Syllabus
1. A brief overview of related topics from previous courses (the Schur decomposition, the QR decomposition, the LU decomposition, the singular value decomposition).
2. Solution of linear approximation problems (the least squares method, the total least squares method, generalizations).
3. Krylov subspaces (the Arnoldi and the Lanczos method for computation of a basis, connections to Jacobi matrices, applications).
4. Krylov subspace methods. Comparison of short a long recurrences (loss of orthogonality, stability, prize), Faber-Manteuffel theorem.
5. The conjugate gradient (CG) method, MINRES method.
6. The generalized minimal residual method (GMRES), FOM method. Overview of other Krylov subspace methods.
7. Matrix functions (definition, evaluation, apllications).
8. Special matrices (definition of selected matrices of special structure and properties, applications).
Annotation
The course is devoted to fundamentals of numerical linear algebra, with the concetration on methods for solving linear algebraic equations, including least squares, and on eigenvalue problems. The course emphasizes formulation of questions, motivation and interconnections. Recommended for bachelor's program in General
Mathematics, specializations Mathematical Modelling and Numerical Analysis, and Stochastics.