\[ \|r_{\text{new}}\|^2_2 \le \bigg[1-\frac{1}{n\kappa^2_2(A)}\bigg] \|r\|^2_2, \] and in 4b) the stopping criterion should be \( \| A_1 x^{(k)} - A_2 x^{(k)} - b \|_2 \le \varepsilon \|b\|_2\). I will take special care when interpreting your answers to these questions and will err on your side. I apologize most profoundly for these and thank you for all the feedback to the exam!
<schedule> <timeslots>08:15|09:15|10:15|11:15|12:15|13:15|14:15|15:15|16:15</timeslots> <entry day:Mo start:3 end:3 color:ntnu2>AE office hour (SBII/1038)</entry> <entry day:We start:6 end:7 color:ntnu2>Lecture (K26)</entry> <entry day:Th start:2 end:3 color:ntnu2>Lecture (K26)</entry> <entry day:We start:8 end:8 color:ntnu2>Exercise (R60)</entry> <entry day:Fr start:2 end:2 color:ntnu2>TR office hour (SBII/1056)</entry> </schedule>
The exam will test a selection of the Learning outcome.
Permitted examination support material: C: Specified, written and handwritten examination support materials are permitted. A specified, simple calculator is permitted. The permitted examination support materials are:
Here is a list of the goals that we will achieve in this course.
The curriculum consists of all topics that have been covered in the lectures, all self-study topics, the semester project, and the exercises with solutions. The lectures and self-study topics are based on the following material. The list is not yet final.
Of the reading material listed below, you should buy the book by Saad, but not necessarily any of the other ones. Saad's book is also available online for free through NTNU's network at SIAM.
S refers to Saad, TB to Trefethen & Bau, GVL to Golub & Van Loan, D to Demmel, and R to Rønquist.
Date | Topic | Reference | Learning outcome |
---|---|---|---|
27.11 | No lecture. Work on project 2 | L2.1, L2.2 | |
26.11 | No lecture. Work on project 2 | L2.1, L2.2 | |
20.11 | Q&A | ||
19.11 | Summary; Q&A | ||
13.11 | The Rayleigh-Ritz method and Lanczos algorithm for eigenvalue computation | D7.1-D7.7 | L1.3, L2.4 |
12.11 | Perturbation of symmetric eigenvalue problems | D5.2 | L1.3 |
06.11 | QR algorithm with shifts, Wilkinson shift | TB29 | L1.3, L2.4 |
05.11 | QR algorithm without shifts, simultaneous iteration | TB28 | L1.3 |
30.10 | Power iteration, inverse iteration, Rayleigh quotient iteration | TB27 | L1.3 |
29.10 | Eigenvalue problems, eigenvalue-revealing factorizations, eigenvalue algorithms | TB24–25, TB26 (self-study), slides | L1.3, L1.9 |
23.10 | Matrix properties via the SVD | TB5 | L1.9 |
22.10 | The singular value decomposition (SVD) | GVL2.5, TB4 | L1.9 |
16.10 | Domain decomposition (DD) methods, Schwarz' alternating procedure, multiplicative and additive overlapping Schwarz, DD as a preconditioner | S14.1, S14.3–14.3.3 | L1.8, L2.3 |
15.10 | Intergrid operators, two-grid cycles, V-cycles and W-cycles, red-black Gauss–Seidel, MG as a preconditioner | S13.3–13.4.3, S12.4.1 | L1.2, L1.8, L2.3 |
09.10 | Introduction to multigrid (MG) methods, weighted Jacobi iteration | S13.1–13.2.2 | L1.2, L1.8 |
08.10 | Preconditioned GMRES and CG. Flexible GMRES. | S9.1, S9.2.1, S9.3.1-9.3.2, 9.4 | L1.2, L1.7, L1.8 |
02.10 | Convergence analysis of MINRES, indefinite case. | Note, S9.1 | L1.6, L2.2, L2.4 |
01.10 | Convergence analysis of CG and GMRES | S6.11.1-6.11.4 (S6.11.2-self study) | L1.6, L2.2, L2.4 |
25.09 | The D-Lanczos algorithm, the conjugate gradient method (CG) | S6.7.1 | L1.2, L1.6 |
24.09 | GMRES (GMRES(k), QGMRES, and DQGMRES), Lanczos method | S6.5.1, S6.5.3, S6.5.5, parts of S6.5.6, S6.6.1 | L1.2, L1.6, L1.9 |
18.09 | Arnoldi's algorithm, the full orthogonalization method (FOM) and variations of FOM | S6.3–6.4.2 | L1.2, L1.5, L1.6, L2.2 |
17.09 | Convergence of steepest descent, minimal residual (MR) iteration, convergence of MR, Krylov subspace methods | S5.3.1–5.3.2, S6.1 | L1.4, L1.5, L1.6, L2.2 |
11.09 | No lecture. Work on project 1 | L2.1, L2.2 | |
10.09 | No lecture. Work on project 1 | L2.1, L2.2 | |
04.09 | Projection methods, error projection, residual projection, steepest descent method | S5.1.1-5.2.2, S5.3-5.3.1 | L1.4 |
03.09 | Orthogonal projections, Jacobi and Gauss-Seidel iteration, convergence of splitting methods, Gershgorin's theorem, the Petrov–Galerkin framework | S1.12.3–1.12.4, S4.1, S4.2–4.2.1, S1.8.4, S4.2.3, S5.1 | L1.1, L1.4 |
28.08 | Perturbation analysis, finite difference methods, diagonalization methods, projection operators | S1.13.1 (self-study), S1.13.2, S2.2–2.2.5 (self-study), R, S1.12.1 | L1.2, L1.4 |
27.08 | Similarity transformations. Normal, Hermitian, and positive definite matrices | S1.8–1.8.1, S1.8.2 (self-study), S1.8.3, S1.9, S1.11 | L1.9 |
21.08 | QR factorizations | S1.7, TB10 | L1.9 |
20.08 | Background in linear algebra | S1.1–1.3 (self-study), S1.4–1.6 |
There will be two projects in this course. The first one, which counts for 10 % of the final grade, is given in September, and the second part, which counts for 20 %, will be given in the end of October/beginning of November. The project can to be done either individually or in groups of two. Hand in your reports in PDF format (perhaps including the snippets of the most important parts of your code) and the full code via itslearning. This is my first time using it for project handins, so please let me know if you experience any technical problems with uploading your answers due to me setting up the project, and I will try to resolve them ASAP. Please submit your report as a PDF and the code as a ZIP file to Torbjørn Ringholm. You may choose which programming language to use. Please do not type your names on the report, only your candidate numbers. If your candidate number is not available yet, you may use your student number.
The project will cover learning outcomes L2.1–2.4, and L3.1.
You will need the note Eigenvalues of tridiagonal Toeplitz matrices in the first semester project.
These exercises are voluntary and should not be handed in. They are here to help you to achieve the learning objectives of the course and you are encouraged to solve them.