Wilkinson J.H. The Algebraic Eigenvalue Problem

Clarendon Press, 1988. — 682 p.The solution of the algebraic eigenvalue problem has for long had a particular fascination for me because it illustrates so well the difference between what might be termed classical mathematics and practical numerical analysis. The eigenvalue problem has a deceptively simple formulation and the background theory has been known for many years; yet the determination of accurate solutions presents a wide variety of challenging problems.
The suggestion that I might write a book on this topic in the series of Monographs on Numerical Analysis was made by Professor L. Fox and Dr. E.T. Goodwin after my early experience with automatic digital computers. It is possible that its preparation would have remained a pious aspiration had it not been for an invitation from Professor J.W. Givens to attend a Matrix Symposium at Detroit in 1957 which led to successive invitations to lecture on 'Practical techniques for solving linear equations and computing eigenvalues and eigenvectors' at Summer Schools held in the University of Michigan. The discipline of providing a set of notes each year for these lectures has proved particularly valuable and much of the material in this book has been presented during its formative stage in this way. It was originally my intention to present a description of most of the known techniques for solving the problem, together with a critical assessment of their merits, supported wherever possible by the relevant error analysis, and by 1961 a manuscript based on these lines was almost complete. However, substantial progress had been made with the eigenvalue problem and error analysis during the period in which this manuscript was prepared and I became increasingly dissatisfied with the earlier chapters. In 1962 I took the decision to redraft the whole book with a modified objective. I felt that it was no longer practical to cover almost all known methods and to give error analyses of them and decided to include mainly those methods of which I had extensive practical experience. I also inserted an additional chapter giving fairly general error analyses which apply to almost all the methods given subsequently. Experience over the years had convinced me that it is by no means easy to give a reliable assessment of a method without using it, and that often comparatively minor changes in the details of a practical process have a disproportionate influence on its effectiveness.
The writer of a book on numerical analysis faces one particularly difficult problem, that of deciding to whom it should be addressed. A practical treatment of the eigenvalue problem is potentially of interest to a wide variety of people, including among others, design engineers, theoretical physicists, classical applied mathematicians and numerical analysts who wish to do research in the matrix field. A book which was primarily addressed to the last class of readers might prove rather inaccessible to those in the first. I have not omitted anything purely on the grounds that some readers might find it too difficult, but have tried to present everything in as elementary terms as the subject matter warrants. The dilemma was at its most acute in the first chapter. I hope the elementary presentation used there will not offend the serious mathematician and that he will treat this merely as a rough indication of the classical material with which he must be familiar if he is to benefit fully from the rest of the book. I have assumed from the start that the reader is familiar with the basic concepts of vector spaces, linear dependence and rank. An admirable introduction to the material in this book is provided by L. Fox's 'An introduction to numerical linear algebra' (Oxford, this series). Research workers in the field will find A. S. Householder's 'The theory of matrices in numerical analysis' (Blaisdell, 1964) an invaluable source of information.Theoretical Background
Perturbation Theory
Error Analysis
Solution of Linear Algebraic Equations
Hermitian Matrices
Reduction of a General Matrix to Condensed Form
Eigenvalues of Matrices of Condensed Forms
The LR and QR Algorithms
Iterative Methods