# Numerical Linear Algebra with Applications

### Using MATLAB

## Description

## Key Features

- Six introductory chapters that thoroughly provide the required background for those who have not taken a course in applied or theoretical linear algebra
- Detailed explanations and examples
- A through discussion of the algorithms necessary for the accurate computation of the solution to the most frequently occurring problems in numerical linear algebra
- Examples from engineering and science applications

## Readership

Graduate or advanced undergraduate students in engineering, science, and mathematics, professionals in engineering and science, such as practicing engineers who want to see how numerical linear algebra problems can be solved using a programming language such as MATLAB, MAPLE, or Mathematica.

## Table of Contents

- Dedication
- List of Figures
- List of Algorithms
- Preface
- Topics
- Intended Audience
- Ways to Use the Book
- Matlab Library
- Supplement
- Acknowledgments

- Chapter 1: Matrices
- Abstract
- 1.1 Matrix Arithmetic
- 1.2 Linear Transformations
- 1.3 Powers of Matrices
- 1.4 Nonsingular Matrices
- 1.5 The Matrix Transpose and Symmetric Matrices
- 1.6 Chapter Summary
- 1.7 Problems

- Chapter 2: Linear Equations
- Abstract
- 2.1 Introduction to Linear Equations
- 2.2 Solving Square Linear Systems
- 2.3 Gaussian Elimination
- 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

- Chapter 3: Subspaces
- Abstract
- 3.1 Introduction
- 3.2 Subspaces of
^{n} - 3.3 Linear Independence
- 3.4 Basis of a Subspace
- 3.5 The Rank of a Matrix
- 3.6 Chapter summary
- 3.7 Problems

- Chapter 4: Determinants
- Abstract
- 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

- Chapter 5: Eigenvalues and Eigenvectors
- Abstract
- 5.1 Definitions and Examples
- 5.2 Selected Properties of Eigenvalues and Eigenvectors
- 5.3 Diagonalization
- 5.4 Applications
- 5.5 Computing Eigenvalues and Eigenvectors Using Matlab
- 5.6 Chapter Summary
- 5.7 Problems

- Chapter 6: Orthogonal Vectors and Matrices
- Abstract
- 6.1 Introduction
- 6.2 The Inner Product
- 6.3 Orthogonal Matrices
- 6.4 Symmetric Matrices and Orthogonality
- 6.5 The
*L*^{2}inner product - 6.6 The Cauchy-Schwarz Inequality
- 6.7 Signal Comparison
- 6.8 Chapter Summary
- 6.9 Problems

- Chapter 7: Vector and Matrix Norms
- Abstract
- 7.1 Vector Norms
- 7.2 Matrix Norms
- 7.3 Submultiplicative Matrix Norms
- 7.4 Computing the Matrix 2-Norm
- 7.5 Properties of the Matrix 2-Norm
- 7.6 Chapter Summary
- 7.7 Problems

- Chapter 8: Floating Point Arithmetic
- Abstract
- 8.1 Integer Representation
- 8.2 Floating-Point Representation
- 8.3 Floating-Point Arithmetic
- 8.4 Minimizing Errors
- 8.5 Chapter summary
- 8.6 Problems

- Chapter 9: Algorithms
- Abstract
- 9.1 Pseudocode Examples
- 9.2 Algorithm Efficiency
- 9.3 The Solution to Upper and Lower Triangular Systems
- 9.4 The Thomas Algorithm
- 9.5 Chapter Summary
- 9.6 Problems

- Chapter 10: Conditioning of Problems and Stability of Algorithms
- Abstract
- 10.1 Why do we need numerical linear algebra?
- 10.2 Computation error
- 10.3 Algorithm stability
- 10.4 Conditioning of a problem
- 10.5 Perturbation analysis for solving a linear system
- 10.6 Properties of the matrix condition number
- 10.7 Matlab computation of a matrix condition number
- 10.8 Estimating the condition number
- 10.9 Introduction to perturbation analysis of eigenvalue problems
- 10.10 Chapter summary
- 10.11 Problems

- Chapter 11: Gaussian Elimination and the
*LU*Decomposition- Abstract
- 11.1
*LU*Decomposition - 11.2 Using
*LU*to Solve Equations - 11.3 Elementary Row Matrices
- 11.4 Derivation of the
*LU*Decomposition - 11.5 Gaussian Elimination with Partial Pivoting
- 11.6 Using the LU Decomposition to Solve Axi=bi,1≤i≤k
- 11.7 Finding
*A*^{–1} - 11.8 Stability and Efficiency of Gaussian Elimination
- 11.9 Iterative Refinement
- 11.10 Chapter Summary
- 11.11 Problems

- Chapter 12: Linear System Applications
- Abstract
- 12.1 Fourier Series
- 12.2 Finite Difference Approximations
- 12.3 Least-Squares Polynomial Fitting
- 12.4 Cubic Spline Interpolation
- 12.5 Chapter Summary
- 12.6 Problems

- Chapter 13: Important Special Systems
- Abstract
- 13.1 Tridiagonal Systems
- 13.2 Symmetric Positive Definite Matrices
- 13.3 The Cholesky Decomposition
- 13.4 Chapter Summary
- 13.5 Problems

- Chapter 14: Gram-Schmidt Orthonormalization
- Abstract
- 14.1 The Gram-Schmidt Process
- 14.2 Numerical Stability of the Gram-Schmidt Process
- 14.3 The QR Decomposition
- 14.4 Applications of The
*QR*Decomposition - 14.5 Chapter Summary
- 14.6 Problems

- Chapter 15: The Singular Value Decomposition
- Abstract
- 15.1 The SVD Theorem
- 15.2 Using the SVD to Determine Properties of a Matrix
- 15.3 SVD and Matrix Norms
- 15.4 Geometric Interpretation of the SVD
- 15.5 Computing the SVD Using MATLAB
- 15.6 Computing
*A*^{–1} - 15.7 Image Compression Using the SVD
- 15.8 Final Comments
- 15.9 Chapter Summary
- 15.10 Problems

- Chapter 16: Least-Squares Problems
- Abstract
- 16.1 Existence and Uniqueness of Least-Squares Solutions
- 16.2 Solving Overdetermined Least-Squares Problems
- 16.3 Conditioning of Least-Squares Problems
- 16.4 Rank-Deficient Least-Squares Problems
- 16.5 Underdetermined Linear Systems
- 16.6 Chapter Summary
- 16.7 Problems

- Chapter 17: Implementing the
*QR*Decomposition- Abstract
- 17.1 Review of the
*QR*Decomposition Using Gram-Schmidt - 17.2 Givens Rotations
- 17.3 Creating a Sequence of Zeros in a Vector Using Givens Rotations
- 17.4 Product of a Givens Matrix with a General Matrix
- 17.5 Zeroing-Out Column Entries in a Matrix Using Givens Rotations
- 17.6 Accurate Computation of the Givens Parameters
- 17.7 THe Givens Algorithm for the
*QR*Decomposition - 17.8 Householder Reflections
- 17.9 Computing the
*QR*Decomposition Using Householder Reflections - 17.10 Chapter Summary
- 17.11 Problems

- Chapter 18: The Algebraic Eigenvalue Problem
- Abstract
- 18.1 Applications of The Eigenvalue Problem
- 18.2 Computation of Selected Eigenvalues and Eigenvectors
- 18.3 The Basic
*QR*Iteration - 18.4 Transformation to Upper Hessenberg Form
- 18.5 The Unshifted Hessenberg
*QR*Iteration - 18.6 The Shifted Hessenberg
*QR*Iteration - 18.7 Schur's Triangularization
- 18.8 The Francis Algorithm
- 18.9 Computing Eigenvectors
- 18.10 Computing Both Eigenvalues and Their Corresponding Eigenvectors
- 18.11 Sensitivity of Eigenvalues to Perturbations
- 18.12 Chapter Summary
- 18.13 Problems

- Chapter 19: The Symmetric Eigenvalue Problem
- Abstract
- 19.1 The Spectral Theorem and Properties of A Symmetric Matrix
- 19.2 The Jacobi Method
- 19.3 The Symmetric
*QR*Iteration Method - 19.4 The Symmetric Francis Algorithm
- 19.5 The Bisection Method
- 19.6 The Divide-And-Conquer Method
- 19.7 Chapter Summary
- 19.8 Problems

- Chapter 20: Basic Iterative Methods
- Abstract
- 20.1 Jacobi Method
- 20.2 The Gauss-Seidel Iterative Method
- 20.3 The Sor Iteration
- 20.4 Convergence of the Basic Iterative Methods
- 20.5 Application: Poisson's Equation
- 20.6 Chapter Summary
- 20.7 Problems

- Chapter 21: Krylov Subspace Methods
- Abstract
- 21.1 Large, Sparse Matrices
- 21.2 The CG Method
- 21.3 Preconditioning
- 21.4 Preconditioning For CG
- 21.5 Krylov Subspaces
- 21.6 The Arnoldi Method
- 21.7 GMRES
- 21.8 The Symmetric Lanczos Method
- 21.9 The Minres Method
- 21.10 Comparison of Iterative Methods
- 21.11 Poisson's Equation Revisited
- 21.12 The Biharmonic Equation
- 21.13 Chapter Summary
- 21.14 Problems

- Chapter 22: Large Sparse Eigenvalue Problems
- Abstract
- 22.1 The Power Method
- 22.2 Eigenvalue Computation Using the Arnoldi Process
- 22.3 The Implicitly Restarted Arnoldi Method
- 22.4 Eigenvalue Computation Using the Lanczos Process
- 22.5 Chapter Summary
- 22.6 Problems

- Chapter 23: Computing the Singular Value Decomposition
- Abstract
- 23.1 Development of the One-Sided Jacobi Method For Computing the Reduced Svd
- 23.2 The One-Sided Jacobi Algorithm
- 23.3 Transforming a Matrix to Upper-Bidiagonal Form
- 23.4 Demmel and Kahan Zero-Shift
*QR*Downward Sweep Algorithm - 23.5 Chapter Summary
- 23.6 Problems

- Appendix A: Complex Numbers
- A.1 Constructing the Complex Numbers
- A.2 Calculating with complex numbers
- A.3 Geometric Representation of
- A.4 Complex Conjugate
- A.5 Complex numbers in matlab
- A.6 Euler’s formula
- A.7 Problems
- A.7.1 MATLAB Problems

- Appendix B: Mathematical Induction
- B.1 Problems

- Appendix C: Chebyshev Polynomials
- C.1 Definition
- C.2 Properties
- C.3 Problems

- Glossary
- Bibliography
- Index

## Product details

- No. of pages: 628
- Language: English
- Copyright: © Academic Press 2014
- Published: September 2, 2014
- Imprint: Academic Press
- Hardcover ISBN: 9780123944351
- eBook ISBN: 9780123947840

## About the Author

### William Ford

William Ford received his undergraduate education at MIT in applied mathematics and a Ph.D. in mathematics from the University of Illinois in 1972. His area of research was the numerical solution of partial differential equations, and the results of his work were published in three top-flight journals. After two years at Clemson University, he accepted an appointment at the University of the Pacific in Stockton, CA, where he taught mathematics and computer science. In 1986, he became a founding member of the Department of Computer Science that is now located in the School of Engineering and Computer Science. He served as Chair of the Department for eleven years and retired in 2014 as a recipient of the Order of Pacific, the highest award the University gives.

Dr. Ford is the co-author of five computer science texts and two commercial software packages. For many years, he has taught computation for engineers, discrete mathematics, and computing theory. His interest in writing a book on numerical linear algebra arose from working with graduate engineering students. There is a tremendous need for engineers to be familiar with numerical linear algebra and its applications. Dr. Ford saw the need for the subject to be taught at the advanced undergraduate as well as the beginning graduate level. Yet, most engineering students have only applied linear algebra available to them, and books in the subject only touch on numerical aspects. The basics of linear algebra are necessary before a study of numerical linear algebra can begin, but few institutions can afford to offer two separate courses. As a result, he developed a book that in the early chapters provides the linear algebra necessary for a study of how to perform accurate computation in such problems such as solving general and specialized square linear systems, least-squares, computation of eigenvalues, and the iterative solution of large, sparse systems. The book contains many computational exercises and carefully chosen written exercises. Some of these written exercises introduce methods not directly covered in the text but, in each case, the method is developed in understandable parts. The same is true of proofs, and hints are provided for the most difficult problems.

#### Affiliations and Expertise

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