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Preface | |
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Acknowledgments | |
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Gaussian Elimination and Its Variants | |
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Matrix Multiplication | |
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Systems of Linear Equations | |
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Triangular Systems | |
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Positive Definite Systems; Cholesky Decomposition | |
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Banded Positive Definite Systems | |
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Sparse Positive Definite Systems | |
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Gaussian Elimination and the LU Decomposition | |
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Gaussian Elimination with Pivoting | |
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Sparse Gaussian Elimination | |
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Sensitivity of Linear Systems | |
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Vector and Matrix Norms | |
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Condition Numbers | |
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Perturbing the Coefficient Matrix | |
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A Posteriori Error Analysis Using the Residual | |
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Roundoff Errors; Backward Stability | |
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Propagation of Roundoff Errors | |
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Backward Error Analysis of Gaussian Elimination | |
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Scaling | |
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Componentwise Sensitivity Analysis | |
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The Least Squares Problem | |
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The Discrete Least Squares Problem | |
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Orthogonal Matrices, Rotators, and Reflectors | |
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Solution of the Least Squares Problem | |
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The Gram-Schmidt Process | |
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Geometric Approach | |
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Updating the QR Decomposition | |
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The Singular Value Decomposition | |
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Introduction | |
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Some Basic Applications of Singular Values | |
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The SVD and the Least Squares Problem | |
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Sensitivity of the Least Squares Problem | |
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Eigenvalues and Eigenvectors I | |
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Systems of Differential Equations | |
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Basic Facts | |
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The Power Method and Some Simple Extensions | |
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Similarity Transforms | |
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Reduction to Hessenberg and Tridiagonal Forms | |
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Francis's Algorithm | |
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Use of Francis's Algorithm to Calculate Eigenvectors | |
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The SVD Revisited | |
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Eigenvalues and Eigenvectors II | |
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Eigenspaces and Invariant Subspaces | |
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Subspace Iteration and Simultaneous Iteration | |
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Krylov Subspaces and Francis's Algorithm | |
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Large Sparse Eigenvalue Problems | |
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Implicit Restarts | |
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The Jacobi-Davidson and Related Algorithms | |
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Eigenvalues and Eigenvectors III | |
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Sensitivity of Eigenvalues and Eigenvectors | |
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Methods for the Symmetric Eigenvalue Problem | |
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Product Eigenvalue Problems | |
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The Generalized Eigenvalue Problem | |
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Iterative Methods for Linear Systems | |
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A Model Problem | |
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The Classical Iterative Methods | |
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Convergence of Iterative Methods | |
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Descent Methods; Steepest Descent | |
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On Stopping Criteria | |
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Preconditioners | |
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The Conjugate-Gradient Method | |
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Derivation of the CG Algorithm | |
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Convergence of the CG Algorithm | |
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Indefinite and Nonsymmetric Problems | |
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References | |
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Index | |
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Index of MATLAB� Terms | |