On the conditioning for heavily damped quadratic eigenvalue problem solved by linearizations

Heavily damped quadratic eigenvalue problem (QEP) is a special class of QEP, which has a large gap between small and large eigenvalues in absolute value. One common way for solving QEP is to linearize it to produce a matrix pencil. 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. Based on the analysis of upper bounds, we introduce applying tropical scaling for the linearizations to reduce the bounds and the condition number ratios. Furthermore, we establish upper bounds for the condition number ratios with tropical scaling and make a comparison with the unscaled bounds. Several numerical experiments are performed to illustrate our results.