in the Neighbourhood of a Singularity
Singular Problems That Are Well Posed
Condition Numbers of Hoelder Singularities
Computability of Ill-Posed Problems
Singularities of z * A * zI
Distances to Singularity
Unfolding of Singularity
Spectral Portraits
Bibliographical Comments
Chapter 5: 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
Chapter 6: 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
Chapter 7: Software Tools for Round-off Error Analysis in Algorithms
A Historical Perspective
Assessment of the Quality of Numerical Software
Backward Error Analysis in Libraries
Sensitivity Analysis
Interval Analysis
Probabilistic Models
Computer Algebra
Bibliographical Comments
Chapter 8: 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
Chapter 9: 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
Chapter 10: 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
Chapter 11: Qualitative Computing. Sensitivity and Pseudosolutions for F (x) = y
Pseudospectra of Matrices
Pseudozeroes of Polynomials
Divergence Portrait for the Complex Logistic Iteration
Qualitative Assessment of a Jordan Form
Beyond Linear Perturbation Theory
Bibliographical Comments
Chapter 12: More Numerical Illustrations with PRECISE
Annex: The Toolbox PRECISE for MATLAB
Bibliography
Index.
Preface
Notation
Chapter 1: General Presentation. Coupling
Chaotic Computations
Computability in Finite Precision
Numerical Quality of Computations
Role of Singularities
Spectral Instability and Nonnormality
Influence on Nonnumerical Software
Qualitative Computing
Experimental Mathematics
Sense of Errors: For a Rehabilitation of Finite Precision Computations
Chapter 2: 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
Chapter 3: 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. Chapter 4: Computation in the Neighbourhood of a Singularity
Singular Problems That Are Well Posed
Condition Numbers of Hoelder Singularities
Computability of Ill-Posed Problems
Singularities of z * A * zI
Distances to Singularity
Unfolding of Singularity
Spectral Portraits
Bibliographical Comments
Chapter 5: 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
Chapter 6: 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
Chapter 7: Software Tools for Round-off Error Analysis in Algorithms
A Historical Perspective
Assessment of the Quality of Numerical Software
Backward Error Analysis in Libraries
Sensitivity Analysis
Interval Analysis
Probabilistic Models
Computer Algebra
Bibliographical Comments
Chapter 8: 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
Chapter 9: 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
Chapter 10: 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
Chapter 11: Qualitative Computing. Sensitivity and Pseudosolutions for F (x) = y
Pseudospectra of Matrices
Pseudozeroes of Polynomials
Divergence Portrait for the Complex Logistic Iteration
Qualitative Assessment of a Jordan Form
Beyond Linear Perturbation Theory
Bibliographical Comments
Chapter 12: More Numerical Illustrations with PRECISE
Annex: The Toolbox PRECISE for MATLAB
Bibliography
Index.