SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1172-1198, January 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.

中文翻译：

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求解$T$-Even 多项式特征值问题的一种有理Even-IRA 算法

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 3 期，第 1172-1198 页，2021 年 1 月。
在这项工作中，我们提出了一种有理 Krylov 子空间方法，用于解决具有 $T$-even（即对称/偏斜对称）结构的真实大规模多项式特征值问题。我们的方法基于 Even-IRA 算法 [V. Mehrmann, C. Schröder 和 V. Simoncini，线性代数应用，436 (2012)，第 4070--4087 页]。为了保留结构，应用了来自块最小基铅笔类的稀疏 $T$-even 线性化（参见 [FM Dopico 等人，Numer. Math., 140 (2018), pp. 373--426）。由于这种线性化，可以以廉价的方式计算 Krylov 基向量。基于 [P. Benner 和 C. Effenberger, Taiwan J. Math., 14 (2010), pp. 805--823]，推导出合理分解，以便我们的方法明确允许迭代期间的移位变化。