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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 | |