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Complexity and Real Computation

  • Lenore Blum
  • Felipe Cucker
  • Michael Shub
  • Steve Smale

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Basic Development

    1. Front Matter
      Pages 1-1
    2. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 3-36
    3. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 37-68
    4. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 69-81
    5. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 83-98
    6. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 99-112
    7. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 113-124
    8. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 125-146
    9. Back Matter
      Pages 147-149
  3. Some Geometry of Numerical Algorithms

    1. Front Matter
      Pages 151-151
    2. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 153-168
    3. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 169-186
    4. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 187-200
    5. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 201-215
    6. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 217-236
    7. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 237-259
    8. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 261-273
    9. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 275-296
    10. Back Matter
      Pages 297-299
  4. Complexity Classes over the Reals

    1. Front Matter
      Pages 301-301
    2. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 303-315
    3. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 317-334
    4. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 335-357
    5. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 359-375
    6. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 377-384
    7. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 385-400
    8. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 401-409
    9. Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
      Pages 411-429
  5. Back Matter
    Pages 431-453

About this book

Introduction

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.

Keywords

algorithms complexity fundamental theorem linear optimization theoretical computer science

Authors and affiliations

  • Lenore Blum
    • 1
    • 2
  • Felipe Cucker
    • 2
    • 3
  • Michael Shub
    • 4
  • Steve Smale
    • 2
  1. 1.International Computer Science InstituteBerkeleyUSA
  2. 2.Department of MathematicsCity University of Hong KongKowloonHong Kong
  3. 3.Universitat Pompeu FabraBarcelonaSpain
  4. 4.IBM T.J. Watson Research CenterYorktown HeightsUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-0701-6
  • Copyright Information Springer-Verlag New York, Inc. 1998
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-6873-4
  • Online ISBN 978-1-4612-0701-6
  • Buy this book on publisher's site
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