Computational theory of iterative methods.

*(English)*Zbl 1147.65313
Studies in Computational Mathematics 15. Amsterdam: Elsevier (ISBN 978-0-444-53162-9/hbk). xv, 487 p. (2007).

Publisher’s description: The book is designed for researchers, students and practitioners interested in using fast and efficient iterative methods to approximate solutions of nonlinear equations. The following four major problems are addressed. Problem 1: Show that the iterates are well defined. Problem 2: concerns the convergence of the sequences generated by a process and the question of whether the limit points are, in fact solutions of the equation. Problem 3: concerns the economy of the entire operations. Problem 4: concerns with how to best choose a method, algorithm or software program to solve a specific type of problem and its description of when a given algorithm succeeds or fails. The book contains applications in several areas of applied sciences including mathematical programming and mathematical economics. There is also a huge number of exercises complementing the theory.

Key Features: Latest convergence results for the iterative methods; Iterative methods with the least computational cost; Iterative methods with the weakest convergence conditions; Open problems on iterative methods.

Table of contents: 1. Linear spaces, 2. Monotone convergence, 3. Contractive fixed point theory, 4. Solving smooth equations, 5. Newton-like methods, 6. More results on Newton’s method, 7. Equations with nonsmooth operators, 8. Applications of the weaker version of the Newton-Kantorovich theorem, 9. The Newton-Kantorovich theorem and mathematical programming, 10. Generalized equations, 11. Monotone convergence of point to set-mapping, 12. Fixed points of point-to-set mappings, 13. Special topics; Bibliography; A Glossary of symbols; Index

Key Features: Latest convergence results for the iterative methods; Iterative methods with the least computational cost; Iterative methods with the weakest convergence conditions; Open problems on iterative methods.

Table of contents: 1. Linear spaces, 2. Monotone convergence, 3. Contractive fixed point theory, 4. Solving smooth equations, 5. Newton-like methods, 6. More results on Newton’s method, 7. Equations with nonsmooth operators, 8. Applications of the weaker version of the Newton-Kantorovich theorem, 9. The Newton-Kantorovich theorem and mathematical programming, 10. Generalized equations, 11. Monotone convergence of point to set-mapping, 12. Fixed points of point-to-set mappings, 13. Special topics; Bibliography; A Glossary of symbols; Index

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

47J25 | Iterative procedures involving nonlinear operators |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |