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stewart g matrix algorithms vol ii eigensystem 2001

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Textbook in PDF and DJVU formats This book, Eigensystems, is the second volume in a projected five-volume series entitled Matrix Algorithms. The first volume treated basic decompositions. The three following this volume will treat iterative methods for linear systems, sparse direct methods, and special topics, including fast algorithms for structured matrices. My intended audience is the nonspecialist whose needs cannot be satisfied by black boxes. It seems to me that these people will be chiefly interested in the methods themselves— how they are derived and how they can be adapted to particular problems. Consequently, the focus of the series is on algorithms, with such topics as rounding- error analysis and perturbation theory introduced impromptu as needed. My aim is to bring the reader to the point where he or she can go to the research literature to augment what is in the series. The series is self-contained. The reader is assumed to have a knowledge of elementary analysis and linear algebra and a reasonable amount of programming experience— about what you would expect from a beginning graduate engineer or an advanced undergraduate in an honors program. Although strictly speaking the individual volumes are not textbooks, they are intended to teach, and my guiding principle has been that if something is worth explaining it is worth explaining fully. This has necessarily restricted the scope of the series, but I hope the selection of topics will give the reader a sound basis for further study. The subject of this volume is computations involving the eigenvalues and eigenvectors of a matrix. The first chapter is devoted to an exposition of the underlying mathematical theory. The second chapter treats the QR algorithm, which in its various manifestations has become the universal workhorse in this field. The third chapter deals with symmetric matrices and the singular value decomposition. The second and third chapters also treat the generalized eigenvalue problem. Up to this point the present volume shares the concern of the first volume with dense matrices and their decompositions. In the fourth chapter the focus shifts to large matrices for which the computation of a complete eigensystem is not possible. The general approach to such problems is to calculate approximations to subspaces corresponding to groups of eigenvalues—eigenspaces or invariant subspaces they are called. Accordingly, in the fourth chapter we will present the algebraic and analytic theory of eigenspaces. We then return to algorithms in the fifth chapter, which treats Krylov sequence methods — in particular the Arnoldi and the Lanczos methods. In the last chapter we consider alternative methods such as subspace iteration and the Jacobi-Davidson method. With this second volume, I have had to come to grips with how the the volumes of this work hang together. Initially, I conceived the series being a continuous exposition of the art of matrix computations, with heavy cross referencing between the volumes. On reflection, I have come to feel that the reader will be better served by semi-independent volumes. However, it is impossible to redevelop the necessary theoretical and computational material in each volume. I have therefore been forced to assume that the reader is familiar, in a general way, with the material in previous volumes. To make life easier, much of the necessary material is slipped unobtrusively into the present volume and more background can be found in the appendix. However, I have not hesitated to reference the first volume—especially its algorithms—where I thought it appropriate. Many methods treated in this volume are summarized by displays of pseudocode (see the list of algorithms following the table of contents). These summaries are for purposes of illustration and should not be regarded as finished implementations. In the first place, they often leave out error checks that would clutter the presentation. Moreover, it is difficult to verify the correctness of algorithms written in pseudocode. Wherever possible, I have checked the algorithms against MATLAB implementations. However, such a procedure is not proof against transcription errors. Be on guard! A word on organization. The book is divided into numbered chapters, sections, and subsections, followed by unnumbered subsubsections. Numbering is by section, so that (3.5) refers to the fifth equations in section three of the current chapter. References to items outside the current chapter are made explicitly—e.g., Theorem 2.7, Chapter 1 . References to Volume I of this series [265] are preceded by the Roman numeral I—e.g., Algorithm 1:3.2.1. Eigensystems The QR Algorithm The Symmetric Eigenvalue Problem Eigenspaces and Their Approximation Krylov Sequence Methods Alternatives

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stewart g matrix algorithms vol ii eigensystem 2001

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