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Interval Analysis

Book Description

This self-contained text is a step-by-step introduction and a complete overview of interval computation and result verification, a subject whose importance has steadily increased over the past many years. The author, an expert in the field, gently presents the theory of interval analysis through many examples and exercises, and guides the reader from the basics of the theory to current research topics in the mathematics of computation.

Contents

  • Preliminaries
  • Real intervals
  • Interval vectors, interval matrices
  • Expressions, P-contraction, ε-inflation
  • Linear systems of equations
  • Nonlinear systems of equations
  • Eigenvalue problems
  • Automatic differentiation
  • Complex intervals

Table of Contents

  1. Cover
  2. Title Page
  3. Front Matter01
  4. Copyright
  5. Front Matter02
  6. Table of Contents
  7. Preface
  8. 1 Preliminaries
    1. 1.1 Notations and basic definitions
    2. 1.2 Metric spaces
    3. 1.3 Normed linear spaces
    4. 1.4 Polynomials
    5. 1.5 Zeros and fixed points of functions
    6. 1.6 Mean value theorems
    7. 1.7 Normal forms of matrices
    8. 1.8 Eigenvalues
    9. 1.9 Nonnegative matrices
    10. 1.10 Particular matrices
  9. 2 Real intervals
    1. 2.1 Intervals, partial ordering
    2. 2.2 Interval arithmetic
    3. 2.3 Algebraic properties, χ-function
    4. 2.4 Auxiliary functions
    5. 2.5 Distance and topology
    6. 2.6 Elementary interval functions
    7. 2.7 Machine interval arithmetic
  10. 3 Interval vectors, interval matrices
    1. 3.1 Basics
    2. 3.2 Powers of interval matrices
    3. 3.3 Particular interval matrices
  11. 4 Expressions, P -contraction, ε -inflation
    1. 4.1 Expressions, range
    2. 4.2 P -contraction
    3. 4.3 ε -inflation
  12. 5 Linear systems of equations
    1. 5.1 Motivation
    2. 5.2 Solution sets
    3. 5.3 Interval hull
    4. 5.4 Direct methods
    5. 5.5 Iterative methods
  13. 6 Nonlinear systems of equations
    1. 6.1 Newton method – one-dimensional case
    2. 6.2 Newton method – multidimensional case
    3. 6.3 Krawczyk method
    4. 6.4 Hansen–Sengupta method
    5. 6.5 Further existence tests
    6. 6.6 Bisection method
  14. 7 Eigenvalue problems
    1. 7.1 Quadratic systems
    2. 7.2 A Krawczyk-like method
    3. 7.3 Lohner method
    4. 7.4 Double or nearly double eigenvalues
    5. 7.5 The generalized eigenvalue problem
    6. 7.6 A method due to Behnke
    7. 7.7 Verification of singular values
    8. 7.8 An inverse eigenvalue problem
  15. 8 Automatic differentiation
    1. 8.1 Forward mode
    2. 8.2 Backward mode
  16. 9 Complex intervals
    1. 9.1 Rectangular complex intervals
    2. 9.2 Circular complex intervals
    3. 9.3 Applications of complex intervals
  17. Final Remarks
  18. Appendix
    1. A Jordan normal form
    2. B Brouwer’s fixed point theorem
    3. C Theorem of Newton–Kantorovich
    4. D The row cyclic Jacobi method
    5. E The CORDIC Algorithm
    6. F The symmetric solution set
    7. G INTLAB
  19. Bibliography
  20. Symbol Index
  21. Author Index
  22. Subject Index