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6FMAI13 Lectures and Seminars
Note The chapters in the book refers to the 3rd edition. There may be small differences compared to the 4th edition.| Activity(*) | Description | Material |
| L1 Introduction |
Introduction. Matrix-Matrix Multiply. Operation counts. Basic concepts such as Rank, Null Space and Transpose. |
GVL 1.1-1.3, 1.4.5-1.4.8, 2.1-2.3. |
| L2 Linear Systems |
The LU and Cholesky decompositions. The Condition number. Sensitivity analysis. Linear Least Squares problems. The QR decomposition. Application: Projections in Computer graphics. |
GVL 3.1-3.2, 5.1 |
| S1 |
First set of handin problems: |
P1.pdf |
| L3 QR Decomposition |
Computing the QR decompositon. Refletions and Rotations. Row updating. Applications: Circle fitting, Tikhonov regularization, Image deblurring. |
GVL 5.1-5.3 |
| S2 |
Finish the first set of handin problems. |
|
| L3.5 Errors |
Model of floating point arithmetic. Error analysis. Orthogonal matrices. Guass transformations. The LU Decomposition and Cholesky decompositions. Special linear systems. |
GVL 3.3-3.5, 4.1-4.2, 4.7 |
| L4 Eigenvalues |
Basic theory of eigenvalues. The Companion Matrix. Rayleigh quotient. The Power method and Inverse iteration. Localization and Sensitivity. |
GVL 7.1-7.3 |
| S3 |
Second set of hand-in problems |
P2.pdf |
| L5 QR Algorithm |
Similarity transformations. Decoupling. The Schur and Hessenberg decompositions. The QR algorithm. Application: Google pageRank. |
GVL 7.4-7.5 |
| L5.5 Symmetric |
The Symmetric Eigenvalue problem. Tridiagonal matrices. Singular Value decomposition. |
GVL 8.1.1, 8.1.5, 8.3, 8.5-8.5.4 |
| S4 |
Finish the second set of problems. |
|
| L8 Singular values |
The singular value decomposition. Fundamental Subspaces. Projections. Computing the SVD. Applications: Low rank approximation, Total least squares, Classification of Hand written digits. |
GVL 8.6, 12.3 |
| L9 Sparse |
Integral Equations. Application: Remote sensing. Sparse matrices. Stationary iterative methods. |
GVL 10.1 |
| S5 |
Third set of hand-in problems. |
P3.pdf |
| L10 Krylov |
The projection method. Optimality Results. Krylov subspaces. GMRES and CG methods. Least Squares problems. |
GVL 10.2-10.4. |
| L10.5 Preconditioning |
Convergence rate for CG and GMRES. Preconditioning. |
|
| S6 |
Finish the third set of hand-in problems. |
Sidansvarig: Fredrik Berntsson
Senast uppdaterad: 2020-06-24