A Rational Even-IRA Algorithm for the Solution of $T$-Even Polynomial Eigenvalue Problems
Peter Benner, Heike Faßbender, Philip Saltenberger · SIAM Journal on Matrix Analysis and Applications · 2021
In this work we present a rational Krylov subspace method for solving real large-scale polynomial eigenvalue problems with $T$-even (that is, symmetric/skew-symmetric) structure. Our method is based on the Even-IRA algorithm [V. Mehrmann, C. Schröder, and V. Simoncini, Linear Algebra Appl., 436 (2012), pp. 4070--4087]. To preserve the structure, a sparse $T$-even linearization from the class of block minimal bases pencils is applied (see [F. M. Dopico et al., Numer. Math., 140 (2018), pp. 373--426). Due to this linearization, the Krylov basis vectors can be computed in a cheap way. Based on the ideas developed in [P. Benner and C. Effenberger, Taiwanese J. Math., 14 (2010), pp. 805--823], a rational decomposition is derived so that our method explicitly allows for changes of the shift during the iteration. This leads to a method that is able to compute parts of the spectrum of a $T$-even matrix polynomial in a fast and reliable way.