Del i serien Software Environments and Tools

Lectures on Finite Precision Computations

Häftad, Engelska, 1997

AvFrançoise Chaitin-Chatelin,Valérie Frayssé

1 399 kr

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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.

Produktinformation

Utgivningsdatum1997-02-28
Mått151 x 228 x 14 mm
Vikt455 g
FormatHäftad
SpråkEngelska
SerieSoftware Environments and Tools
Antal sidor251
FörlagSociety for Industrial & Applied Mathematics,U.S.
ISBN9780898713589

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ForewordPrefaceNotationChapter 1: General Presentation. CouplingChaotic ComputationsComputability in Finite PrecisionNumerical Quality of ComputationsRole of SingularitiesSpectral Instability and NonnormalityInfluence on Nonnumerical SoftwareQualitative ComputingExperimental MathematicsSense of Errors: For a Rehabilitation of Finite Precision ComputationsChapter 2: Computability in Finite PrecisionWell-Posed ProblemsApproximationsConvergence in Exact ArithmeticComputability in Finite PrecisionGaussian EliminationForward Error AnalysisThe Influence of SingularitiesNumerical Stability in Exact ArithmeticComputability in Finite Precision for Iterative and Approximate MethodsThe Limit of Numerical Stability in Finite PrecisionArithmetically Robust ConvergenceThe Computed LogisticBibliographical CommentsChapter 3: Measures of Stability for Regular ProblemsChoice of Data and Class of PerturbationsChoice of Norms: ScalingConditioning of Regular ProblemsSimple Roots of PolynomialsFactorizations of a Complex MatrixSolving Linear SystemsFunctions of a Square MatrixConcluding RemarksBibliographical Comments. Chapter 4: Computation in the Neighbourhood of a SingularitySingular Problems That Are Well PosedCondition Numbers of Hölder SingularitiesComputability of Ill-Posed ProblemsSingularities of z * A * zIDistances to SingularityUnfolding of SingularitySpectral PortraitsBibliographical CommentsChapter 5: Arithmetic Quality of Reliable AlgorithmsForward and Backward AnalysesBackward ErrorQuality of Reliable SoftwareFormulae for Backward ErrorsInfluence of the Class of PerturbationsIterative Refinement for Backward StabilityRobust Reliability and Arithmetic QualityBibliographical CommentsChapter 6: Numerical Stability in Finite PrecisionIterative and Approximate MethodsNumerical Convergence of Iterative SolversStopping Criteria in Finite PrecisionRobust ConvergenceThe Computed Logistic RevisitedCare of UseBibliographical CommentsChapter 7: Software Tools for Round-off Error Analysis in AlgorithmsA Historical PerspectiveAssessment of the Quality of Numerical SoftwareBackward Error Analysis in LibrariesSensitivity AnalysisInterval AnalysisProbabilistic ModelsComputer AlgebraBibliographical CommentsChapter 8: The Toolbox PRECISE for Computer ExperimentationWhat is PRECISE?Module for Backward Error AnalysisSample SizeBackward Analysis with PRECISEDangerous Border and Unfolding of a SingularitySummary of Module 1Bibliographical CommentsChapter 9: Experiments with PRECISE. Format of the ExamplesBackward Error Analysis for Linear SystemsComputer Unfolding of SingularityDangerous Border and Distance to SingularityRoots of PolynomialsEigenvalue ProblemsConclusionBibliographical CommentsChapter 10: Robustness to NonnormalityNonnormality and Spectral InstabilityNonnormality in Physics and TechnologyConvergence of Numerical Methods in Exact ArithmeticInfluence on Numerical SoftwareBibliographical CommentsChapter 11: Qualitative Computing. Sensitivity and Pseudosolutions for F (x) = yPseudospectra of MatricesPseudozeroes of PolynomialsDivergence Portrait for the Complex Logistic IterationQualitative Assessment of a Jordan FormBeyond Linear Perturbation TheoryBibliographical CommentsChapter 12: More Numerical Illustrations with PRECISEAnnex: The Toolbox PRECISE for MATLABBibliographyIndex.