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 D-1(lambda,P) and D-k(lambda,P) in the "standard basis" for the vector space DL(P) of block-symmetric pencils to solve the symmetric/Hermitian polynomial eigenvalue problem P(lambda)x=0. When the polynomial P(lambda) 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(lambda), while D1(lambda;P)} and Dk(lambda;P) are not. However, T-P is not defined for even degree matrix polynomials. In this paper we consider the case when P(lambda) has even degree. It is believed that the eigenpair backward errors for the linearization D1(lambda;P) and Dk(lambda;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(lambda) is well-scaled because of the ill-conditioning of the eigenvectors of D1(lambda;P) and Dk(lambda;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(lambda;P)and Dk(lambda;P).