Computational Complexity and Feasibility of Data Processing and Interval Computa

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Computational Complexity and Feasibility of Data Processing and Interval Computations by V. Kreinovich, A.V. Lakeyev, J. Rohn, P.T. Kahl

Targeted audience • Specialists in numerical computations, especially in numerical optimiza­ tion, who are interested in designing algorithms with automatie result ver­ ification, and who would therefore be interested in knowing how general their algorithms caIi in principle be.

FORMAT
Hardcover LANGUAGE
English CONDITION
Brand New Publisher Description

The input data for data processing algorithms come from measurements and are therefore not precise. It is therefore necessary to estimate the accuracy of the results of data processing. Even for the simplest data processing algorithms, this problem is, however, intractable. This book describes for what classes of problems interval computations (i.e. data processing with automatic results verification) are feasible, and when they are intractable. This knowledge is important for algorithm developers, because it will enable them to concentrate on the classes of problems for which general algorithms are possible.

Table of Contents

1 Informal Introduction: Data Processing, Interval Computations, and Computational Complexity.- 2 The Notions of Feasibility and NP-Hardness: Brief Introduction.- 3 In the General Case, the Basic Problem of Interval Computations is Intractable.- 4 Basic Problem of Interval Computations for Polynomials of a Fixed Number of Variables.- 5 Basic Problem of Interval Computations for Polynomials of Fixed Order.- 6 Basic Problem of Interval Computations for Polynomials with Bounded Coefficients.- 7 Fixed Data Processing Algorithms, Varying Data: Still NP-Hard.- 8 Fixed Data, Varying Data Processing Algorithms: Still Intractable.- 9 What if We only Allow some Arithmetic Operations in Data Processing?.- 10 For Fractionally-Linear Functions, a Feasible Algorithm Solves the Basic Problem of Interval Computations.- 11 Solving Interval Linear Systems is NP-Hard.- 12 Interval Linear Systems: Search for Feasible Classes.- 13 Physical Corollary: Prediction is not Always Possible, Even for Linear Systems with Known Dynamics.- 14 Engineering Corollary: Signal Processing is NP-Hard.- 15 Bright Sides of NP-Hardness of Interval Computations I: NP-Hard Means That Good Interval Heuristics can Solve other Hard Problems.- 16 If Input Intervals are Narrow Enough, Then Interval Computations are Almost Always Easy.- 17 Optimization — a First Example of a Numerical Problem in which Interval Methods are used: Computational Complexity and Feasibility.- 18 Solving Systems of Equations.- 19 Approximation of Interval Functions.- 20 Solving Differential Equations.- 21 Properties of Interval Matrices I: Main Results.- 22 Properties of Interval Matrices II: Proofs and Auxiliary Results.- 23 Non-Interval Uncertainty I: Ellipsoid Uncertainty and its Generalizations.- 24 Non-Interval Uncertainty II:Multi-Intervals and Their Generalizations.- 25 What if Quantities are Discrete?.- 26 Error Estimation for Indirect Measurements: Interval Computation Problem is (Slightly) Harder than a Similar Probabilistic Computational Problem.- A In Case of Interval (Or More General) Uncertainty, no Algorithm can Choose the Simplest Representative.- B Error Estimation for Indirect Measurements: Case of Approximately Known Functions.- C From Interval Computations to Modal Mathematics.- D Beyond NP: Two Roots Good, one Root Better.- E Does "NP-Hard"Really Mean "Intractable"?.- F Bright Sides of NP-Hardness of Interval Computations II: Freedom of Will?.- G The Worse, The Better: Paradoxical Computational Complexity of Interval Computations and Data Processing.- References.

Review

`The book is very well organized. The book is readable, very detailed, and intelligible. Moreover, it is exciting and gripping because the reader is often surprised by unexpected results. Everyone who is interested in modern aspects of numerical computation will enjoy reading this interesting and informative work. The book is a must for everyone who is doing research in this field.'
Mathematical Reviews, 98m

Long Description

Targeted audience * Specialists in numerical computations, especially in numerical optimiza

Review Text

' The book is very well organized. The book is readable, very detailed, and intelligible. Moreover, it is exciting and gripping because the reader is often surprised by unexpected results. Everyone who is interested in modern aspects of numerical computation will enjoy reading this interesting and informative work. The book is a must for everyone who is doing research in this field. ' Mathematical Reviews, 98m

Review Quote

'The book is very well organized. The book is readable, very detailed, and intelligible. Moreover, it is exciting and gripping because the reader is often surprised by unexpected results. Everyone who is interested in modern aspects of numerical computation will enjoy reading this interesting and informative work. The book is a must for everyone who is doing research in this field.' Mathematical Reviews, 98m

Details ISBN0792348656 Short Title COMPUTATIONAL COMPLEXITY & FEA Series Applied Optimization Language English ISBN-10 0792348656 ISBN-13 9780792348658 Media Book Format Hardcover Series Number 10 Year 1997 Publication Date 1997-12-31 Place of Publication Dordrecht Residence US Affiliation University of Texas Pages 459 Illustrations XII, 459 p. DOI Author P.T. Kahl Edition Description 1998 ed. Edition 1998th Alternative 9781441947857 DEWEY 004.151 Audience Postgraduate, Research & Scholarly Country of Origin US Product Class Description Mathematics Publisher Kluwer Academic Publishers Imprint Kluwer Academic Publishers Country of Publication United States US Release Date 1997-12-31
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