论文标题
结构化强$ \ boldsymbol {\ ell} $ - 在单个基础上用于结构矩阵多项式
Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis
论文作者
论文摘要
在多项式特征值问题的框架中，在应用中产生的大多数矩阵多项式是结构化的多项式（即（skew-）对称，（偏斜）Hermitian，（抗）（抗）（抗）palindromic或交替使用）。解决多项式特征值问题的标准方法是通过线性化。最常用的线性化属于一般构造，对所有固定度的矩阵多项式有效，称为{\ em companion linearizations}。然而，众所周知，不可能构建伴随线性化来保留均匀程度的矩阵多项式的任何先前结构。这激发了搜索更一般的伴侣表格，尤其是{\ em companion $ \ ell $ --ifications}。在本文中，我们首次介绍了一个（广义）伴侣$ \ ell $的家庭，可以保留这些结构中的任何一个，用于$ k =（2d+1）\ ell $的矩阵多项式。我们还展示了如何在这个家庭中构建稀疏的$ \ ell $。最后，我们证明没有针对四分之一矩阵多项式的结构化伴侣四个。
In the framework of Polynomial Eigenvalue Problems, most of the matrix polynomials arising in applications are structured polynomials (namely (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve Polynomial Eigenvalue Problems is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as {\em companion linearizations}. It is well known, however, that is not possible to construct companion linearizations that preserve any of the previous structures for matrix polynomials of even degree. This motivates the search for more general companion forms, in particular {\em companion $\ell$-ifications}. In this paper, we present, for the first time, a family of (generalized) companion $\ell$-ifications that preserve any of these structures, for matrix polynomials of degree $k=(2d+1)\ell$. We also show how to construct sparse $\ell$-ifications within this family. Finally, we prove that there are no structured companion quadratifications for quartic matrix polynomials.
