On why using $${{\mathbb {D}}}{{\mathbb {L}}}(P)$$ for the symmetric polynomial eigenvalue problem might need to be reconsidered
On why using $${{\mathbb {D}}}{{\mathbb {L}}}(P)$$ for the symmetric polynomial eigenvalue problem might need to be reconsidered
复制标题
为什么可能需要重新考虑使用 $${{mathbb {D}}}{{mathbb {L}}}(P)$$ 来解决对称多项式特征值问题
DOI：
10.1007/s10092-022-00483-4
复制
发表时间：
2022
期刊：
影响因子：
1.7
通讯作者：
Rogers, S.
中科院分区：
文献类型：
--
作者：
Bueno, M. I.;Pérez, J.;Rogers, S.
In the literature it is common to use the first and last pencilsandin the “standard basis” for the vector spaceof block-symmetric pencils to solve the symmetric/Hermitian polynomial eigenvalue problem. When the polynomialhas odd degree, it was proven in recent years that the use of an alternative linearizationis more convenient because it has better numerical properties and its use is more universal sinceis a strong linearization of any matrix polynomial, whileandare not. However,is not defined for even degree matrix polynomials. In this paper we consider the case whenhas even degree. It is believed that the eigenpair backward errors for the linearizationandcannot differ much from the backward error of the original problem. We show that this is not the case, even when the polynomialis well-scaled because of the ill-conditioning of the eigenvectors ofand. We introduce two block-symmetric linearizations for even degree matrix polynomials that overcome this problem and become an appropriate alternative to the traditional use ofand.
