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A Rational Even-IRA Algorithm for the Solution of $T$-Even Polynomial Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications ( IF 1.5 ) Pub Date : 2021-07-28 , DOI: 10.1137/20m1364485
Peter Benner , Heike Fassbender , Philip Saltenberger

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:

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。
更新日期:2021-07-29
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