Matrix iterative analysis.
1st softcover printing of the 2nd revised and expanded ed. 2000.

*(English)*Zbl 1216.65042
Springer Series in Computational Mathematics 27. Dordrecht: Springer (ISBN 978-3-642-05154-8/pbk; 978-3-642-05156-2/ebook). x, 358 p. (2009).

This book is a revised and expanded edition of the classic “Matrix Iterative Analysis”, which was first published in 1962 by Prentice Hall. It has had an enormous influence on the field of numerical mathematics and in particular numerical linear algebra.

The main topic is the numerical solution of linear partial differential equations. The Dirichlet problem for the unit square is used as a model. Discretizing this problem by difference methods (or finite elements later on) leads to very large linear systems which can only be solved iteratively. As the solution of the linear system is anyhow only an approximation of the solution of the partial differential equation, an iterative approach, i.e., in each step improving a given approximation, makes sense.

This book discusses in all mathematical detail two classes of methods which were very new at that time and improved the (then) classical methods such as the Jacobi and the Gauss-Seidel method. The new methods, namely the successive overrelaxation iterative methods (SOR) and the alternating direction implicit iterative methods (ADI), made it then possible to calculate more realistic examples.

The new methods depend on parameters which have a strong influence on their speed of convergence. Their determination is an important part of this book. Varga also points out that several other aspects of the numerical calculation are not treated here.

While the technical numerical aspects of this book lost their importance because of further developments and quite a few new books in this field, the description of the mathematical background of this sort of iterative methods and the ways of analysing such methods are still important and of great influence.

In particular, the stressing of the role of (elementwise) nonnegative matrices and graph theory should be mentioned. Wielandt’s proof of the Perron-Frobenius theorems is given here and made this approach popular. Also, results from rational approximation are used to discuss convergence rates.

It seems to me that this is the reason why the expanded version of this book has been published. Indeed, the new version is not very different.

In its “Preface to the revised edition” a list of items newly treated here is given. It contains among other things ovals of Cassini, \(H\)-matrices and weak regular splittings, ultrametric matrices, and matrix rational approximations to \(\exp(-z)\).

These topics had been strongly influenced by Varga in the last decades.

Finally, we give for completeness a list of the contents:

Only Chapters 4, 5 and 8 have been changed in the new version. Each chapter ends with a bibliography and very interesting discussions describing new developments.

The main topic is the numerical solution of linear partial differential equations. The Dirichlet problem for the unit square is used as a model. Discretizing this problem by difference methods (or finite elements later on) leads to very large linear systems which can only be solved iteratively. As the solution of the linear system is anyhow only an approximation of the solution of the partial differential equation, an iterative approach, i.e., in each step improving a given approximation, makes sense.

This book discusses in all mathematical detail two classes of methods which were very new at that time and improved the (then) classical methods such as the Jacobi and the Gauss-Seidel method. The new methods, namely the successive overrelaxation iterative methods (SOR) and the alternating direction implicit iterative methods (ADI), made it then possible to calculate more realistic examples.

The new methods depend on parameters which have a strong influence on their speed of convergence. Their determination is an important part of this book. Varga also points out that several other aspects of the numerical calculation are not treated here.

While the technical numerical aspects of this book lost their importance because of further developments and quite a few new books in this field, the description of the mathematical background of this sort of iterative methods and the ways of analysing such methods are still important and of great influence.

In particular, the stressing of the role of (elementwise) nonnegative matrices and graph theory should be mentioned. Wielandt’s proof of the Perron-Frobenius theorems is given here and made this approach popular. Also, results from rational approximation are used to discuss convergence rates.

It seems to me that this is the reason why the expanded version of this book has been published. Indeed, the new version is not very different.

In its “Preface to the revised edition” a list of items newly treated here is given. It contains among other things ovals of Cassini, \(H\)-matrices and weak regular splittings, ultrametric matrices, and matrix rational approximations to \(\exp(-z)\).

These topics had been strongly influenced by Varga in the last decades.

Finally, we give for completeness a list of the contents:

- {\(\bullet\)}
- Matrix properties and concepts;
- {\(\bullet\)}
- Nonnegative matrices;
- {\(\bullet\)}
- Basic iterative methods and comparison theorems;
- {\(\bullet\)}
- Successive overrelaxation iterative methods;
- {\(\bullet\)}
- Semi-iterative methods;
- {\(\bullet\)}
- Derivation and solution of elliptic difference equations;
- {\(\bullet\)}
- Alternating-direction implicit iterative methods;
- {\(\bullet\)}
- Matrix methods for parabolic partial differential equations;
- {\(\bullet\)}
- Estimation of acceleration parameters.

Only Chapters 4, 5 and 8 have been changed in the new version. Each chapter ends with a bibliography and very interesting discussions describing new developments.

Reviewer: Ludwig Elsner (Bielefeld)