Buongiorno,
vi invio gli estremi del seminario di Raf Vandebril.
Dario
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Giorno: 6 Aprile, ore 10:00
Luogo: Aula Magna, DIpartimento di Matematica
Autore: Raf Vandebril, KU Leuven
Titolo: On classical, extended, and rational Krylov and the associated
QR algorithms
Abstract:
In this lecture we will discuss three main classes of Krylov and QR type
methods: the classical, the extended, and the rational versions. We will
initially focus on the classical QR method and reintepret all
ingredients such as: associated Krylov spaces, associated Hessenberg
matrices, implicit theorems, bulge chasing, eigenvalue swapping, and
subspace iteration. Another look at these essentials will allow us to
straightforwardly generalize all of this to the extended and rational case.
More precisely, we will start by deducing the structure of the
recurrence pencil associated to all Krylov methods. It will be shown
that the rational Krylov method produces the most general pencil, namely
two Hessenberg matrices, which implicitly store the poles of the
rational Krylov method. We will revisit the implicit Q-Theorem and link
it to its dual theorem, the implicit HK-Theorem, allowing us to provide
a pole chasing algorithm for two Hessenberg matrices. This generalizes
the classical and extended QR algorithms.
Correctness will follow directly from the previous analysis of the
classical QR algorithm. The rational QR algorithm allows for additional
freedom, which is the introduction of poles. Even without a good
pole-strategy we will see that the iteration count reduces with almost
10 procent.
