De Teran, Fernando and Dopico, Froilan M. and Mackey, D. Steven (2010) Structured Linearizations for Palindromic Matrix Polynomials of Odd Degree. [MIMS Preprint]
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Abstract
The standard way to solve polynomial eigenvalue problems $P(\la)x=0$ is to convert the matrix polynomial $P(\la)$ into a matrix pencil that preserves its spectral information-- a process known as linearization. When $P(\la)$ is palindromic, the eigenvalues, elementary divisors, and minimal indices of $P(\la)$ have certain symmetries that can be lost when using the classical first and second companion linearizations for numerical computations, since these linearizations do not preserve the palindromic structure. Recently new families of linearizations have been introduced with the goal of finding linearizations that retain whatever structure that the original $P(\la)$ might possess, with particular attention paid to the preservation of palindromic structure. However, no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of linearizations for odd degree polynomials $P(\la)$ which are palindromic whenever $P(\la)$ is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family.
Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | matrix polynomial, matrix pencil, minimal indices, palindromic, structured linearization |
Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis |
Depositing User: | Dr. D. Steven Mackey |
Date Deposited: | 19 Apr 2010 |
Last Modified: | 08 Nov 2017 18:18 |
URI: | http://eprints.maths.manchester.ac.uk/id/eprint/1441 |
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- Structured Linearizations for Palindromic Matrix Polynomials of Odd Degree. (deposited 19 Apr 2010) [Currently Displayed]
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