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 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.
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.
'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, CHOICE
