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 FOM, IOM, DIOM | YS 1.12, 6.1, 6.2, 6.3 |
| Week 40 | GMRES, QGMRES 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 PCG | 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 Regularisation 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 | TB27-29 |
| Summary | ||
| 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.