Syllabus
1. Idea and basic principles of iterative methods. Introduction to work with sparse and structured matrices.
2. Methods for solving systems with symmetric matrices.
3. Methods for solving systems with nonsymmetric matrices based on orthogonality and long recurrences, and based on biorthogonality and short recurrences.
4. Methods for solving linear approximation and ill-posed problems.
5. Generalizations for problems with multiple observations - block and band methods.
6. Preconditioning - idea, selection, construction.
7. Convergence and numerical stability - comparison and examples.
8. Multigrid - idea.
Annotation
The course is devoted to the most widely used iterative methods for solving systems of linear algebraic equations, linear approximation problems, eigenvalue problems, etc., including preconditioning.
The emphasis is put especially on effective algorithmic realization and study of convergence properties.