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.Round-off 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 when describing physical phenomena.Special Features:Discusses the influence of nonnormality on the reliability of algorithms and methods in relation to physics and technology.Shows rounding errors to be treatable by classical analysis due to the framework of backward error analysis.Presents a unified theory of convergence and stability by means of elementary mathematics.Illustrates how to take advantage of modern programming environments to do experimental investigation of the stability of a problem or of the reliability of an algorithm or a numerical method.Contains, for the first time in a book, a unified survey of normwise/componentwise error analysis for linear algebra (linear systems, least squares, and eigenproblems) and roots of polynomials.
