Structured matrices in (rational) Krylov subspace methods

• Date: 17. 2. 2022
• Time: 14:00 - 15:30
• Address:
  Sokolovská 83, Praha
• Room: K3
• Speaker: Niel Van Buggenhout

Structured matrices play a key role in the development of effective Krylov subspace methods and their theoretical study. Standard Krylov subspaces are built using consecutive powers of a matrix multiplied by a fixed vector.The associated structured matrices are well known, these are Hessenberg and tridiagonal matrices. In recent years there has been a growing interest in rational Krylov subspaces. Rational Krylov subspaces allow multiplication with a matrix and with a shifted and inverted version of this matrix. This freedom allows more flexibility than the standard Krylov subspaces. For example, in eigenvalue computation rational Krylov subspace methods allow us to specify different regions of interest throughout the whole complex plane. This can speed up the convergence of Ritz values to eigenvalues within these regions. Before any successful numerical methods can be developed, it is necessary to have a theoretical framework. This seminar will focus on identifying all possible matrix structures that can occur in the context of rational Krylov subspaces. The main result shows that a tridiagonal matrix pencil suffices to represent biorthonormal bases for any rational Krylov subspace. This leads to a rational generalization of the Lanczos iteration. As an application of the developed theory, novel methods are discussed to generate (formal) orthogonal polynomials and (bi)orthogonal rational functions.