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2004 | Buch

Supplements for Linear Algebra Kapitel lesen Erstes Kapitel lesen

Projectors and Projection Methods

verfasst von: Aurél Galántai

Verlag: Springer US

Buchreihe : Advances in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Supplements for Linear Algebra
Abstract
The purpose of this chapter is to collect the notation, concepts and results of linear algebra that are usually not part of the standard linear algebra courses and which we need to develop our material. We use the following standard notation and concepts.
Aurél Galántai
Chapter 2. Projections
Abstract
This chapter is a collection of various results on projections, where the well-known facts are supplemented by less-known ones. The material is selected so that it supports the subsequent parts of the book. Several researchers, mainly in the last decade, investigated the properties of projections and achieved a large number of interesting and important new results. Since we could include only a fraction of these, we encourage the interested reader to consult with the cited works as well.
Aurél Galántai
Chapter 3. Finite Projection Methods for Linear Systems
Abstract
We can define projection methods in several ways. Consider the equation
$$Ax = y,$$
(3.1)
, where \(A \in {\mathbb{R}^{m\; \times \;n}}\).
Aurél Galántai
Chapter 4. Iterative Projection Methods for Linear Algebraic Systems
Abstract
We consider linear algebraic systems of the form
$$Ax = b\left( {A \in {\mathbb{R}^{n \times n}},b \in {\mathbb{R}^n}} \right)$$
(4.1)
provided that A is nonsingular.
Aurél Galántai
Chapter 5. Projection Methods for Nonlinear Algebraic Equations
Abstract
We investigate two types of projection methods for solving nonlinear algebraic equations of the form
$$F\left( x \right) = 0\;\left( {F:{\mathbb{R}^m} \to {\mathbb{R}^m}} \right),$$
(5.1)
, where
$$F\left( x \right) = {\left[ {{f_1}\left( x \right), \ldots ,{f_m}\left( x \right)} \right]^T}.$$
(5.2)
.
Aurél Galántai
Chapter 6. Projection Methods in Optimization
Abstract
Projections appear in many areas of optimization. In linear programming we mention the interior gradient projection method of Pyle and Cline [299], [298] and the interior point method of Karmarkar. Papers [349], [328], [166], [211] on projections are related to the Karmarkar algorithm. In nonlinear programming the gradient projection methods and the convex feasibility problem are perhaps the most important applications of projections. Further typical applications can be found in papers [185], [238], [192], [77], [78], [313], [82], [99].
Aurél Galántai
Chapter 7. Projection Methods for Linear Equations in Hilbert Spaces
Abstract
Projections and projection methods in abstract vector spaces show similarities and differences when compared to the finite-dimensional case. The difference stems from the change from finite- to infinite-dimensional vector spaces. Since a deeper study requires a strong background in functional analysis we restrict the subject to results that can be understood easily. We concentrate on the alternating projection method that is behind the many algorithms we studied in earlier chapters. For easier understanding we summarize the necessary basic concepts and results from functional analysis. For deeper details the reader can consult with any of the books [177], [6], [301], [236], [217], [113], [128], [27] and the references therein.
Aurél Galántai
Backmatter
Metadaten
Titel
Projectors and Projection Methods
verfasst von
Aurél Galántai
Copyright-Jahr
2004
Verlag
Springer US
Electronic ISBN
978-1-4419-9180-5
Print ISBN
978-1-4613-4825-2
DOI
https://doi.org/10.1007/978-1-4419-9180-5