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SOURCE: Layton W., Sussman M. Numerical Linear Algebra 2020
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Textbook in PDF format Preface Introduction Sources of Arithmetical Error Measuring Errors: The Trademarked Quantities Linear Systems and Finite Precision Arithmetic Vectors and Matrices Eigenvalues and Singular Values Properties of eigenvalues Error and Residual When is a Linear System Solvable? When is an N×N Matrix Numerically Singular? Gaussian Elimination Elimination + Backsubstitution Algorithms and Pseudocode The Gaussian Elimination Algorithm Computational Complexity and Gaussian Elimination Pivoting Strategies Tridiagonal and Banded Matrices The LU Decomposition Norms and Error Analysis FENLA and Iterative Improvement Vector Norms Norms that come from inner products Matrix Norms A few proofs Error, Residual and Condition Number Backward Error Analysis The general case The MPP and the Curse of Dimensionality Derivation 1D Model Poisson Problem Difference approximations Reduction to linear equations Complexity of solving the 1D MPP The 2D MPP The 3D MPP The Curse of Dimensionality Computing a residual grows slowly with dimension Iterative Methods Introduction to Iterative Methods Iterative methods three standard forms Three quantities of interest Mathematical Tools Convergence of FOR Optimization of ρ Geometric analysis of the min-max problem How many FOR iterations? Better Iterative Methods The Gauss–Seidel Method Relaxation Gauss–Seidel with over-relaxation = Successive Over Relaxation Three level over-relaxed FOR Algorithmic issues: storing a large, sparse matrix Dynamic Relaxation Splitting Methods Parameter selection Connection to dynamic relaxation The ADI splitting Solving Ax = b by Optimization The Connection to Optimization Application to Stationary Iterative Methods Application to Parameter Selection The Steepest Descent Method The Conjugate Gradient Method The CG Algorithm Algorithmic options CG’s two main convergence theorems Analysis of the CG Algorithm Convergence by the Projection Theorem The Gram–Schmidt algorithm Orthogonalization of moments instead of Gram–Schmidt A simplified CG method Error Analysis of CG Preconditioning CGN for Non-SPD Systems Eigenvalue Problems Introduction and Review of Eigenvalues Properties of eigenvalues Gershgorin Circles Perturbation Theory of Eigenvalues Perturbation bounds The Power Method Convergence of the power method Symmetric matrices Inverse Power, Shifts and Rayleigh Quotient Iteration The inverse power method Rayleigh Quotient Iteration The QR Method Appendix A An Omitted Proof Appendix B Tutorial on Basic MatLAB Programming Objective MatLAB Files Variables, Values and Arithmetic Variables Are Matrices Matrix and Vector Operations Flow Control Script and Function Files MatLAB Linear Algebra Functionality Solving matrix systems in MatLAB Condition number of a matrix Matrix factorizations Eigenvalues and singular values Debugging Execution Speed Initializing vectors and matrices in MatLAB Array notation and efficiency in MatLAB Bibliography Index
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