Lectures on Finite Precision Computations

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 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; 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.