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| Introduction | |
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| Discretization of a Differential Equation | |
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| Least Squares Fitting | |
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| Vibrations of a Mechanical System | |
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| The Vibrating String | |
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| Image Compression by the SVD Factorization | |
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| Definition and Properties of Matrices | |
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| Gram-Schmidt Orthonormalization Process | |
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| Matrices | |
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| Trace and Determinant | |
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| Special Matrices | |
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| Rows and Columns | |
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| Row and Column Permutation | |
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| Block Matrices | |
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| Spectral Theory of Matrices | |
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| Matrix Triangularization | |
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| Matrix Diagonalization | |
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| Min-Max Principle | |
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| Singular Values of a Matrix | |
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| Exercises | |
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| Matrix Norms, Sequences, and Series | |
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| Matrix Norms and Subordinate Norms | |
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| Subordinate Norms for Rectangular Matrices | |
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| Matrix Sequences and Series | |
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| Exercises | |
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| Introduction to Algorithmics | |
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| Algorithms and pseudolanguage | |
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| Operation Count and Complexity | |
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| The Strassen Algorithm | |
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| Equivalence of Operations | |
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| Exercises | |
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| Linear Systems | |
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| Square Linear Systems | |
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| Over- and Underdetermined Linear Systems | |
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| Numerical Solution | |
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| Floating-Point System | |
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| Matrix Conditioning | |
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| Conditioning of a Finite Difference Matrix | |
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| Approximation of the Condition Number | |
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| Preconditioning | |
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| Exercises | |
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| Direct Methods for Linear Systems | |
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| Gaussian Elimination Method | |
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| LU Decomposition Method | |
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| Practical Computation of the LU Factorization | |
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| Numerical Algorithm | |
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| Operation Count | |
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| The Case of Band Matrices | |
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| Cholesky Method | |
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| Practical Computation of the Cholesky Factorization | |
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| Numerical Algorithm | |
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| Operation Count | |
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| QR Factorization Method | |
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| Operation Count | |
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| Exercises | |
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| Least Squares Problems | |
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| Motivation | |
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| Main Results | |
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| Numerical Algorithms | |
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| Conditioning of Least Squares Problems | |
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| Normal Equation Method | |
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| QR Factorization Method | |
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| Householder Algorithm | |
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| Exercises | |
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| Simple Iterative Methods | |
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| General Setting | |
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| Jacobi, Gauss-Seidel, and Relaxation Methods | |
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| Jacobi Method | |
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| Gauss-Seidel Method | |
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| Successive Overrelaxation Method (SOR) | |
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| The Special Case of Tridiagonal Matrices | |
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| Discrete Laplacian | |
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| Programming Iterative Methods | |
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| Block Methods | |
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| Exercises | |
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| Conjugate Gradient Method | |
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| The Gradient Method | |
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| Geometric Interpretation | |
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| Some Ideas for Further Generalizations | |
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| Theoretical Definition of the Conjugate Gradient Method | |
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| Conjugate Gradient Algorithm | |
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| Numerical Algorithm | |
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| Number of Operations | |
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| Convergence Speed | |
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| Preconditioning | |
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| Chebyshev Polynomials | |
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| Exercises | |
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| Methods for Computing Eigenvalues | |
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| Generalities | |
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| Conditioning | |
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| Power Method | |
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| Jacobi Method | |
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| Givens-Householder Method | |
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| QR Method | |
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| Lanczos Method | |
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| Exercises | |
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| Solutions and Programs | |
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| Exercises of Chapter 2 | |
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| Exercises of Chapter 3 | |
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| Exercises of Chapter 4 | |
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| Exercises of Chapter 5 | |
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| Exercises of Chapter 6 | |
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| Exercises of Chapter 7 | |
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| Exercises of Chapter 8 | |
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| Exercises of Chapter 9 | |
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| Exercises of Chapter 10 | |
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| References | |
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| Index | |
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| Index of Programs | |