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大型線性系統與特徵值問題/台大數學系教授
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摘要
1
Chapter 1: Introduction (課程講義)
一、Vectors and matrices
二、Eigenvalues and Eigenvectors
三、Norms and eigenvalues
四、Backward error and Forward error（一）
五、Backward error and Forward error（二）
2
Chapter 2:Gaussian Elimination for Linear Systems (課程講義)
一、Error analysis for Gaussian algorithm
3
Iterative Methods for Solving Large Linear Systems (課程講義)
Iterative Methods for Solving Large Linear Systems
一、General procedures for the construction of iterative methods
二、Determination of the Optimal Parameter ω for 2-consistly Ordered Matrices
4
Conjugate Gradient Method (課程講義)
一、A Variational Problem, Steepest Descent Method
二、Conjugate gradient method
三、Practical Implementation
四、Convergence of CG-method
5
GCG-type Methods for Nonsymmetric Linear Systems(課程講義)
GCG-type Methods for Nonsymmetric Linear Systems
一、GCG method(Generalized Conjugate Gradient)
二、BCG method (A: unsymmetric)
三、The polynomial equivalent method of the CG method
四、Bi-CGSTAB(SISC, 1992, Van der Vorst)
6
GMRES: Generalized Minimal Residual Algorithm forSolving Nonsymmetric Linear Systems (課程講義)
GMRES: Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
一、The generalized minimal residual (GMRES) algorithm
7
CG-Method as an Iterative Method, Preconditioning (課程講義)
CG-Method as an Iterative Method, Preconditioning
一、A new point of view of PCG
二、Chebychev Semi-Iteration Acceleration Method
三、Some theorems and definitions
四、Sufficient conditions for convergence of TSM and SSM
五、Incomplete Cholesky Decomposition
8
Recent Advance in Numerical Algorithms for Large/Sparse Eigenvalue Problems (課程講義)
Recent Advance in Numerical Algorithms for Large/Sparse Eigenvalue Problems
一、Power Methods
二、Inverse Power Iteration
三、Connection with Newton method
9
Krylov Subspace Methods for Large/Sparse Eigenvalue Problems (I) (課程講義)
Krylov Subspace Methods for Large/Sparse Eigenvalue Problems (I)
一、Orthogonal projection methods
二、Krylov Subspaces
三、Householder transformation
四、Arnoldi Method
10
Krylov Subspace Methods for Large/Sparse Eigenvalue Problems (II) (課程講義)
Krylov Subspace Methods for Large/Sparse Eigenvalue Problems (II)
一、Restarting method
二、Generalized eigenvalue problem
11
Polynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems (課程講義)
Polynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems
一、Jacobi’s orthogonal component correction, 1846
二、Davidson’s method (1975)
三、Jacobi-Davidson method (1996)
四、Polynomial Jacobi-Davidson method
五、Non-equivalence deflation of quadratic eigenproblems
12
QR Algorithm for Dense Eigenvalue Problems (課程講義)
QR Algorithm for Dense Eigenvalue Problems
一、QR Algorithm
二、The Practical QR Algorithm
三、Single-shift QR-iteration
四、Double Shift QR iteration