Lehrinhalte
The mathematical principles of Krylov subspace methods, which form an important class of iterative
methods for solving linear algebraic systems and eigenvalue problems, will be discussed. Different ap-
proaches to the de nition and implementation of such methods will be presented, and the connections
to various areas of mathematics will be explored. This will lead to a uni ed view of important con-
cepts of analysis and algebra. Among the topics are: Projection methods, mathematical characterization
of Krylov subspace methods by their projection properties, implementation of different methods using
orthogonalization algorithms (in particular Arnoldi and Lanczos algorithms), orthogonal polynomials,
Gauss-Christoffel quadrature methods, solution of moment problems, continued fractions, Jacobi matrices
and their algebraic properties, cyclic subspaces and the existence of short recurrences for orthogonalizing
Krylov sequences, error bounds and error estimation in Krylov subspace methods.
