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2010 | Book

G.W. Stewart

Selected Works with Commentaries

Editors: Misha E. Kilmer, Dianne P. O’Leary

Publisher: Birkhäuser Boston

Book Series : Contemporary Mathematicians

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About this book

Published in honor of his 70th birthday, this volume explores and celebrates the work of G.W. (Pete) Stewart, a world-renowned expert in computational linear algebra. It is widely accepted that Stewart is the successor to James Wilkinson, the first giant in the field, taking up the perturbation theory research that Wilkinson so ably began and using it as a foundation for algorithmic insights. Stewart’s results in many areas of computational linear algebra broke new ground and are still widely used in an increasing number of applications. Stewart’s papers, widely cited, are characterized by elegance in theorems and algorithms and clear, concise, and beautiful exposition. His six popular textbooks are excellent sources of knowledge and history. Stewart is a member of the National Academy of Engineering and has received numerous additional honors, including the Bauer Prize. This volume includes: forty-four of Stewart's most influential research papers in two subject areas: matrix algorithms, and rounding and perturbation theory; a biography of Stewart; a complete list of his publications, students, and honors; selected photographs; and commentaries on his works in collaboration with leading experts in the field. G.W. Stewart: Selected Works with Commentaries will appeal to graduate students, practitioners, and researchers in computational linear algebra and the history of mathematics.

Table of Contents

Frontmatter

G. W. Stewart

Frontmatter
1. Biography of G. W. Stewart
Abstract
If one is asked to name the most influential people in numerical linear algebra, then Pete (G.W.) Stewart would come very high on the list. Pete has had a major influence on the field and, in several ways, on my own career. It is with great pleasure that I pen these words as a biography and tribute to him. I am grateful to Pete not only for spending the time to discuss his life with me but also for going carefully over a draft and finding many instances where accuracy had been sacrificed for the sake of the narrative. I should, however, stress that any rounding errors that remain are purely my responsibility but hopefully do not now contaminate the result.
Iain S. Duff
2. Publications, Honors, and Students
Abstract
Dissertation: G. W. Stewart III, “Some Topics in Numerical Analysis,” University of Tennessee. Published as Technical Report ORNL-4303, Oak Ridge National Laboratory, September 1968.
Misha E. Kilmer, Dianne P. O’Leary

Commentaries

Frontmatter
3. Introduction to the Commentaries
Abstract
In research spanning over 40 years, G.W. (Pete) Stewart has made foundational contributions to numerical linear algebra.
Misha E. Kilmer, Dianne P. O’Leary
4. Matrix Decompositions: Linpack and Beyond
Abstract
Stewart’s thesis advisor Alston Householder was a strong and effective advocate for using factorizations in explaining and in solving matrix problems [73]. Stewart adopted a similar viewpoint in his expository work; see [GWS-B1,GWS-B3], etc. Through his research, Stewart brought new theoretical insights into matrix factorizations and worked on definitive software implementations of factorization algorithms. In this chapter we focus on some of his key contributions to LU, QR, Cholesky, and singular value decompositions (SVD). Additional contributions to eigendecompositions, SVD, and updating of factorizations will be discussed in later chapters.
Charles F. Van Loan, Misha E. Kilmer, Dianne P. O’Leary
5. Updating and Downdating Matrix Decompositions
Abstract
The Sherman–Morrison–Woodbury formula ([GWS-B7], p. 328) is a recipe for constructing the inverse of a matrix after it has been modified by a low-rank correction. For a matrix of size n ×n that has been so modified, it enables the inverse of this matrix to be updated in time proportional to kn, where k is the rank of the correction, rather than the n 3 time usually necessary to compute the inverse directly. This important fact has enabled a variety of algorithms, from early implementations of the simplex method for linear optimization [29] to algorithms for solving least squares problems when new data arrive.
Lars Eldén, Misha E. Kilmer, Dianne P. O’Leary
6. Least Squares, Projections, and Pseudoinverses
Abstract
These papers form a small sample of Stewart’s work on least squares, projec- 14 tions, and generalized inverses; see also [GWS-J23, GWS-J28, GWS-J60, GWS-J97, 15 GWS-J119, GWS-N5, GWS-N24], for example. Stewart’s focus in this area was to 16 put the understanding of generalized inverses on as firm a footing as that of the 17 usual matrix inverse. He proceeded by establishing continuity properties, effects of 18 scaling, and perturbation theory, using orthogonal projectors as a unifying frame- 19 work. 20
Misha E. Kilmer, Dianne P. O’Leary
7. The Eigenproblem and Invariant Subspaces: Perturbation Theory
Abstract
In this collection of papers, Pete Stewart established the foundations for the perturbation theory of invariant subspaces. He introduced two crucial concepts that allow a systematic approach toward such a perturbation theory: subspace rotation and operator separation. These two concepts form the guiding principle in most of these papers.
Ilse C. F. Ipsen
8. The SVD, Eigenproblem, and Invariant Subspaces: Algorithms
Abstract
These papers form a sample of Stewart’s work on algorithms for the eigenvalue problem, in particular computing selected invariant subspaces and corresponding eigenvalues and eigenvectors of both Hermitian and non-Hermitian matrices. We also include some contributions to the SVD. Algorithms for the generalized nonsymmetric eigenproblem, in particular the QZ algorithm, are discussed in Sect. 9.2.
James W. Demmel
9. The Generalized Eigenproblem
Abstract
These papers describe Stewart’s original and fundamental contributions on the generalized matrix eigenvalue problem. In these papers, Stewart systematically presented perturbation theory and sensitivity analysis for the problem, and (with Moler) introduced a landmark algorithm, namely the QT algorithm, for computing eigenvalues and eigenvectors.
Zhaojun Bai
10. Krylov Subspace Methods for the Eigenproblem
Abstract
These papers comprise some of Stewart’s recent contributions to the development and analysis of iterative algorithms based on Krylov subspace methods for computing eigenvalues.
Howard C. Elman, Dianne P. O’Leary
11. Other Contributions
Abstract
The preceding seven chapters of this commentary had outlined some of Stewart’s important contributions to matrix algorithms and matrix perturbation theory.
Misha E. Kilmer, Dianne P. O’Leary
Backmatter

Reprints

Frontmatter
12. Papers on Matrix Decompositions
Misha E. Kilmer, Dianne P. O’Leary
13. Papers on Updating and Downdating Matrix Decompositions
Misha E. Kilmer, Dianne P. O’Leary
14. Papers on Least Squares, Projections, and Generalized Inverses
Misha E. Kilmer, Dianne P. O’Leary
15. Papers on the Eigenproblem and Invariant Subspaces: Perturbation Theory
Misha E. Kilmer, Dianne P. O’Leary
16. Papers on the SVD, Eigenproblem and Invariant Subspaces: Algorithms
Misha E. Kilmer, Dianne P. O’Leary
17. Papers on the Generalized Eigenproblem
Misha E. Kilmer, Dianne P. O’Leary
18. Papers on Krylov Subspace Methods for the Eigenproblem
Misha E. Kilmer, Dianne P. O’Leary
Metadata
Title
G.W. Stewart
Editors
Misha E. Kilmer
Dianne P. O’Leary
Copyright Year
2010
Publisher
Birkhäuser Boston
Electronic ISBN
978-0-8176-4968-5
Print ISBN
978-0-8176-4967-8
DOI
https://doi.org/10.1007/978-0-8176-4968-5

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