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Complexity and Real Computation

  • Textbook
  • © 1998

Overview

Authors:
  1. Lenore Blum

    1. International Computer Science Institute, Berkeley, USA
      Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

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  2. Felipe Cucker

    1. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
      Universitat Pompeu Fabra, Barcelona, Spain

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  3. Michael Shub

    1. IBM T.J. Watson Research Center, Yorktown Heights, USA

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  4. Steve Smale

    1. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

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  • Unique work on this core topic * Written by internationally recognised specialists in mathematics and computing * Provides the basics for numerous practical industrial applications, e.g. AI, robotics, digital cash

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Table of contents (23 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Basic Development

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 3-36

    3. Definitions and First Properties of Computation

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 37-68

    4. Computation over a Ring

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 69-81

    5. Decision Problems and Complexity over a Ring

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 83-98

    6. The Class NP and NP-Complete Problems

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 99-112

    7. Integer Machines

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 113-124

    8. Algebraic Settings for the Problem “P ≠ NP?”

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 125-146

  3. Back Matter

    Pages 147-149
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  4. Some Geometry of Numerical Algorithms

    1. Front Matter

      Pages 151-151
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    2. Newton’s Method

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 153-168

    3. Fundamental Theorem of Algebra: Complexity Aspects

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 169-186

    4. Bézout’s Theorem

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 187-200

    5. Condition Numbers and the Loss of Precision of Linear Equations

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 201-215

    6. The Condition Number for Nonlinear Problems

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 217-236

    7. The Condition Number in ℙ(H (d))

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 237-259

    8. Complexity and the Condition Number

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 261-273

    9. Linear Programming

      • Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 275-296

  5. Back Matter

    Pages 297-299
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Keywords

About this book

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.

Authors and Affiliations

  • International Computer Science Institute, Berkeley, USA

    Lenore Blum

  • Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Lenore Blum, Felipe Cucker, Steve Smale

  • Universitat Pompeu Fabra, Barcelona, Spain

    Felipe Cucker

  • IBM T.J. Watson Research Center, Yorktown Heights, USA

    Michael Shub

Bibliographic Information

  • Book Title: Complexity and Real Computation

  • Authors: Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale

  • DOI: https://doi.org/10.1007/978-1-4612-0701-6

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1998

  • Hardcover ISBN: 978-0-387-98281-6

  • Softcover ISBN: 978-1-4612-6873-4

  • eBook ISBN: 978-1-4612-0701-6

  • Edition Number: 1

  • Number of Pages: XVI, 453

  • Topics: Theory of Computation, Mathematical Logic and Foundations

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