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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

First set of handin problems:

L3 QR Decomposition

Computing the QR decompositon. Refletions and Rotations. Row updating. Applications: Circle fitting, Tikhonov regularization, Image deblurring.

GVL 5.1-5.3

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

Second set of hand-in problems

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

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

Third set of hand-in problems.

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.


Finish the third set of hand-in problems.

Sidansvarig: Fredrik Berntsson
Senast uppdaterad: 2020-06-24