Discrete Iterations A Metric Study

Discrete Iterations A Metric Study / by Francois Robert.

a c 9 h In presenting this monograph, I would like to indicate both its orientation as well as my personal reasons for being interested in discrete iterations (that is, iterations on a generally very large,jinite set). While working in numerical analysis I have been interested in two main aspects: -...

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Bibliographic Details
Main Author: Robert, Francois (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1986.
Edition:1st ed. 1986.
Series:Springer Series in Computational Mathematics, 6
Springer eBook Collection.
Subjects:
Numerical analysis.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1. Discrete Iterations and Automata Networks: Basic Concepts
  • 1. Discrete Iterations and Their Graphs
  • 2. Examples
  • 3. Connectivity Graphs and Incidence Matrices
  • 4. Interpretations in Terms of Automata Networks
  • 5. Serial Operation and the Gauss-Seidel Operator
  • 6. Serial-Parallel Modes of Operation and the Associated Operators
  • 2. A Metric Tool
  • 1. The Boolean Vector Distance d
  • 2. Some Basic Inequalities
  • 3. First Applications
  • 4. Serial-Parallel Operators. An Outline
  • 5. Other Possible Metric Tools
  • 3. The Boolean Perron-Frobenius and Stein-Rosenberg Theorems
  • 1. Eigenelements of a Boolean Matrix
  • 2. The Boolean Perron-Frobenius Theorem
  • 3. The Boolean Stein-Rosenberg Theorems
  • 4. Conclusion
  • 4. Boolean Contraction and Applications
  • 1. Boolean Contraction
  • 2. A Fixed Point Theorem
  • 3. Examples
  • 4. Serial Mode: Gauss-Seidel Iteration for a Contracting Operator
  • 5. Examples
  • 6. Comparison of Operating Modes for a Contracting Operator
  • 7. Examples
  • 8. Rounding-Off: Successive Gauss- Seidelisations
  • 9. Conclusions
  • 5.Comparison of Operating Modes
  • 1. Comparison of Serial and Parallel Operating Modes
  • 2. Examples
  • 3. Extension to the Comparison of Two Serial-Parallel Modes of Operation
  • 4. Examples
  • 5. Conclusions
  • 6. The Discrete Derivative and Local Convergence
  • 1. The Discrete Derivative
  • 2. The Discrete Derivative and the Vector Distance
  • 3. Application: Characterization of the Local Convergence in the Immediate Neighbourhood of a Fixed Point
  • 4. Interpretation in Terms of Automata Networks
  • 5. Application: Local Convergence in a Massive Neighbourhood of a Fixed Point
  • 6. Gauss-Seidel
  • 7. The Derivative of a Function Composition
  • 8. The Study of Cycles: Attractive Cycles
  • 9. Conclusions
  • 7. A Discrete Newton Method
  • 1. Context
  • 2. Two Simple Examples
  • 3. Interpretation in Terms of Automata
  • 4. The Study of Convergence: The Case of the Simplified Newton Method
  • 5. The Study of Convergence, The General Case
  • 6. The Efficiency of an Iterative Method on a Finite Set
  • 7. Numerical Experiments
  • 8. Conclusions
  • General Conclusion
  • Appendix 2. The Number of Regular n x n Matrices with Elements in Z/p (p prime)
  • Appendix 4. Continuous Iterations-Discrete Iterations
  • Inde.

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