三次特征值问题的迭代shift-and-invert Arnoldi算法（英文）

An iterated shift-and-invert Arnoldi algorithm for cubic eigenvalue problems

对于求解大规模二次特征值问题,叶强提出了一种迭代shift-and-invert Arnoldi投影算法(Ye Q.An iterated shift-and-invert Arnoldi algorithm for quadratic matrix eigenvalue problems.Appl Math Compt,2006,172:818-827).将这一策略推广到求解大规模三次特征值问题,基于改进的Krylov子空间,给出了求解大规模三次特征值问题的一种迭代shiftand-invert Arnoldi算法.结果表明,结合shift-and-invert技术,这是一种具有快速收敛性的高效算法.数值试验结果验证了算法的有效性.

To solve the large scale quadratic eigenvalue problem （QEP） L（λ）x ：= （λ2A ＋ λB ＋ C）x = 0, Ye proposed an iterated shift-and-invert Arnoldi projection algorithm based on the Krylov subspaces solely generated by the matrix A-1B （Ye Q. An iterated shift-and-invert Arnoldi algorithm for quadratic matrix eigenvalue problems. Appl Math Compt, 2006, 172： 818-827）. In this paper, we generalize this strategy for solving the large scale cubic eigenvalue problem （CEP）L（λ）x ：= （λ2A ＋ λB ＋ C）x = 0. Namely, we propose an extension of the iterated shift-and-invert Arnoldi algorithm from the QEP to the CEP. It is shown that, when iteratively combined with the shift-and-invert technique, it results in a fast converging algorithm for the CEP, which has the similar behavior of Ye＇s algorithm for the QEP. Numerical experiments are presented to illustrate this algorithm.