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Complexity and Real Computation by Felipe Cucker, Michael Shub, Lenore Blum and Steve Smale (1997, Hardcover)
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Greenworld Books (32280)
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About this product
Product Identifiers
PublisherSpringer New York
ISBN-100387982817
ISBN-139780387982816
eBay Product ID (ePID)853556
Product Key Features
Number of PagesXvi, 453 Pages
LanguageEnglish
Publication NameComplexity and Real Computation
Publication Year1997
SubjectMachine Theory, Programming / Algorithms, Computer Science, General, Data Processing, Logic
TypeTextbook
Subject AreaMathematics, Computers
AuthorFelipe Cucker, Michael Shub, Lenore Blum, Steve Smale
FormatHardcover
Dimensions
Item Height0.4 in
Item Weight33.7 Oz
Item Length9.3 in
Item Width6.1 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN97-022859
Dewey Edition21
Number of Volumes1 vol.
IllustratedYes
Dewey Decimal511.3
Table Of Content1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bézout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.
SynopsisThis book develops a formal theory of computation which integrates major themes of the classical theory and which applies directly to problems in mathematics, numerical analysis, and scientific computing. Following an extensive introduction are proofs and extensions of relevant theorems. Along the way, the authors consider such fundamental questions as: Is the Mandelbrot set decidable? Is the Hilbert Nullstellensatz intractable? And, What is the real complexity of Newton's method?, 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., Computational complexity theory provides a framework for understanding the cost of solving computational problems, as measured by the requirement for resources such as time and space. The objects of study are algorithms defined within a formal model of computation. Upper bounds on the computational complexity of a problem are usually derived by constructing and analyzing specific algorithms. Meaningful lower bounds on computational complexity are harder to come by, and are not available for most problems of interest. The dominant approach in complexity theory is to consider algorithms as oper- ating on finite strings of symbols from a finite alphabet. Such strings may represent various discrete objects such as integers or algebraic expressions, but cannot rep- resent real or complex numbers, unless the numbers are rounded to approximate values from a discrete set. A major concern of the theory is the number of com- putation steps required to solve a problem, as a function of the length of the input string., Computational complexity theory provides a framework for understanding the cost of solving computational problems, as measured by the requirement for resources such as time and space. The objects of study are algorithms defined within a formal model of computation. Upper bounds on the computational complexity of a problem are usually derived by constructing and analyzing specific algorithms. Meaningful lower bounds on computational complexity are harder to come by, and are not available for most problems of interest. The dominant approach in complexity theory is to consider algorithms as oper ating on finite strings of symbols from a finite alphabet. Such strings may represent various discrete objects such as integers or algebraic expressions, but cannot rep resent real or complex numbers, unless the numbers are rounded to approximate values from a discrete set. A major concern of the theory is the number of com putation steps required to solve a problem, as a function of the length of the input string.
LC Classification NumberQA75.5-76.95
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