Numerical Linear Algebra with Applications

Numerical Linear Algebra with Applications

by William Ford
Released September 2014
Publisher(s): Academic Press
ISBN: 9780123947840

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Book description

Numerical Linear Algebra with Applications is designed for those who want to gain a practical knowledge of modern computational techniques for the numerical solution of linear algebra problems, using MATLAB as the vehicle for computation. The book contains all the material necessary for a first year graduate or advanced undergraduate course on numerical linear algebra with numerous applications to engineering and science. With a unified presentation of computation, basic algorithm analysis, and numerical methods to compute solutions, this book is ideal for solving real-world problems.

The text consists of six introductory chapters that thoroughly provide the required background for those who have not taken a course in applied or theoretical linear algebra. It explains in great detail the algorithms necessary for the accurate computation of the solution to the most frequently occurring problems in numerical linear algebra. In addition to examples from engineering and science applications, proofs of required results are provided without leaving out critical details. The Preface suggests ways in which the book can be used with or without an intensive study of proofs.

This book will be a useful reference for graduate or advanced undergraduate students in engineering, science, and mathematics. It will also appeal to 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.

  • 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

Table of contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. List of Figures
  7. List of Algorithms
  8. Preface
    1. Topics
    2. Intended Audience
    3. Ways to Use the Book
    4. Matlab Library
    5. Supplement
    6. Acknowledgments
  9. Chapter 1: Matrices
    1. Abstract
    2. 1.1 Matrix Arithmetic
    3. 1.2 Linear Transformations
    4. 1.3 Powers of Matrices
    5. 1.4 Nonsingular Matrices
    6. 1.5 The Matrix Transpose and Symmetric Matrices
    7. 1.6 Chapter Summary
    8. 1.7 Problems
  10. Chapter 2: Linear Equations
    1. Abstract
    2. 2.1 Introduction to Linear Equations
    3. 2.2 Solving Square Linear Systems
    4. 2.3 Gaussian Elimination
    5. 2.4 Systematic Solution of Linear Systems
    6. 2.5 Computing the Inverse
    7. 2.6 Homogeneous Systems
    8. 2.7 Application: A Truss
    9. 2.8 Application: Electrical Circuit
    10. 2.9 Chapter Summary
    11. 2.10 Problems
  11. Chapter 3: Subspaces
    1. Abstract
    2. 3.1 Introduction
    3. 3.2 Subspaces of n
    4. 3.3 Linear Independence
    5. 3.4 Basis of a Subspace
    6. 3.5 The Rank of a Matrix
    7. 3.6 Chapter summary
    8. 3.7 Problems
  12. Chapter 4: Determinants
    1. Abstract
    2. 4.1 Developing the Determinant of A 2 × 2 and A 3 × 3 matrix
    3. 4.2 Expansion by Minors
    4. 4.3 Computing a Determinant Using Row Operations
    5. 4.4 Application: Encryption
    6. 4.5 Chapter Summary
    7. 4.6 Problems
  13. Chapter 5: Eigenvalues and Eigenvectors
    1. Abstract
    2. 5.1 Definitions and Examples
    3. 5.2 Selected Properties of Eigenvalues and Eigenvectors
    4. 5.3 Diagonalization
    5. 5.4 Applications
    6. 5.5 Computing Eigenvalues and Eigenvectors Using Matlab
    7. 5.6 Chapter Summary
    8. 5.7 Problems
  14. Chapter 6: Orthogonal Vectors and Matrices
    1. Abstract
    2. 6.1 Introduction
    3. 6.2 The Inner Product
    4. 6.3 Orthogonal Matrices
    5. 6.4 Symmetric Matrices and Orthogonality
    6. 6.5 The L2 inner product
    7. 6.6 The Cauchy-Schwarz Inequality
    8. 6.7 Signal Comparison
    9. 6.8 Chapter Summary
    10. 6.9 Problems
  15. Chapter 7: Vector and Matrix Norms
    1. Abstract
    2. 7.1 Vector Norms
    3. 7.2 Matrix Norms
    4. 7.3 Submultiplicative Matrix Norms
    5. 7.4 Computing the Matrix 2-Norm
    6. 7.5 Properties of the Matrix 2-Norm
    7. 7.6 Chapter Summary
    8. 7.7 Problems
  16. Chapter 8: Floating Point Arithmetic
    1. Abstract
    2. 8.1 Integer Representation
    3. 8.2 Floating-Point Representation
    4. 8.3 Floating-Point Arithmetic
    5. 8.4 Minimizing Errors
    6. 8.5 Chapter summary
    7. 8.6 Problems
  17. Chapter 9: Algorithms
    1. Abstract
    2. 9.1 Pseudocode Examples
    3. 9.2 Algorithm Efficiency
    4. 9.3 The Solution to Upper and Lower Triangular Systems
    5. 9.4 The Thomas Algorithm
    6. 9.5 Chapter Summary
    7. 9.6 Problems
  18. Chapter 10: Conditioning of Problems and Stability of Algorithms
    1. Abstract
    2. 10.1 Why do we need numerical linear algebra?
    3. 10.2 Computation error
    4. 10.3 Algorithm stability
    5. 10.4 Conditioning of a problem
    6. 10.5 Perturbation analysis for solving a linear system
    7. 10.6 Properties of the matrix condition number
    8. 10.7 Matlab computation of a matrix condition number
    9. 10.8 Estimating the condition number
    10. 10.9 Introduction to perturbation analysis of eigenvalue problems
    11. 10.10 Chapter summary
    12. 10.11 Problems
  19. Chapter 11: Gaussian Elimination and the LU Decomposition
    1. Abstract
    2. 11.1 LU Decomposition
    3. 11.2 Using LU to Solve Equations
    4. 11.3 Elementary Row Matrices
    5. 11.4 Derivation of the LU Decomposition
    6. 11.5 Gaussian Elimination with Partial Pivoting
    7. 11.6 Using the LU Decomposition to Solve Axi=bi,1≤i≤k
    8. 11.7 Finding A–1
    9. 11.8 Stability and Efficiency of Gaussian Elimination
    10. 11.9 Iterative Refinement
    11. 11.10 Chapter Summary
    12. 11.11 Problems
  20. Chapter 12: Linear System Applications
    1. Abstract
    2. 12.1 Fourier Series
    3. 12.2 Finite Difference Approximations
    4. 12.3 Least-Squares Polynomial Fitting
    5. 12.4 Cubic Spline Interpolation
    6. 12.5 Chapter Summary
    7. 12.6 Problems
  21. Chapter 13: Important Special Systems
    1. Abstract
    2. 13.1 Tridiagonal Systems
    3. 13.2 Symmetric Positive Definite Matrices
    4. 13.3 The Cholesky Decomposition
    5. 13.4 Chapter Summary
    6. 13.5 Problems
  22. Chapter 14: Gram-Schmidt Orthonormalization
    1. Abstract
    2. 14.1 The Gram-Schmidt Process
    3. 14.2 Numerical Stability of the Gram-Schmidt Process
    4. 14.3 The QR Decomposition
    5. 14.4 Applications of The QR Decomposition
    6. 14.5 Chapter Summary
    7. 14.6 Problems
  23. Chapter 15: The Singular Value Decomposition
    1. Abstract
    2. 15.1 The SVD Theorem
    3. 15.2 Using the SVD to Determine Properties of a Matrix
    4. 15.3 SVD and Matrix Norms
    5. 15.4 Geometric Interpretation of the SVD
    6. 15.5 Computing the SVD Using MATLAB
    7. 15.6 Computing A–1
    8. 15.7 Image Compression Using the SVD
    9. 15.8 Final Comments
    10. 15.9 Chapter Summary
    11. 15.10 Problems
  24. Chapter 16: Least-Squares Problems
    1. Abstract
    2. 16.1 Existence and Uniqueness of Least-Squares Solutions
    3. 16.2 Solving Overdetermined Least-Squares Problems
    4. 16.3 Conditioning of Least-Squares Problems
    5. 16.4 Rank-Deficient Least-Squares Problems
    6. 16.5 Underdetermined Linear Systems
    7. 16.6 Chapter Summary
    8. 16.7 Problems
  25. Chapter 17: Implementing the QR Decomposition
    1. Abstract
    2. 17.1 Review of the QR Decomposition Using Gram-Schmidt
    3. 17.2 Givens Rotations
    4. 17.3 Creating a Sequence of Zeros in a Vector Using Givens Rotations
    5. 17.4 Product of a Givens Matrix with a General Matrix
    6. 17.5 Zeroing-Out Column Entries in a Matrix Using Givens Rotations
    7. 17.6 Accurate Computation of the Givens Parameters
    8. 17.7 THe Givens Algorithm for the QR Decomposition
    9. 17.8 Householder Reflections
    10. 17.9 Computing the QR Decomposition Using Householder Reflections
    11. 17.10 Chapter Summary
    12. 17.11 Problems
  26. Chapter 18: The Algebraic Eigenvalue Problem
    1. Abstract
    2. 18.1 Applications of The Eigenvalue Problem
    3. 18.2 Computation of Selected Eigenvalues and Eigenvectors
    4. 18.3 The Basic QR Iteration
    5. 18.4 Transformation to Upper Hessenberg Form
    6. 18.5 The Unshifted Hessenberg QR Iteration
    7. 18.6 The Shifted Hessenberg QR Iteration
    8. 18.7 Schur's Triangularization
    9. 18.8 The Francis Algorithm
    10. 18.9 Computing Eigenvectors
    11. 18.10 Computing Both Eigenvalues and Their Corresponding Eigenvectors
    12. 18.11 Sensitivity of Eigenvalues to Perturbations
    13. 18.12 Chapter Summary
    14. 18.13 Problems
  27. Chapter 19: The Symmetric Eigenvalue Problem
    1. Abstract
    2. 19.1 The Spectral Theorem and Properties of A Symmetric Matrix
    3. 19.2 The Jacobi Method
    4. 19.3 The Symmetric QR Iteration Method
    5. 19.4 The Symmetric Francis Algorithm
    6. 19.5 The Bisection Method
    7. 19.6 The Divide-And-Conquer Method
    8. 19.7 Chapter Summary
    9. 19.8 Problems
  28. Chapter 20: Basic Iterative Methods
    1. Abstract
    2. 20.1 Jacobi Method
    3. 20.2 The Gauss-Seidel Iterative Method
    4. 20.3 The Sor Iteration
    5. 20.4 Convergence of the Basic Iterative Methods
    6. 20.5 Application: Poisson's Equation
    7. 20.6 Chapter Summary
    8. 20.7 Problems
  29. Chapter 21: Krylov Subspace Methods
    1. Abstract
    2. 21.1 Large, Sparse Matrices
    3. 21.2 The CG Method
    4. 21.3 Preconditioning
    5. 21.4 Preconditioning For CG
    6. 21.5 Krylov Subspaces
    7. 21.6 The Arnoldi Method
    8. 21.7 GMRES
    9. 21.8 The Symmetric Lanczos Method
    10. 21.9 The Minres Method
    11. 21.10 Comparison of Iterative Methods
    12. 21.11 Poisson's Equation Revisited
    13. 21.12 The Biharmonic Equation
    14. 21.13 Chapter Summary
    15. 21.14 Problems
  30. Chapter 22: Large Sparse Eigenvalue Problems
    1. Abstract
    2. 22.1 The Power Method
    3. 22.2 Eigenvalue Computation Using the Arnoldi Process
    4. 22.3 The Implicitly Restarted Arnoldi Method
    5. 22.4 Eigenvalue Computation Using the Lanczos Process
    6. 22.5 Chapter Summary
    7. 22.6 Problems
  31. Chapter 23: Computing the Singular Value Decomposition
    1. Abstract
    2. 23.1 Development of the One-Sided Jacobi Method For Computing the Reduced Svd
    3. 23.2 The One-Sided Jacobi Algorithm
    4. 23.3 Transforming a Matrix to Upper-Bidiagonal Form
    5. 23.4 Demmel and Kahan Zero-Shift QR Downward Sweep Algorithm
    6. 23.5 Chapter Summary
    7. 23.6 Problems
  32. Appendix A: Complex Numbers
    1. A.1 Constructing the Complex Numbers
    2. A.2 Calculating with complex numbers
    3. A.3 Geometric Representation of
    4. A.4 Complex Conjugate
    5. A.5 Complex numbers in matlab
    6. A.6 Euler’s formula
    7. A.7 Problems
    8. A.7.1 MATLAB Problems
  33. Appendix B: Mathematical Induction
    1. B.1 Problems
  34. Appendix C: Chebyshev Polynomials
    1. C.1 Definition
    2. C.2 Properties
    3. C.3 Problems
  35. Glossary
  36. Bibliography
  37. Index

Product information

  • Title: Numerical Linear Algebra with Applications
  • Author(s): William Ford
  • Release date: September 2014
  • Publisher(s): Academic Press
  • ISBN: 9780123947840