## Abstract

In the literature it is common to use the first and last pencils D_{1}(λ , P) and D_{k}(λ , P) in the “standard basis” for the vector space DL(P) of block-symmetric pencils to solve the symmetric/Hermitian polynomial eigenvalue problem P(λ) x= 0. When the polynomial P(λ) has odd degree, it was proven in recent years that the use of an alternative linearization T_{P} is more convenient because it has better numerical properties and its use is more universal since T_{P} is a strong linearization of any matrix polynomial P(λ) , while D_{1}(λ ; P) and D_{k}(λ ; P) are not. However, T_{P} is not defined for even degree matrix polynomials. In this paper we consider the case when P(λ) has even degree. It is believed that the eigenpair backward errors for the linearization D_{1}(λ ; P) and D_{k}(λ ; P) cannot differ much from the backward error of the original problem. We show that this is not the case, even when the polynomial P(λ) is well-scaled because of the ill-conditioning of the eigenvectors of D_{1}(λ ; P) and D_{k}(λ ; P). We introduce two block-symmetric linearizations for even degree matrix polynomials that overcome this problem and become an appropriate alternative to the traditional use of D_{1}(λ ; P) and D_{k}(λ ; P).

Original language | English |
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Article number | 48 |

Journal | Calcolo |

Volume | 59 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2022 |

## Keywords

- Backward errors
- Eigenvalue
- Eigenvector
- Polynomial eigenvalue problem
- symmetric/Hermitian matrix polynomial