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A Contemporary Study of Iterative Methods

Á. Alberto Magreñán and Ioannis K. Argyros
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is an encyclopedia of iterative methods for calculating approximate solutions of problems. The problems fall generally into two categories: minimization and equation-solving. The context is usually operators on Banach or Hilbert spaces. The problems of interest are those for which no exact solution is available, and we seek a numerical approximation. The book does not deal with iterative methods for proving existence of mathematical objects with particular properties, and so for example generally does not deal with fixed-point theorems.

Iterative methods go back thousands of years. Heron’s method for finding square roots is an ancient example. Most of us are familiar with Newton’s iterative method for finding roots. Almost as old is the Gauss–Newton method for solving non-linear least-squares problems. Many variations of both of these appear in the book, along with lots of other approaches. These methods have become important as the problems become more complicated and computers become more powerful.

“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.

The big weakness of the book is that there is no overview or survey chapter. Each chapter covers one particular method and is almost independent of the other chapters, so unless you are already familiar with the subject area, you have to browse through the chapters looking for something you can use.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

  1. The majorization method in the Kantorovich theory
  2. Directional Newton methods
  3. Newton’s method
  4. Generalized equations
  5. Gauss–Newton method
  6. Gauss–Newton method for convex optimization
  7. Proximal Gauss–Newton method
  8. Multistep modified Newton–Hermitian and Skew-Hermitian Splitting method
  9. Secant-like methods in chemistry
  10. Robust convergence of Newton’s method for cone inclusion problem
  11. Gauss–Newton method for convex composite optimization
  12. Domain of parameters
  13. Newton’s method for solving optimal shape design problems
  14. Osada method
  15. Newton’s method to solve equations with solutions of multiplicity greater than one
  16. Laguerre-like method for multiple zeros
  17. Traub’s method for multiple roots
  18. Shadowing lemma for operators with chaotic behavior
  19. Inexact two-point Newton-like methods
  20. Two-step Newton methods
  21. Introduction to complex dynamics
  22. Convergence and the dynamics of Chebyshev–Halley type methods
  23. Convergence planes of iterative methods
  24. Convergence and dynamics of a higher order family of iterative methods
  25. Convergence and dynamics of iterative methods for multiple zeros