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Projectors and Projection Methods : Advances in Mathematics - Aurel Galantai

Projectors and Projection Methods

Advances in Mathematics

Hardcover Published: 31st December 2003
ISBN: 9781402075728
Number Of Pages: 288

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The projectors are considered as simple but important type of matrices and operators. Their basic theory can be found in many books, among which Hal- mas [177], [178] are of particular significance. The projectors or projections became an active research area in the last two decades due to ideas generated from linear algebra, statistics and various areas of algorithmic mathematics. There has also grown up a great and increasing number of projection meth- ods for different purposes. The aim of this book is to give a unified survey on projectors and projection methods including the most recent results. The words projector, projection and idempotent are used as synonyms, although the word projection is more common. We assume that the reader is familiar with linear algebra and mathemati- cal analysis at a bachelor level. The first chapter includes supplements from linear algebra and matrix analysis that are not incorporated in the standard courses. The second and the last chapter include the theory of projectors. Four chapters are devoted to projection methods for solving linear and non- linear systems of algebraic equations and convex optimization problems.

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From the reviews:

"The author presents an excellent unified survey of projectors and projection methods, including the most recent results. ... There is an extensive bibliography. This book fills a much needed of an excellent survey of old and new results in the area." (R. P. Tewarson, Zentralblatt MATH, Vol. 1055 (6), 2005)

Prefacep. ix
Supplements for Linear Algebrap. 1
Schur complementsp. 4
Factorizations and decompositionsp. 6
Normsp. 13
Generalized inversesp. 18
Linear least squares problemsp. 23
Variational characterizations of eigenvaluesp. 23
Angles between subspaces and the CS decompositionp. 25
Properties of triangular factorizationsp. 29
Perturbations of triangular factorizationsp. 30
The Gram matrixp. 35
Projectionsp. 37
Basic definitions and propertiesp. 37
Properties of orthogonal projectionsp. 40
Representations of projectionsp. 44
Operations with projectionsp. 50
Operations with two projectionsp. 50
Sums of projectionsp. 54
Generalized inverses and projectionsp. 57
The gap between two subspaces and norms of projectionsp. 59
Bounds for projectionsp. 65
Estimations for projectionsp. 67
Perturbations of projectionsp. 75
Further resultsp. 81
Finite Projection Methods for Linear Systemsp. 83
The Galerkin-Petrov projection methodp. 84
The conjugate direction methodsp. 87
Conjugation proceduresp. 93
The rank reduction procedurep. 93
Rank reduction and factorizationsp. 99
Rank reduction and conjugationp. 102
Other conjugation proceduresp. 104
Stewart's conjugation algorithmp. 104
The ABS conjugation algorithmp. 105
Perturbation analysisp. 106
The stability of conjugate direction methodsp. 106
The stability of conjugationp. 110
A survey of particular methodsp. 112
Iterative Projection Methods for Linear Algebraic Systemsp. 117
Construction principlesp. 118
Classical projection methods and their extensionsp. 119
The method of Kaczmarzp. 119
The method of Cimminop. 126
The method of Altmanp. 127
The projection methods of Gastinelp. 130
The projection methods of Householder and Bauerp. 133
Mixed methodsp. 139
Relaxed Kaczmarz methodsp. 139
Relaxed Cimmino methodsp. 146
Relaxed Householder, Bauer methodsp. 150
Projection Methods for Nonlinear Algebraic Equationsp. 155
Extensions of classical iterative projection methodsp. 157
Nonlinear conjugate direction methodsp. 163
Particular methodsp. 168
Methods with fixed direction matricesp. 168
Methods with Quasi-Newton update matricesp. 169
The nonlinear ABS methodsp. 171
Monotone convergencep. 172
Projection Methods in Optimizationp. 181
Introductionp. 181
Methods for constrained optimization problemsp. 188
Methods for convex feasibility problemsp. 200
Projection Methods for Linear Equations in Hilbert Spacesp. 215
Introductionp. 215
Projections of Hilbert spacesp. 223
Angles between subspacesp. 239
General convergence theoremsp. 244
The method of alternating projectionsp. 250
Convergence resultsp. 251
Estimates for the convergence speedp. 259
Referencesp. 265
Indexp. 285
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781402075728
ISBN-10: 1402075723
Series: Advances in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 288
Published: 31st December 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 1.32