Noferini, Vanni and Sharify, Meisam and Tisseur, Francoise (2014) Tropical roots as approximations to eigenvalues of matrix polynomials. [MIMS Preprint]
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Abstract
The tropical roots of $tp(x) = \max_{0\le j\le d}\A_j\x^j$ are points at which the maximum is attained at least twice. These roots, which can be computed in only $O(d)$ operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial $P(\lambda)=\sum_{j=0}^d \lambda^j A_j$, in particular when the norms of the matrices $A_j$ vary widely. Our aim is to investigate this observation and its applications. We start by providing annuli defined in terms of the tropical roots of $tp(x)$ that contain the eigenvalues of $P(\lambda)$. Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of $P(\lambda)$. Our tropical localization of eigenvalues are less tight than eigenvalue localization results derived from a generalized matrix version of Pellet's theorem but they are easier to interpret. Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the EhrlichAberth method and our results further justify this choice. Our results provide the basis for analyzing the effect of Gaubert and Sharify's tropical scalings for $P(\lambda)$ on (a) the conditioning of linearizations of tropically scaled $P(\lambda)$ and (b) the backward stability of eigensolvers based on linearizations of tropically scaled $P(\lambda)$. We anticipate that the tropical roots of $tp(x)$, on which the tropical scalings are based, will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems such as that implemented in the MATLAB function \texttt{polyeig}.
Item Type:  MIMS Preprint 

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 Françoise Tisseur 
Date Deposited:  21 Apr 2014 
Last Modified:  08 Nov 2017 18:18 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2126 
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Tropical roots as approximations to eigenvalues of matrix polynomials. (deposited 31 Mar 2014)
 Tropical roots as approximations to eigenvalues of matrix polynomials. (deposited 21 Apr 2014) [Currently Displayed]
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