A Contemporary Study of Iterative Methods: Convergence, Dynamics and Applications

ISBN-13: 9780128092149 / Angielski / Miękka / 2018 / 400 str.

A. Alberto Magrenan; Ioannis Argyros

Opis

A Contemporary Study of Iterative Methods evaluates and compares advances in iterative techniques, and their numerous applications in applied mathematics, engineering, mathematical economics, mathematical biology, and other applied sciences. Oftentimes, dynamic systems are modeled by difference or differential equations, and their solutions represent the states of the systems. Similar equations are also used in the case of discrete systems. The 'unknowns' of engineering equations can be functions (difference, differential, and integral equations), vectors, (systems of linear and nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). Except in special cases, the most commonly used solution methods are iterative - when starting from one or several approximations a sequence is constructed that converges to a solution of the equation. Iteration methods are also applied for solving optimization problems. In such cases the iteration sequences converge to an optimal solution of the problem at hand. Since all these methods have the same recursive structure, they can be introduced and discussed in a general framework.

• Contains recent results on the convergence analysis of numerical algorithms in both finite-dimensional and infinite-dimensional spaces, and present several applications and connections with fixed point theory
• Encompasses the novel tool of dynamic analysis for iterative methods, including new developments in Smale stability theory and polynomiography
• Explores the uses of computation of iterative methods across nonlinear analysis, placing the useful technique before non-specialist users from many application fields
• Uniquely places discussion of derivative-free methods in context of other discoveries, aiding comparison and contrast between options

Recenzje

"Contemporary in the title means that the coverage is state-of-the-art, with all currently-useful methods being shown. The level of detail is reasonable for an encyclopedia, and each chapter is extensively footnoted with references to research papers. Usually each chapter describes the method, quotes some theorems about the conditions under which it will succeed (occasionally with proofs), and usually a contrived numeric example to show how it works. There's usually some discussion of convergence speed." --MAA Reviews

Spis treści

1. The majorization method in the Kantorovich theory2. Directional Newton methods3. Newton's method4. Generalized equations5. Gauss-Newton method6. Gauss-Newton method for convex optimization7. Proximal Gauss-Newton method8. Multistep modified Newton-Hermitian and Skew-Hermitian Splitting method9. Secant-like methods in chemistry10. Robust convergence of Newton's method for cone inclusion problem11. Gauss-Newton method for convex composite optimization12. Domain of parameters13. Newton's method for solving optimal shape design problems14. Osada method15. Newton's method to solve equations with solutions of multiplicity greater than one16. Laguerre-like method for multiple zeros17. Traub's method for multiple roots18. Shadowing lemma for operators with chaotic behavior19. Inexact two-point Newton-like methods20. Two-step Newton methods21. Introduction to complex dynamics22. Convergence and the dynamics of Chebyshev-Halley type methods23. Convergence planes of iterative methods24. Convergence and dynamics of a higher order family of iterative methods25. Convergence and dynamics of iterative methods for multiple zeros

O autorze

Professor Alberto Magreñán (Department of Mathematics, Universidad Internacional de La Rioja, Spain). Magreñán has published 43 documents. He works in operator theory, computational mathematics, Iterative methods, dynamical study and computation. Professor Ioannis Argyros (Department of Mathematical Sciences Cameron University, Lawton, OK, USA) has published 329 indexed documents and 25 books. Argyros is interested in theories of inequalities, operators, computational mathematics and iterative methods, and banach spaces.