| Preface Outline 1 Linear Systems 1.1 Solution of Triangular Systems 1.2 Gaussian Elimination 1.3 Pivoting Strategies and Iterative Refinement 1.4 Cholesky Decomposition for Symmetric Positive Definite Matrices Exercises 2 Error Analysis 2.1 Sources of Errors 2.2 Condition of Problems 2.2.1 Normwise Condition Analysis 2.2.2 Componentwise Condition Analysis 2.3 Stability of Algorithms 2.3.1 Stability Concepts 2.3.2 Forward Analysis 2.3.3 Backward Analysis 2.4 Application to Linear Systems 2.4.1 A Zoom into Solvability 2.4.2 Backward Analysis of Gaussian Elimination 2.4.3 Assessment of Approximate Solutions Exercises 3 Linear Least-Squares Problems 3.1 Least-Squares Method of Gauss 3.1.1 Formulation of the Problem 3.1.2 Normal Equations 3.1.3 Condition 3.1.4 Solution of Normal Equations 3.2 Orthogonalization Methods 3.2.1 Givens Rotations 3.2.2 Householder Reflections 3.3 Generalized Inverses Exercises 4 Nonlinear Systems and Least-Squares Problems 4.1 Fixed-Point Iterations 4.2 Newton Methods for Nonlinear Systems 4.3 Gauss-Newton Method for Nonlinear Least-Squares Problems 4.4 Nonlinear Systems Depending on Parameters 4.4.1 Solution Structure 4.4.2 Continuation Methods Exercises 5 Linear Eigenvalue Problems 5.1 Condition of General Eigenvalue Problems 5.2 Power Method 5.3 QR-Algorithm for Symmetric Eigenvalue Problems 5.4 Singular Value Decomposition 5.5 Stochastic Eigenvalue Problems Exercises 6 Three-Term Recurrence Relations 6.1 Theoretical Background 6.1.1 Orthogonality and Three-Term Recurrence Relations 6.1.2 Homogeneous and Inhomogeneous Recurrence Relations 6.2 Numerical Aspects 6.2.1 Condition Number 6.2.2 Idea of the Miller Algorithm 6.3 Adjoint Summation …… 7 Interpolation and Approximation 8 Large Symmetric Systems of Equations and Eigenvalue Problems 9 Definite Integrals References Software Index |
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