TY - JOUR

T1 - On why using DL(P) for the symmetric polynomial eigenvalue problem might need to be reconsidered

AU - Bueno, M. I.

AU - Pérez, J.

AU - Rogers, S.

N1 - Funding Information:
The research of S. Rogers and M. I. Bueno was partially supported by NSF grant DMS-1358884. The research of M. I. Bueno was also partially supported by “Ministerio de Economía, Industria y Competitividad (MINECO)” of Spain and “Fondo Europeo de Desarrollo Regional (FEDER)” of EU through grant MTM2015-65798-P and by the “Proyecto financiado por la Agencia Estatal de Investigación de España” (PID2019-106362GB-I00 / AEI / 10.13039/501100011033).”
Publisher Copyright:
© 2022, The Author(s) under exclusive licence to Istituto di Informatica e Telematica (IIT).

PY - 2022/11

Y1 - 2022/11

N2 - In the literature it is common to use the first and last pencils D1(λ , P) and Dk(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 TP is more convenient because it has better numerical properties and its use is more universal since TP is a strong linearization of any matrix polynomial P(λ) , while D1(λ ; P) and Dk(λ ; P) are not. However, TP 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 Dk(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 Dk(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 Dk(λ ; P).

AB - In the literature it is common to use the first and last pencils D1(λ , P) and Dk(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 TP is more convenient because it has better numerical properties and its use is more universal since TP is a strong linearization of any matrix polynomial P(λ) , while D1(λ ; P) and Dk(λ ; P) are not. However, TP 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 Dk(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 Dk(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 Dk(λ ; P).

KW - Backward errors

KW - Eigenvalue

KW - Eigenvector

KW - Polynomial eigenvalue problem

KW - symmetric/Hermitian matrix polynomial

UR - http://www.scopus.com/inward/record.url?scp=85141935325&partnerID=8YFLogxK

U2 - 10.1007/s10092-022-00483-4

DO - 10.1007/s10092-022-00483-4

M3 - Article

AN - SCOPUS:85141935325

SN - 0008-0624

VL - 59

JO - Calcolo

JF - Calcolo

IS - 4

M1 - 48

ER -