A Rational Shira Method for the Hamiltonian Eigenvalue Problem

Abstract

The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skewHamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structureinduced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method rational SHIRA. A numerical example is presented to demonstrate its efficiency.

<inline-formula><tex-math id="M1">\begin{document} $\mathcal{L}_{∞}$ \end{document}</tex-math></inline-formula>-norm computation for large-scale descriptor systems using structured iterative eigensolvers

“…As already mentioned before, in the future we might use an alternative approach making use of a structured rational Krylov method like the rational SHIRA method suggested in [4]. When implemented with a criterion deciding whether or not there exists a purely imaginary eigenvalue in a neighborhood of a given target shift, one could cover the imaginary axis by a certain distribution of shifts and associated neighborhoods including an interval from zero to an upper bound on the eigenvalue of largest magnitude of the structured matrix pencil.…”

“…Therefore, we do not pursue this approach here any further and leave this to future work. 4. Implementation and Numerical Examples.…”

“…The data for the numerical examples was taken from [26,20,17,15], and is available on the website of Joost Rommes 2 and on the SLICOT webpage 3 . Furthermore, we use the MATLAB implementation for even IRA from Christian Schröder 4 . All examples are stable, so we indeed compute the H ∞ -norm in all tests.…”

“…Alternatively, one can use shiftand-invert variants of the Hamiltonian Lanczos algorithm [5,6]. In particular, the method discussed in [4] can be adapted here. The latter approach, based on the rational Krylov idea of Ruhe [28], would allow to change the shift on the fly (without restarting the Lanczos recursion) and thus, to search along the imaginary axis by incrementing the shift from 0 to an upper bound based on an (estimated) spectral bound of the shift-and-invert operator.…”

A rational Even-IRA algorithm for the solution of T-even polynomial eigenvalue problems

“…The method we propose is an implicitly-restarted rational Krylov-Schur approach based on the Even-IRA algorithm introduced in [24] (see also [12]). In contrast to the Even-IRA algorithm and motivated by [3], our approach explicitly allows for changes of the shift parameter during the iteration. This leads to a flexible and adjustable rational Krylov algorithm.…”