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Complexity and Real Computation - Lenore Blum

Book with Other Items Published: 30th October 1997
ISBN: 9780387982816
Number Of Pages: 453

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The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers.  The later parts of the book develop  a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.

Foreword
Preface
Basic Development
Introductionp. 3
Definitions and First Properties of Computationp. 37
Computation over a Ringp. 69
Decision Problems and Complexity over a Ringp. 83
The Class NP and NP-Complete Problemsp. 99
Integer Machinesp. 113
Algebraic Settings for the Problem [actual symbol not reproducible]p. 125
Basic Notions of Algebraic Geometryp. 147
Additional Comments and Bibliographical Remarksp. 149
Some Geometry of Numerical Algorithms
Newton's Methodp. 153
Fundamental Theorem of Algebra: Complexity Aspectsp. 169
Bezout's Theoremp. 187
Condition Numbers and the Loss of Precision of Linear Equationsp. 201
The Condition Number for Nonlinear Problemsp. 217
The Condition Number in P(H[subscript (d)])p. 237
Complexity and the Condition Numberp. 261
Linear Programmingp. 275
The Main Theorem of Elimination Theoryp. 297
Additional Comments and Bibliographical Remarksp. 299
Complexity Classes over the Reals
Deterministic Lower Boundsp. 303
Probabilistic Machinesp. 317
Parallel Computationsp. 335
Some Separations of Complexity Classesp. 359
Weak Machinesp. 377
Additive Machinesp. 385
Nonuniform Complexity Classesp. 401
Descriptive Complexityp. 411
Referencesp. 431
Indexp. 447
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387982816
ISBN-10: 0387982817
Audience: Professional
Format: Book with Other Items
Language: English
Number Of Pages: 453
Published: 30th October 1997
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 0.95