# Fundamentals of Matrix Computations, 2nd Edition

# Fundamentals of Matrix Computations, 2nd Edition

ISBN: 978-0-471-46167-8 August 2004 640 Pages

## Description

A significantly revised and improved introduction to a critical aspect of scientific computation

Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying theory clearly and in detail, along with useful insights.

This Second Edition of a popular text has now been revised and improved to appeal to the needs of practicing scientists and graduate and advanced undergraduate students. New to this edition is the use of MATLAB for many of the exercises and examples, although the Fortran exercises in the First Edition have been kept for those who want to use them. This new edition includes:

* Numerous examples and exercises on applications including electrical circuits, elasticity (mass-spring systems), and simple partial differential equations

* Early introduction of the singular value decomposition

* A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems

* An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods

With in-depth discussions of such other topics as modern componentwise error analysis, reorthogonalization, and rank-one updates of the QR decomposition, Fundamentals of Matrix Computations, Second Edition will prove to be a versatile companion to novice and practicing mathematicians who seek mastery of matrix computation.

Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying theory clearly and in detail, along with useful insights.

This Second Edition of a popular text has now been revised and improved to appeal to the needs of practicing scientists and graduate and advanced undergraduate students. New to this edition is the use of MATLAB for many of the exercises and examples, although the Fortran exercises in the First Edition have been kept for those who want to use them. This new edition includes:

* Numerous examples and exercises on applications including electrical circuits, elasticity (mass-spring systems), and simple partial differential equations

* Early introduction of the singular value decomposition

* A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems

* An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods

With in-depth discussions of such other topics as modern componentwise error analysis, reorthogonalization, and rank-one updates of the QR decomposition, Fundamentals of Matrix Computations, Second Edition will prove to be a versatile companion to novice and practicing mathematicians who seek mastery of matrix computation.

## Table of contents

Preface.

Acnowledgments.

Gaussian Elimination and its Variants.

Sensitivity of Linear Systems.

The Least Squares Problem.

The Singular Value Decomposition.

Eigenvalues and Eigenvectors I.

Eigenvalues and Eigenvectors II.

Iterative Methods for Linear Systems.

Appendix: Some Sources of Software for Matrix Computations.

References.

Index.

Index of MATLAB Terms.

Acnowledgments.

Gaussian Elimination and its Variants.

Sensitivity of Linear Systems.

The Least Squares Problem.

The Singular Value Decomposition.

Eigenvalues and Eigenvectors I.

Eigenvalues and Eigenvectors II.

Iterative Methods for Linear Systems.

Appendix: Some Sources of Software for Matrix Computations.

References.

Index.

Index of MATLAB Terms.

## New To This Edition

- Uses MATLAB to solve many of the exercises and illustrate many of the examples.
- Includes more applications in electrical circuits, mass-spring systems, and simple partial differential equations.
- Earlier introduction of the Singular Value Decomposition (SVD).
- Includes a chapter on interative methods.
- A backward error analysis of Gaussian elimination, including a discussion of the modern componentwise error analysis.
- A discussion of reorthogonalization, a practical means of obtaining numerically orthonormal vectors.
- A discussion of how to update the
*QR*decomposition when a row or column is added to or deleted from the data matrix, as happens in signal processing and data analysis applications. - A section on introducting new methods for the symmetric eigenvalue problem that have been developed since the first edition was published.

## Features

- Contains exercises of varying level, both interspersed with the text and collected at the end of each section.
- Includes exercises and examples which us MATLAB, an easy to use, very high-level language that allows users to perform very elaborate computational experiments.
- Includes examples, and exercises on applications at the beginning of each chapter, dealing with elecrical circuits, mass-spring systems, and simple partial differential equations.
- Singular Value Decomposition (SVD) is introduced in the middle of the book.
- Contains a chapter on iterative methods for solving large, sparse systems of linear equations.