How to Use This Book

This book is a guide to the numerical solution of eigenvalue problems. It attempts to present the many available methods in an organized fashion, to make it easier for you to identify the most promising methods.

Chapter 1

gives the motivation for this book and the use of templates.

Chapter 2

provides the top level of a decision tree for classifying eigenvalue problems, and their corresponding solution methods, into six categories. It is important to read enough of this chapter to make sure you pick the most efficient and accurate method for your problem.

Chapter 3

summarizes the two mathematical principles used by most algorithms for large eigenvalue problems: projection onto subspaces and spectral transformations.

Chapters 4 through 9 give details for each of the six categories of eigenvalue problems identified in Chapter 2, including algorithm templates and pointers to available software. These chapters form the core of the book.

Chapter 4:

Hermitian eigenvalue problems.

Chapter 5:

Generalized Hermitian eigenvalue problems.

Chapter 6:

Singular value decomposition.

Chapter 7:

Non-Hermitian eigenvalue problems.

Chapter 8:

Generalized non-Hermitian eigenvalue problems.

Chapter 9:

Nonlinear eigenvalue problems.

Chapter 10 describes common issues of sparse matrix representation and computation, both sequentially and in parallel, shared by all algorithms. Chapter 11 describes some preconditioning techniques that are the subject of current research.

It is inevitable that a large amount of important material needs to be excluded from this book. In the appendix, we list some of the subjects not covered in this book, giving references for the reader who is interested in pursuing a particular subject in depth.

The field of numerical methods for solving eigenvalue problems is in constant flux, with new methods and approaches continually being created, modified, tuned, and eventually, in some cases, discarded. We expect the material in this book to undergo changes from time to time as some of these new methods mature and become the state of the art. Therefore, we plan to update the material included in this book periodically in future editions. We welcome your comments and criticisms of this work to help us in that updating process. Please send your comments and questions by email to etemplates@cs.utk.edu.

The book's Web site describes how to access software discussed in this book and many other related topics. The URL for the book's homepage will be referred to as ETHOME throughout the text, which stands for http://www.netlib.org/etemplates/