Discrete iterations: A metric study

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.