Discrete Iterations

1. Front Matter

   Pages I-XIV

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2. Discrete Iterations and Automata Networks: Basic Concepts

   • François Robert

   Pages 1-25

3. A Metric Tool

   • François Robert

   Pages 27-41

4. The Boolean Perron-Frobenius and Stein-Rosenberg Theorems

   • François Robert

   Pages 43-55

5. Boolean Contraction and Applications

   • François Robert

   Pages 57-78

6. Comparison of Operating Modes

   • François Robert

   Pages 79-93

7. The Discrete Derivative and Local Convergence

   • François Robert

   Pages 95-129

8. A Discrete Newton Method

   • François Robert

   Pages 131-166

9. General Conclusion

   • François Robert

   Pages 166-166

10. Back Matter

    Pages 167-195

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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: - the algorithmic aspect: an iterative algorithm is a mathematical entity which behaves in a dynamic fashion. Even if it is started far from a solution, it will often tend to get closer and closer. - the mathematical aspect: this consists of a coherent and rigorous analy­ sis of convergence, with the aid of mathematical tools (these tools are mainly the use of norms for convergence proofs, the use of matrix algebra and so on). One may for example refer to the algorithmic and mathematical aspects of Newton's method in JRn as well as to the QR algorithm for eigenvalues of matrices. These two algorithms seem to me to be the most fascinating algorithms in numerical analysis, since both show a remarkable practical efficiency even though there exist relatively few global convergence results for them.

• Eigenvalue
• Mathematica
• Matrix
• Newton's method
• algebra
• algorithms
• boundary element method
• convergence
• eXist
• efficiency
• iteration
• numerical analysis
• proof
• set
• tool

• University of Grenoble, Institut IMAG, Saint Martin d’Heres, France

  François Robert

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