Week: Lecture Recordings: Numerical Linear Algebra | Computational Mathematics for Learning and Data Analysis - AA 2024/25 | INF - e-learning

Section outline

- Select activity 2024-09-19: Recap of linear algebra: linear combinations, matrix products, coordinates
  
  2024-09-19: Recap of linear algebra: linear combinations, matrix products, coordinates File
- Select activity l1
  
  l1 File
- Select activity 2024-09-20: Orthogonality, eigenvectors, positive definiteness and semidefiniteness
  
  2024-09-20: Orthogonality, eigenvectors, positive definiteness and semidefiniteness File
- Select activity l2
  
  l2 File
- Select activity 2024-09-27: introduction to least squares problems. Some applications: linear estimation, polynomial fitting. Uniqueness of solution. Method of normal equations. Pseudoinverse.
  
  2024-09-27: introduction to least squares problems. Some applications: linear estimation, polynomial fitting. Uniqueness of solution. Method of normal equations. Pseudoinverse. File
- Select activity l3
  
  l3 File
- Select activity 2024-10-02: Singular value decomposition. Matrix norms. Eckhart-Young theorem (statement)
  
  2024-10-02: Singular value decomposition. Matrix norms. Eckhart-Young theorem (statement) File
- Select activity l4
  
  l4 File
- Select activity 2024-10-04: sparse matrices. Conjugate gradient: introduction, subspace optimality properties. Krylov subspaces and their relation to gradient-type methods.
  
  2024-10-04: sparse matrices. Conjugate gradient: introduction, subspace optimality properties. Krylov subspaces and their relation to gradient-type methods. File
- Select activity l5
  
  l5 File
- Select activity 2024-10-16: Q-norm; orthogonality and convergence properties of CG (convergence in terms of polynomial approximation, worst-case bound, both without proof)
  
  2024-10-16: Q-norm; orthogonality and convergence properties of CG (convergence in terms of polynomial approximation, worst-case bound, both without proof) File
- Select activity l6
  
  l6 File
- Select activity 2024-10-18: convergence of CG and polynomial approximation. Data analysis with the SVD: images, student scores, text (latent semantic analysis)
  
  2024-10-18: convergence of CG and polynomial approximation. Data analysis with the SVD: images, student scores, text (latent semantic analysis) File
- Select activity l7
  
  l7 File
- Select activity 2024-10-25: dimensionality reduction with PCA; PCA of the Yale faces dataset, used also for image recognition
  
  2024-10-25: dimensionality reduction with PCA; PCA of the Yale faces dataset, used also for image recognition File
- Select activity l8
  
  l8 File
- Select activity 2024-10-30: Householder reflectors. QR factorization. Different ways to handle Q: thin QR, returning the Householder vectors.
  
  2024-10-30: Householder reflectors. QR factorization. Different ways to handle Q: thin QR, returning the Householder vectors. File
- Select activity l9
  
  l9 File
- Select activity 2024-11-08: solving least-squares problems with the QR factorization and with the SVD. Singular least squares problems. The effect of noise; regularization via truncated SVD.
  
  2024-11-08: solving least-squares problems with the QR factorization and with the SVD. Singular least squares problems. The effect of noise; regularization via truncated SVD. File
- Select activity l10
  
  l10 File
- Select activity 2024-11-13: Tikhonov regularization. Condition number. The condition number of solving linear equations and least-squares problems.
  
  2024-11-13: Tikhonov regularization. Condition number. The condition number of solving linear equations and least-squares problems. File
- Select activity l11
  
  l11 File
- Select activity 2024-11-15: stability of floating point computations. Backward stability, with an example. A posteriori stability tests for linear systems and LS problems.
  
  2024-11-15: stability of floating point computations. Backward stability, with an example. A posteriori stability tests for linear systems and LS problems. File
- Select activity l12
  
  l12 File
- Select activity 2024-11-22: backward stability of QR factorization. Backward stability properties of algorithms to solve LS problems and their comparison. Introduction to the Arnoldi algorithm.
  
  2024-11-22: backward stability of QR factorization. Backward stability properties of algorithms to solve LS problems and their comparison. Introduction to the Arnoldi algorithm. File
- Select activity l13
  
  l13 File
- Select activity 2024-11-27: Arnoldi algorithm, GMRES. Convergence, computational and implementation remarks. The symmetric version: MINRES.
  
  2024-11-27: Arnoldi algorithm, GMRES. Convergence, computational and implementation remarks. The symmetric version: MINRES. File
- Select activity l14
  
  l14 File
- Select activity 2024-12-06: overview of direct methods for linear systems: LU, LDL, Cholesky, remarks about sparsity
  
  2024-12-06: overview of direct methods for linear systems: LU, LDL, Cholesky, remarks about sparsity File
- Select activity l15
  
  l15 File
- Select activity 2024-12-16: examples of solutions of large-scale linear systems. Reordering in direct methods. A quick introduction to preconditioners for iterative methods.
  
  2024-12-16: examples of solutions of large-scale linear systems. Reordering in direct methods. A quick introduction to preconditioners for iterative methods. File
- Select activity l16
  
  l16 File