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.
Foreword Iain S. Duff; Preface; General Presentation Notations. Part I. 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 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 Holder-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; Conclusion; 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; Index; Bibliography.
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