# Effect of tropical scaling on linearizations of matrix polynomials: backward error and conditioning

Sharify, Meisam and Tisseur, Francoise (2017) Effect of tropical scaling on linearizations of matrix polynomials: backward error and conditioning. [MIMS Preprint]

## Abstract

The \textit{tropical scaling} algorithm has experimentally shown to generate accurate results in computing the eigenpairs of a matrix polynomial $P(\l)=\sum_{k=1}^\d \l^k A_k$. This algorithm scales $P(\l)$ by using certain quantities known as \textit{tropical roots}, then, for each tropical roots, it constructs a linearization of the tropically scaled polynomial, computes its eigenpairs and extracts eigenpairs of $P(\l)$ from those of the linearizations. In this work we analyse this algorithm in terms of backward error and conditioning. We show that when the tropical roots are well separated, for an eigenpair, the backward error of the scale matrix polynomial is bounded by the backward error of its linearization times the condition number of certain coefficients of $P(\l)$. These coefficients determine the abscissae of the nodes of the {\it Newton polygon} associated with the matrix polynomial. Similar results show that for an eigenvalue, the condition number of the linearization of a scaled matrix polynomial is bounded by the condition number of the scaled matrix polynomial times the condition number of these coefficients. Our results show that when these coefficients are well conditioned the eigenpairs can be computed without any numerical difficulty. These analysis is supported by the experiments which are provided at the end of the paper.

Item Type: MIMS Preprint MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Dr Françoise Tisseur 02 May 2017 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2547