## Lectures on Finite Precision ComputationsDevoted 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. |

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### Contents

Measures of Stability for Regular Problems 39 | 52 |

Computation in the Neighbourhood of a Singularity | 57 |

Experiments with PRECISE 125 | 135 |

Robustness to Nonnormality | 159 |

Arithmetic Quality of Reliable Algorithms 71 | 162 |

Software Tools for Roundoff Error Analysis in Algorithms 97 | 175 |

More Numerical Illustrations with PRECISE | 199 |

The Toolbox PRECISE for MATLAB | 211 |

219 | |

233 | |

### Common terms and phrases

algorithm approximate Arnoldi ascent backward error analysis backward stability behaviour Bennani Chapter Chatelin companion matrix componentwise perturbations computability in finite computed solution condition number constant corresponding dangerous border decreases defective eigenvalue defined Definition distance to singularity double root eigenvector equation error estimation exact arithmetic Example Figure finite precision arithmetic finite precision computation formulae forward error GEPP highly nonnormal Holder condition Holder condition number ill conditioned ill-posed problem IMGS indicators invariant subspace iterative methods Jordan block Jordan structure LAPACK linear system machine precision mathematical matrix inversion neighbourhood nonnormal matrices norm normwise and componentwise normwise perturbations numerical analysis numerical methods numerical software numerical stability orthogonal parameter perturbed spectra polynomial precision arithmetic pseudospectra QR algorithm reliability interval residual resp right-hand side round-off round-off errors sample sensitivity simple eigenvalue simple root spectral instability spectral portraits spectrum theoretical Trefethen upper bound vector Wilkinson