# Numerical Linear Algebra

Springer Science & Business Media, Dec 17, 2008 - Mathematics - 271 pages

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.

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### Contents

 Introduction 1 12 Least Squares Fitting 4 13 Vibrations of a Mechanical System 8 14 The Vibrating String 10 15 Image Compression by the SVD Factorization 12 Deﬁnition and Properties of Matrices 15 22 Matrices 17 221 Trace and Determinant 19
 72 Main Results 126 73 Numerical Algorithms 128 732 Normal Equation Method 131 733 QR Factorization Method 132 734 Householder Algorithm 136 74 Exercises 140 Simple Iterative Methods 143 82 Jacobi GaussSeidel and Relaxation Methods 147

 222 Special Matrices 20 223 Rows and Columns 21 224 Row and Column Permutation 22 23 Spectral Theory of Matrices 23 24 Matrix Triangularization 26 25 Matrix Diagonalization 28 26 MinMax Principle 31 27 Singular Values of a Matrix 33 28 Exercises 38 Matrix Norms Sequences and Series 45 32 Subordinate Norms for Rectangular Matrices 52 33 Matrix Sequences and Series 54 34 Exercises 57 Introduction to Algorithmics 61 42 Operation Count and Complexity 64 43 The Strassen Algorithm 65 44 Equivalence of Operations 67 45 Exercises 69 Linear Systems 71 52 Over and Underdetermined Linear Systems 75 53 Numerical Solution 76 531 FloatingPoint System 77 532 Matrix Conditioning 79 533 Conditioning of a Finite Difference Matrix 85 534 Approximation of the Condition Number 88 535 Preconditioning 91 54 Exercises 92 Direct Methods for Linear Systems 97 62 LU Decomposition Method 103 621 Practical Computation of the LU Factorization 107 622 Numerical Algorithm 108 624 The Case of Band Matrices 110 63 Cholesky Method 112 631 Practical Computation of the Cholesky Factorization 113 632 Numerical Algorithm 114 633 Operation Count 115 64 QR Factorization Method 116 641 Operation Count 118 65 Exercises 119 Least Squares Problems 125
 822 GaussSeidel Method 148 823 Successive Overrelaxation Method SOR 149 83 The Special Case of Tridiagonal Matrices 150 84 Discrete Laplacian 154 85 Programming Iterative Methods 156 86 Block Methods 157 87 Exercises 159 Conjugate Gradient Method 163 92 Geometric Interpretation 165 93 Some Ideas for Further Generalizations 168 94 Theoretical Deﬁnition of the Conjugate Gradient Method 171 95 Conjugate Gradient Algorithm 174 951 Numerical Algorithm 178 952 Number of Operations 179 953 Convergence Speed 180 954 Preconditioning 182 955 Chebyshev Polynomials 186 96 Exercises 189 Methods for Computing Eigenvalues 191 102 Conditioning 192 103 Power Method 194 104 Jacobi Method 198 105 GivensHouseholder Method 203 106 QR Method 209 107 Lanczos Method 214 108 Exercises 219 Solutions and Programs 223 112 Exercises of Chapter 3 234 113 Exercises of Chapter 4 237 114 Exercises of Chapter 5 241 115 Exercises of Chapter 6 250 116 Exercises of Chapter 7 257 117 Exercises of Chapter 8 258 118 Exercises of Chapter 9 260 119 Exercises of Chapter 10 262 References 265 Index 266 Index of Programs 272 Copyright