Lectures on Finite Precision Computations (Software, Environments and Tools, Series Number 1) by Françoise Chaitin-Chatelin (PDF)

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. 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.

User’s Reviews

Editorial Reviews: Review ‘Chaitin-Chatelin and Frayssé 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 Book Description This book combines techniques from engineering and mathematics to describe the rigorous and novel theory of computability in finite precision.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Interesting premise: instead of formulating mathematical problems in Real (or complex) space, formulate them instead in the space containing only those numbers representable on a computer. Surely, this is a very important point and one that deserves the *rigorous* mathematical treatment given in this book. Using computers forces us to have finite numbers, which introduces several little quirks when performing calculations on a computer- Chaitin-Chatelin and Fraysse make a stab at trying to quantify and understand said quirks by developing a mathematical framework in which to discuss the various issues that pop up. Taking what they gleaned from the theory, the latter half of the book is full of numerical experiments with a MATLAB tool written by the authors, investigating singularities, polynomial roots, and eigenvalues.Fair warning: this text is very heavy on the math. I had to break out my numerical analysis book a few times. Throughout most of the book, I found myself thinking, “why bother?” as I often do when reading outer-space math works. Down here with the rest of us non-mathematicians, users of most practical computers codes don’t care about the 15th significant figure of results. Nevertheless, it was still a fascinating read for me- and the subject matter should, of course, always be kept in mind when writing/working with number crunching codes.

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