### Available:*

Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
---|---|---|---|---|---|

PSZ JB | 30000010159157 | QA185.D37 A44 2008 | Open Access Book | Searching... |

### On Order

### Summary

### Summary

This book distinguishes itself from the many other textbooks on the topic of linear algebra by including mathematical and computational chapters along with examples and exercises with Matlab. In recent years, the use of computers in many areas of engineering and science has made it essential for students to get training in numerical methods and computer programming. Here, the authors use both Matlab and SciLab software as well as covering core standard material. It is intended for libraries; scientists and researchers; pharmaceutical industry.

### Author Notes

Sidi Mahmoud Kaber is an associate professor at the Universite Pierre et Marie Curie, France.

### Reviews 1

### Choice Review

Most numerical computations in applied mathematical fields involve numerical linear algebra. These problems typically result in requiring a series of matrix computations. The most prevalent problems involve solving systems of linear equations and finding eigenvalues and eigenvectors of matrices. This book emphasizes practical algorithms for solving these problems utilizing MATLAB and SCILAB. Allaire (Ecole Polytechnique, France) and Kaber (Universite Pierre et Marie Curie, France) taught a numerical linear algebra course to third-year undergraduates; this work, initially published in French, is the culmination of that course. Readers must understand the material presented in an undergraduate linear algebra course prior to studying this book. However, students do not need previous numerical analysis experience. What sets this book apart from others on the subject is its experimental approach in all exercises. For many exercises, the book provides complete solutions including MATLAB scripts; the last chapter is entirely devoted to complete solutions of selected exercises. Teachers and professors can request solutions to other exercises as well. Summing Up: Recommended. Upper-division undergraduates, researchers/faculty, and professionals/practitioners. J. T. Zerger Catawba College

### Table of Contents

1 Introduction | p. 1 |

1.1 Discretization of a Differential Equation | p. 1 |

1.2 Least Squares Fitting | p. 4 |

1.3 Vibrations of a Mechanical System | p. 8 |

1.4 The Vibrating String | p. 10 |

1.5 Image Compression by the SVD Factorization | p. 12 |

2 Definition and Properties of Matrices | p. 15 |

2.1 Gram-Schmidt Orthonormalization Process | p. 15 |

2.2 Matrices | p. 17 |

2.2.1 Trace and Determinant | p. 19 |

2.2.2 Special Matrices | p. 20 |

2.2.3 Rows and Columns | p. 21 |

2.2.4 Row and Column Permutation | p. 22 |

2.2.5 Block Matrices | p. 22 |

2.3 Spectral Theory of Matrices | p. 23 |

2.4 Matrix Triangularization | p. 26 |

2.5 Matrix Diagonalization | p. 28 |

2.6 Min-Max Principle | p. 31 |

2.7 Singular Values of a Matrix | p. 33 |

2.8 Exercises | p. 38 |

3 Matrix Norms, Sequences, and Series | p. 45 |

3.1 Matrix Norms and Subordinate Norms | p. 45 |

3.2 Subordinate Norms for Rectangular Matrices | p. 52 |

3.3 Matrix Sequences and Series | p. 54 |

3.4 Exercises | p. 57 |

4 Introduction to Algorithmics | p. 61 |

4.1 Algorithms and pseudolanguage | p. 61 |

4.2 Operation Count and Complexity | p. 64 |

4.3 The Strassen Algorithm | p. 65 |

4.4 Equivalence of Operations | p. 67 |

4.5 Exercises | p. 69 |

5 Linear Systems | p. 71 |

5.1 Square Linear Systems | p. 71 |

5.2 Over- and Underdetermined Linear Systems | p. 75 |

5.3 Numerical Solution | p. 76 |

5.3.1 Floating-Point System | p. 77 |

5.3.2 Matrix Conditioning | p. 79 |

5.3.3 Conditioning of a Finite Difference Matrix | p. 85 |

5.3.4 Approximation of the Condition Number | p. 88 |

5.3.5 Preconditioning | p. 91 |

5.4 Exercises | p. 92 |

6 Direct Methods for Linear Systems | p. 97 |

6.1 Gaussian Elimination Method | p. 97 |

6.2 LU Decomposition Method | p. 103 |

6.2.1 Practical Computation of the LU Factorization | p. 107 |

6.2.2 Numerical Algorithm | p. 108 |

6.2.3 Operation Count | p. 108 |

6.2.4 The Case of Band Matrices | p. 110 |

6.3 Cholesky Method | p. 112 |

6.3.1 Practical Computation of the Cholesky Factorization | p. 113 |

6.3.2 Numerical Algorithm | p. 114 |

6.3.3 Operation Count | p. 115 |

6.4 QR Factorization Method | p. 116 |

6.4.1 Operation Count | p. 118 |

6.5 Exercises | p. 119 |

7 Least Squares Problems | p. 125 |

7.1 Motivation | p. 125 |

7.2 Main Results | p. 126 |

7.3 Numerical Algorithms | p. 128 |

7.3.1 Conditioning of Least Squares Problems | p. 128 |

7.3.2 Normal Equation Method | p. 131 |

7.3.3 QR Factorization Method | p. 132 |

7.3.4 Householder Algorithm | p. 136 |

7.4 Exercises | p. 140 |

8 Simple Iterative Methods | p. 143 |

8.1 General Setting | p. 143 |

8.2 Jacobi, Gauss-Seidel, and Relaxation Methods | p. 147 |

8.2.1 Jacobi Method | p. 147 |

8.2.2 Gauss-Seidel Method | p. 148 |

8.2.3 Successive Overrelaxation Method (SOR) | p. 149 |

8.3 The Special Case of Tridiagonal Matrices | p. 150 |

8.4 Discrete Laplacian | p. 154 |

8.5 Programming Iterative Methods | p. 156 |

8.6 Block Methods | p. 157 |

8.7 Exercises | p. 159 |

9 Conjugate Gradient Method | p. 163 |

9.1 The Gradient Method | p. 163 |

9.2 Geometric Interpretation | p. 165 |

9.3 Some Ideas for Further Generalizations | p. 168 |

9.4 Theoretical Definition of the Conjugate Gradient Method | p. 171 |

9.5 Conjugate Gradient Algorithm | p. 174 |

9.5.1 Numerical Algorithm | p. 178 |

9.5.2 Number of Operations | p. 179 |

9.5.3 Convergence Speed | p. 180 |

9.5.4 Preconditioning | p. 182 |

9.5.5 Chebyshev Polynomials | p. 186 |

9.6 Exercises | p. 189 |

10 Methods for Computing Eigenvalues | p. 191 |

10.1 Generalities | p. 191 |

10.2 Conditioning | p. 192 |

10.3 Power Method | p. 194 |

10.4 Jacobi Method | p. 198 |

10.5 Givens-Householder Method | p. 203 |

10.6 QR Method | p. 209 |

10.7 Lanczos Method | p. 214 |

10.8 Exercises | p. 219 |

11 Solutions and Programs | p. 223 |

11.1 Exercises of Chapter 2 | p. 223 |

11.2 Exercises of Chapter 3 | p. 234 |

11.3 Exercises of Chapter 4 | p. 237 |

11.4 Exercises of Chapter 5 | p. 241 |

11.5 Exercises of Chapter 6 | p. 250 |

11.6 Exercises of Chapter 7 | p. 257 |

11.7 Exercises of Chapter 8 | p. 258 |

11.8 Exercises of Chapter 9 | p. 260 |

11.9 Exercises of Chapter 10 | p. 262 |

References | p. 265 |

Index | p. 267 |

Index of Programs | p. 272 |