Computational Methods for Matrix Eigenproblems: A.R. Gourlay, G.A. Watson

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From the Preface:
Since 1947, numerical analysis has enjoyed dedicated effort from mathematicians (both pure and applied), computer scientists, engineers and the like, all striving to devise and analyse processes aimed at being of use and value in the solution of the problems of the outside world. One of the topic areas of numerical analysis in which this effort has led to rich rewards is that of computational linear algebra. Our understanding of matrix problems is much more complete than that of almost any other subset of numerical analysis. We have powerful algorithms for solving linear equations, and for tackling nearly all the commonly occurring eigenproblems. More important perhaps is the existence of reliable and robust computer routines for these algorithms. Much of this knowledge is contained in the monumental work by Wilkinson (The Algebraic Eigenvalue Problem, O.U.P., 1965). The rest is in the more recent literature.

The topics, numerical solution of linear systems, and numerical solution of matrix eigenproblems, are fundamental to courses in numerical analysis. Therefore, the availability of suitable textbooks for such courses is desirable. For the first topic, the book by Forsythe and Moler {Computer Solution of Linear Systems, Prentice Hall, 1967) is very suitable. Its style of short, single topic chapters is one which should appeal to lecturers and students alike. The present text is an attempt to provide a similar book for the second topic. The content is based on lectures given by the authors to M.Sc. students at I he University of Dundee. Some of the material has also been used in undergraduate courses at the same University. All these students would have attended a basic course in matrix algebra and be familiar with the fundamental concepts, for example the definitions of matrix multiplication, inversion and determinants. This book is written for students with such a background. The material is suitable for courses to students both in mathematical disciplines und in the more applied subjects such as engineering, and, because of the particular format in which the material is presented, it could form the basis for any level of course in its subject matter. Each chapter could be covered in one or two lectures, and thus the whole text could be suitable for a course of some fifteen to twenty lectures (including an allocation of time for consideration of some of the relevant chapters in the book by Forsythe and Moler). The subjects have been chosen so as to present only the more commonly tiHcd and more reliable techniques for computing solutions to eigenproblems.

The aim is primarily to describe the techniques; therefore little will be said of the error analysis of each method, although the conclusions to be drawn from the relevant error analyses will of course be stressed. It is hoped that any student whose interest may be roused by the material covered here will turn to the book by Wilkinson for the 'complete story'.

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