The aim of the course is to enable the students to name, state and classify state-of-the-art Krylov subspace methods for the solution of the core problems of the engineering sciences, namely, eigenvalue problems, solution of linear systems, and model reduction; they gather basic knowledge about approaches for the solution of matrix equations.
Part A (Krylov subspace methods): derivation (Richardson and power method); computation of a basis; Ritz, OR, and MR approaches; Arnoldi-based methods (Arnoldi, GMRes); Lanczos-based methods (Lanczos, CG, BiCG, QMR, SymmLQ, PvL); Sonneveld-based methods (IDR, BiCGStab, TFQMR, IDR(s)). Part B (matrix equations): Sylvester equation; Lyapunov equation; algebraic Riccati equation.