TANA15 Lectures and Seminars
(*) For Lectures the slides are available as PDFs and the the material are chapters in the course book. The seminars are for problem solving and the exercises are found in the problem collection.Activity(*)  Description  Material 
L1 Introduction 
Introduction. MatrixMatrix Multiply. Operation counts. Basic concepts such as Rank and Null Space. 
Wendland 1.11.4, 2.2 
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. 
Wendland 3.13.6, 2.32.4 
S1 

1.11.2, 1.41.6, 1.10, 2.22.5. 3.13.3. 
L3 QR Decomposition 
Computing the QR decompositon. Refletions and Rotations. Row updating. Applications: Circle fitting, Tikhonov regularization, Image deblurring. 
Wendland 3.6 
L4 Eigenvalues 
Basic theory of eigenvalues. Rayleigh quotient. The Power method and Inverse iteration. Localization and Sensitivity. 
Wendland 5.15.4 
S2 

3.43.6, 3.8, 3.9, 3.123.13, 4.1, 4.3, 4.5, 4.10 
L5 QR Algorithm 
Similarity transformations. Decoupling. The Schur and Hessenberg decompositions. The QR algorithm. Application: Google pageRank. 
Wendland 3.7, 5.55.6 
S3 

4.12,4.144.18 4.204.22 
L6 NonLinear 
Nonlinear equations. The Contraction mapping theorem. Taylor series and linearization. Newton's method. Updating methods. Application: Image inpainting. Trajectory of a Soccer ball. 

S4 

5.25.5 
L7 Optimization 
NonLinear Least Squares. Existance and Uniquenesas results. The Newton and GaussNewton methods. Application: Data Assimilation. The Singular Value Decomposition. Application: Linear Systems of Equations. The Condition Number. 
Heath 6.56.6 
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. 
Wendland 1.5, 5.7. 
S5 

5.1, 5.75.9, 6.16.5 
L9 Sparse 
Integral Equations. Application: Remote sensing. Sparse matrices. Stationary iterative methods. 
Wendland 4.14.3 
S6 

6.66.13, 7.17.2 
L10 Krylov 
The projection method. Optimality Results. Krylov subspaces. GMRES and CG methods. Least Squares problems. 
Wendland 6.16.3 
S7 

7.35, 7.77.8 
Page responsible: Fredrik Berntsson
Last updated: 20230113