Author | Robert, Franรงois. author |
---|---|
Title | Discrete Iterations [electronic resource] : A Metric Study / by Franรงois Robert |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1986 |
Connect to | http://dx.doi.org/10.1007/978-3-642-61607-5 |
Descript | XVI, 198 p. online resource |
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