## 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. Matrix-Matrix Multiply. Operation counts. Basic concepts such as Rank and Null Space. Wendland 1.1-1.4, 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 3.1-3.6, 2.3-2.4 S1 1.1-1.2, 1.4-1.6, 1.10, 2.2-2.5. 3.1-3.3. 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 5.1-5.4 S2 3.4-3.6, 3.8, 3.9, 3.12-3.13, 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, 5.5-5.6 S3 4.12,4.14-4.18 4.20-4.22 L6 Non-Linear Non-linear equations. The Contraction mapping theorem. Taylor series and linearization. Newton's method. Updating methods. Application: Image inpainting. Trajectory of a Soccer ball. S4 5.2-5.5 L7 Optimization Non-Linear Least Squares. Existance and Uniquenesas results. The Newton and Gauss-Newton methods. Application: Data Assimilation. The Singular Value Decomposition. Application: Linear Systems of Equations. The Condition Number. Heath 6.5-6.6 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, 5.7-5.9, 6.1-6.5 L9 Sparse Integral Equations. Application: Remote sensing. Sparse matrices. Stationary iterative methods. Wendland 4.1-4.3 S6 6.6-6.13, 7.1-7.2 L10 Krylov The projection method. Optimality Results. Krylov subspaces. GMRES and CG methods. Least Squares problems. Wendland 6.1-6.3 S7 7.3-5, 7.7-7.8

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Last updated: 2023-01-13