## Numerical Linear AlgebraThis 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

1 | |

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 |

265 | |

266 | |

272 | |

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### Common terms and phrases

ai,j band matrix columns cond2(A condition number conjugate gradient method deduce defined Deﬁnition denote diagonal entries diagonalizable diﬀerent dimension eﬃcient eigenvalues eigenvectors equal equivalent error exists ﬁnd ﬁnite ﬁrst formula Gauss Gauss-Seidel method Gaussian elimination Gram–Schmidt Hermitian induction integer inverse iterative method Jacobi method least squares problem Lemma linear system linear system Ax lower triangular matrix LU factorization Matlab matrix norm method converges minimal Mn(C multiplication nonsingular matrix nonzero Nop(n number of operations obtain orthogonal matrix orthonormal basis permutation pivot polynomial positive definite power method preconditioning Proposition prove QR factorization QR method real numbers real symmetric Remark result right-hand side scalar product sequence singular values Solution of Exercise solving a linear solving the linear square matrix SVD factorization symmetric matrix tridiagonal unique unitary matrix upper triangular matrix vector norm Write a function xk+1 zero