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 allenstenger.com. His mathematical interests are number theory and classical analysis.