| The polynomial eigenvalue problem is an important research topic in the field of numerical algebra and is of great importance in the field of scientific and engineering computing.Heavily damped quadratic eigenvalue problem is a special class of the polynomial eigenvalue problem,which has a large gap between small and large eigenvalues in absolute value.One common way to solve the polynomial eigenvalue problem is to recast it by linearization,as a generalized eigenvalue problem.In this paper,we analyze the numerical stability of solving the polynomial eigenvalue problem via linearization in terms of both backward error and condition number and combine the linearization with tropical scaling to increase the numerical stability of the polynomial eigenvalue problem solved by a linearization.We study backward errors of approximated eigenvalues of the polynomial eigenvalue problem via linearization and use the companion linearization without and with tropical scaling to construct global upper bounds for the backward errors of approximated eigenvalues of the polynomial eigenvalue problem.Moreover,we investigate upper bounds for the conditioning of eigenvalues of linearizations of four common forms relative to that of the quadratic and compare them with the previous studies and establish upper bounds for the condition number ratios with tropical scaling and make a comparison with the unscaled bounds.The numerical stability of the polynomial eigenvalue problem,in terms of backward error and condition number,can be successfully improved by tropical scaling and well predicted by upper bounds,as shown in numerical experiments. |
