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Numerical Liner Algebra with Applications.pdf
Front Cover
Numerical Linear Algebra with Applications
Copyright
Dedication
Contents
List of Figures
List of Algorithms
Preface
Matrices
Matrix Arithmetic
Matrix Product
The Trace
MATLAB Examples
Linear Transformations
Rotations
Powers of Matrices
Nonsingular Matrices
The Matrix Transpose and Symmetric Matrices
Chapter Summary
Problems
MATLAB Problems
Linear Equations
Introduction to Linear Equations
Solving Square Linear Systems
Gaussian Elimination
Upper-Triangular Form
Systematic Solution of Linear Systems
Computing the Inverse
Homogeneous Systems
Application: A Truss
Application: Electrical Circuit
Chapter Summary
Problems
MATLAB Problems
Subspaces
Introduction
Subspaces of Rn
Linear Independence
Basis of a Subspace
The Rank of a Matrix
Chapter Summary
Problems
MATLAB Problems
Determinants
Developing the Determinant of a 2bold0mu mumu section2 and a 3bold0mu mumu section3 Matrix
Expansion by Minors
Computing a Determinant Using Row Operations
Application: Encryption
Chapter Summary
Problems
MATLAB Problems
Eigenvalues and Eigenvectors
Definitions and Examples
Selected Properties of Eigenvalues and Eigenvectors
Diagonalization
Powers of Matrices
Applications
Electric Circuit
Irreducible Matrices
Ranking of Teams Using Eigenvectors
Computing Eigenvalues and Eigenvectors using MATLAB
Chapter Summary
Problems
MATLAB Problems
Orthogonal Vectors and Matrices
Introduction
The Inner Product
Orthogonal Matrices
Symmetric Matrices and Orthogonality
The L2 Inner Product
The Cauchy-Schwarz Inequality
Signal Comparison
Chapter Summary
Problems
MATLAB Problems
Vector and Matrix Norms
Vector Norms
Properties of the 2-Norm
Spherical Coordinates
Matrix Norms
The Frobenius Matrix Norm
Induced Matrix Norms
Submultiplicative Matrix Norms
Computing the Matrix 2-Norm
Properties of the Matrix 2-Norm
Chapter Summary
Problems
MATLAB Problems
Floating Point Arithmetic
Integer Representation
Floating-Point Representation
Mapping from Real Numbers to Floating-Point Numbers
Floating-Point Arithmetic
Relative Error
Rounding Error Bounds
Addition
Multiplication
Matrix Operations
Minimizing Errors
Avoid Adding a Huge Number to a Small Number
Avoid Subtracting Numbers That Are Close
Chapter Summary
Problems
MATLAB Problems
Algorithms
Pseudocode Examples
Inner Product of Two Vectors
Computing the Frobenius Norm
Matrix Multiplication
Block Matrices
Algorithm Efficiency
Smaller Flop Count Is Not Always Better
Measuring Truncation Error
The Solution to Upper and Lower Triangular Systems
Efficiency Analysis
The Thomas Algorithm
Efficiency Analysis
Chapter Summary
Problems
MATLAB Problems
Conditioning of Problems and Stability of Algorithms
Why Do We Need Numerical Linear Algebra?
Computation Error
Forward Error
Backward Error
Algorithm Stability
Examples of Unstable Algorithms
Conditioning of a Problem
Perturbation Analysis for Solving a Linear System
Properties of the Matrix Condition Number
MATLAB Computation of a Matrix Condition Number
Estimating the Condition Number
Introduction to Perturbation Analysis of Eigenvalue Problems
Chapter Summary
Problems
MATLAB Problems
Gaussian Elimination and the LU Decomposition
LU Decomposition
Using LU to Solve Equations
Elementary Row Matrices
Derivation of the LU Decomposition
Colon Notation
The LU Decomposition Algorithm
LU Decomposition Flop Count
Gaussian Elimination with Partial Pivoting
Derivation of PA=LU
Algorithm for Gaussian Elimination with Partial Pivoting
Using the LU Decomposition to Solve Axi = bi, 1i k
Finding A–1
Stability and Efficiency of Gaussian Elimination
Iterative Refinement
Chapter Summary
Problems
MATLAB Problems
Linear System Applications
Fourier Series
The Square Wave
Finite Difference Approximations
Steady-State Heat and Diffusion
Least-Squares Polynomial Fitting
Normal Equations
Cubic Spline Interpolation
Chapter Summary
Problems
MATLAB Problems
Important Special Systems
Tridiagonal Systems
Symmetric Positive Definite Matrices
Applications
The Cholesky Decomposition
Computing the Cholesky Decomposition
Efficiency
Solving Ax = b If A Is Positive Definite
Stability
Chapter Summary
Problems
MATLAB Problems
Gram-Schmidt Orthonormalization
The Gram-Schmidt Process
Numerical Stability of the Gram-Schmidt Process
The QR Decomposition
Efficiency
Stability
Applications of the QR Decomposition
Computing the Determinant
Finding an Orthonormal Basis for the Range of a Matrix
Chapter Summary
Problems
MATLAB Problems
The Singular Value Decomposition
The SVD Theorem
Using the SVD to Determine Properties of a Matrix
The Four Fundamental Subspaces of a Matrix
SVD and Matrix Norms
Geometric Interpretation of the SVD
Computing the SVD Using MATLAB
Computing A–1
Image Compression Using the SVD
Image Compression Using MATLAB
Additional Uses
Final Comments
Chapter Summary
Problems
MATLAB Problems
Least-Squares Problems
Existence and Uniqueness of Least-Squares Solutions
Existence and Uniqueness Theorem
Normal Equations and Least-Squares Solutions
The Pseudoinverse, m n
The Pseudoinverse, m
Solving Overdetermined Least-Squares Problems
Using the Normal Equations
Efficiency
Computational Note
Using the QR Decomposition
Efficiency
Using the SVD
Efficiency
Remark on Curve Fitting
Conditioning of Least-Squares Problems
Sensitivity when using the Normal Equations
Rank-Deficient Least-Squares Problems
Efficiency
Underdetermined Linear Systems
Efficiency
Chapter Summary
Problems
MATLAB Problems
Implementing the QR Decomposition
Review of the QR Decomposition Using Gram-Schmidt
Givens Rotations
Zeroing a Particular Entry in a Vector
Creating a Sequence of Zeros in a Vector Using Givens Rotations
Product of a Givens Matrix with a General Matrix
Zeroing-Out Column Entries in a Matrix Using Givens Rotations
Accurate Computation of the Givens Parameters
The Givens Algorithm for the QR Decomposition
The Reduced QR Decomposition
Efficiency
Householder Reflections
Matrix Column Zeroing Using Householder Reflections
Implicit Computation with Householder Reflections
Computing the QR Decomposition Using Householder Reflections
Efficiency and Stability
Chapter Summary
Problems
MATLAB Problems
The Algebraic Eigenvalue Problem
Applications of the Eigenvalue Problem
Vibrations and Resonance
The Leslie Model in Population Ecology
Buckling of a Column
Computation of Selected Eigenvalues and Eigenvectors
Additional Property of a Diagonalizable Matrix
The Power Method for Computing the Dominant Eigenvalue
Computing the Smallest Eigenvalue and Corresponding Eigenvector
The Basic QR Iteration
Transformation to Upper Hessenberg Form
Efficiency and Stability
The Unshifted Hessenberg QR Iteration
Efficiency
The Shifted Hessenberg QR Iteration
A Single Shift
Schur's Triangularization
The Francis Algorithm
Francis Iteration of Degree One
Preparation for Understanding the Iteration
Demonstration of the Francis Iteration of Degree One
Francis Iteration of Degree Two
Computing Eigenvectors
Hessenberg Inverse Iteration
Computing Both Eigenvalues and TheirCorresponding Eigenvectors
Sensitivity of Eigenvalues to Perturbations
Sensitivity of Eigenvectors
Chapter Summary
Problems
MATLAB Problems
The Symmetric Eigenvalue Problem
The Spectral Theorem and Properties of a Symmetric Matrix
Properties of a Symmetric Matrix
The Jacobi Method
Computing Eigenvectors Using the Jacobi Iteration
The Cyclic-by-Row Jacobi Algorithm
The Symmetric QR Iteration Method
Tridiagonal Reduction of a Symmetric Matrix
Efficiency
Orthogonal Transformation to a Diagonal Matrix
The Symmetric Francis Algorithm
Theoretical Overview and Efficiency
The Bisection Method
Efficiency
Matrix A Is Not Unreduced
The Divide-and-Conquer Method
Using dconquer
Chapter Summary
Problems
MATLAB Problems
Basic Iterative Methods
Jacobi Method
The Gauss-Seidel Iterative Method
The SOR Iteration
Convergence of the Basic Iterative Methods
Matrix Form of the Jacobi Iteration
Matrix Form of the Gauss-Seidel Iteration
Matrix Form for SOR
Conditions Guaranteeing Convergence
The Spectral Radius and Rate of Convergence
Convergence of the Jacobi and Gauss-Seidel Methods for Diagonally Dominant Matrices
Choosing for SOR
Application: Poisson's Equation
Chapter Summary
Problems
MATLAB Problems
Krylov Subspace Methods
Large, Sparse Matrices
Storage of Sparse Matrices
The CG Method
The Method of Steepest Descent
From Steepest Descent to CG
Convergence
Preconditioning
Preconditioning for CG
Incomplete Cholesky Decomposition
SSOR Preconditioner
Krylov Subspaces
The Arnoldi Method
Efficiency
An Alternative Formulation of the Arnoldi Decomposition
GMRES
Convergence
Preconditioned GMRES
The Symmetric Lanczos Method
Loss of Orthogonality with the Lanczos Process
The MINRES Method
Convergence
Comparison of Iterative Methods
Poisson's Equation Revisited
The Biharmonic Equation
Chapter Summary
Problems
MATLAB Problems
Large Sparse Eigenvalue Problems
The Power Method
Eigenvalue Computation Using the Arnoldi Process
Estimating Eigenvalues Without Restart or Deflation
Estimating Eigenvalues Using Restart
A Restart Method Using Deflation
Restart Strategies
The Implicitly Restarted Arnoldi Method
Convergence of the Arnoldi Iteration
Eigenvalue Computation Using the Lanczos Process
Mathematically Provable Properties
Chapter Summary
Problems
MATLAB Problems
Computing the Singular Value Decomposition
Development of the One-Sided Jacobi Methodfor Computing the Reduced SVD
Stability of Singular Value Computation
The One-Sided Jacobi Algorithm
Faster and More Accurate Jacobi Algorithm
Transforming a Matrix to Upper-Bidiagonal Form
Demmel and Kahan Zero-Shift QR Downward Sweep Algorithm
Chapter Summary
Problems
MATLAB Problems
Complex Numbers
Constructing the Complex Numbers
Calculating with Complex Numbers
Geometric Representation of C
Complex Conjugate
Complex Numbers in MATLAB
Euler's Formula
Problems
MATLAB Problems
Mathematical Induction
Problems
Chebyshev Polynomials
Definition
Properties
Problems
MATLAB Problems
Glossary
Bibliography
Index
Numerical Linear Algebra with Applications
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Numerical Linear Algebra
with Applications
Using MATLAB
By
William Ford
Department of Computer Science
University of the Pacific
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier
Academic Press is an imprint of Elsevier
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The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK
First edition 2015
Copyright © 2015 Elsevier Inc. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
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This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be
noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes
in research methods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information,
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and the safety of others, including parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or
damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods,
products, instructions, or ideas contained in the material herein.
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A catalogue record for this book is available from the British Library
For information on all Academic Press publications
visit our web site at store.elsevier.com
Printed and bound in USA
ISBN: 978-0-12-394435-1
Dedication
Dedicated to my friend the late Paul Burdick
for all the wonderful conversations we had
and to
Dr. Ravi Jain, a great Dean and friend
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Contents
List of Figures
List of Algorithms
Preface
1. Matrices
1.1. Matrix Arithmetic
1.1.1. Matrix Product
1.1.2. The Trace
1.1.3. MATLAB Examples
1.2. Linear Transformations
1.2.1. Rotations
1.3. Powers of Matrices
1.4. Nonsingular Matrices
1.5. The Matrix Transpose and Symmetric
Matrices
1.6. Chapter Summary
1.7. Problems
1.7.1. MATLAB Problems
2. Linear Equations
Introduction to Linear Equations
2.1.
2.2. Solving Square Linear Systems
2.3. Gaussian Elimination
2.3.1. Upper-Triangular Form
2.4. Systematic Solution of Linear Systems
2.5. Computing the Inverse
2.6. Homogeneous Systems
2.7. Application: A Truss
2.8. Application: Electrical Circuit
2.9. Chapter Summary
2.10. Problems
2.10.1. MATLAB Problems
3. Subspaces
3.1. Introduction
3.2. Subspaces of Rn
3.3. Linear Independence
3.4. Basis of a Subspace
3.5. The Rank of a Matrix
3.6. Chapter Summary
3.7. Problems
3.7.1. MATLAB Problems
xiii
xvii
xix
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1
2
5
6
7
7
11
13
16
18
19
22
25
25
27
28
29
31
34
36
37
39
40
42
43
47
47
47
49
50
51
55
56
57
4. Determinants
4.1. Developing the Determinant of a 2 × 2
and a 3 × 3 Matrix
4.2. Expansion by Minors
4.3. Computing a Determinant Using Row
Operations
4.4. Application: Encryption
4.5. Chapter Summary
4.6. Problems
4.6.1. MATLAB Problems
5. Eigenvalues and Eigenvectors
5.1. Definitions and Examples
5.2. Selected Properties of Eigenvalues
and Eigenvectors
5.3. Diagonalization
5.3.1. Powers of Matrices
5.4. Applications
5.4.1. Electric Circuit
5.4.2. Irreducible Matrices
5.4.3. Ranking of Teams Using
Eigenvectors
5.5. Computing Eigenvalues and Eigenvectors
using MATLAB
5.6. Chapter Summary
5.7. Problems
5.7.1. MATLAB Problems
6. Orthogonal Vectors
and Matrices
6.1. Introduction
6.2. The Inner Product
6.3. Orthogonal Matrices
6.4. Symmetric Matrices and Orthogonality
6.5. The L2 Inner Product
6.6. The Cauchy-Schwarz Inequality
6.7. Signal Comparison
6.8. Chapter Summary
6.9. Problems
6.9.1. MATLAB Problems
59
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79
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