2020/10/12 by Fernando De Terán, De Terán, Fernando, Carla Hernando +3
Computer Science · Mathematics · #15A18 #15A21 #15A22 #15B57 #65F15 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2010.06033
openalex publication_date 2020/10/12 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In the framework of Polynomial Eigenvalue Problems, most of the matrix\npolynomials arising in applications are structured polynomials (namely\n(skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The\nstandard way to solve Polynomial Eigenvalue Problems is by means of\nlinearizations. The most frequently used linearizations belong to general\nconstructions, valid for all matrix polynomials of a fixed degree, known as\n em companion linearizations. It is well known, however, that is not possible\nto construct companion linearizations that preserve any of the previous\nstructures for matrix polynomials of even degree. This motivates the search for\nmore general companion forms, in particular em companion \ℓ-ifications.\nIn this paper, we present, for the first time, a family of (generalized)\ncompanion \ℓ-ifications that preserve any of these structures, for matrix\npolynomials of degree k=(2d+1)\ℓ. We also show how to construct sparse\n\ℓ-ifications within this family. Finally, we prove that there are no\nstructured companion quadratifications for quartic matrix polynomials.\n