Lecture plan

The lecture plan will be continuously updated.

Date Topics Reading
Introduction and basics of linear algebra
Week 34 Introduction
Inner products, norms, and matrix norms
Orthogonality and QR-decomposition
YS 1.1-1.7
TB 10
Week 36 QR-decomposition
Jordan and Schur canonical forms
Normal and Hermitian matrices
Values of a matrix
Courant-Fischer Theorem
Positive definite matrices
TB 10
YS 1.8 (not 1.8.4), 1.9, 1.11
Solving Poisson equations Einar Rønquist's note
Please read YS 2.2-2.2.5 if you are not familiar with finite difference methods.
Week 37 Fast Poisson solvers
Perturbation analysis
Einar Rønquist's note
YS 1.13.2
Basic iterative methods for linear systems
Jacobi, Gauss-Seidel, and SOR
Convergence analysis
YS 4.1, 4.2
Week 38 Basics of projection methods
Steepest Descent and MR
YS 5.1, 5.2, 5.3
Krylov subspace methods
Krylov subspaces YS 6.2
Week 39 More on Krylov subspaces
Projection operators
Arnoldi's method
YS 1.12, 6.1, 6.2, 6.3
Symmetric Lanczos algorithm
CG method
YS 6.5.1-6.5.7, 6.6, 6.7
Week 41 CG method
Convergence of CG and GMRES
Idea of preconditioning
Examples of preconditioners
YS 6.11, 10.1, 10.2
Week 42 Preconditionend GMRES
YS 10.1
Multigrid and domain decomposition
Basics of multigrid methods
Weighted Jacobi method
Prolongation and interpolation
V-cycles and W-cycles
YS 13.1-13.4
Week 43 V-cycles
Red-black Gauss-Seidel method
Multigrid as preconditioner
Domain decomposition
Additive and multiplicative Schwarz sweeps
Project assignment
YS 13.4, 12.4.1-12.4.2, 14.1, 14.3
Singular value and eigenvalue decompositions
Week 44 Singular value decomposition (SVD)
Matrix properties via the SVD
SVD and matrix inversion
TB4, TB5 GvL 2.5.2, note on the SVD
Week 45 Truncated SVD
Similarity transforms and eigenvalues
Reductions to Hessenberg form
Power method and inverse power method
note on the SVD, TB25-27
Week 46 Rayleigh quotient iteration
Simultaneous power method
QR-method for eigenvalue computations
QR-method with shifts
Week 47 Summary of the lecture (Monday)
Question session (Wednesday)

YS refers to the book Iterative Methods for Sparse Linear Systems by Yousef Saad.
TB refers to the book Numerical Linear Algebra by L. N. Trefethen and D. Bau; GvL refers to the book Matrix Computations by G. Golub and C. Van Loan. Scans of the chapters used in the course can be found on itslearning.

2016-11-14, Markus Grasmair