On why using DL(P) for the symmetric polynomial eigenvalue problem might need to be reconsidered

In the literature it is common to use the first and last pencils D₁(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 D₁(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 D₁(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 D₁(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 D₁(λ ; P) and D_k(λ ; P).