Solving the quadratic eigenvalue problem expressed in non-monomial bases by the tropical scaling

In this paper, we consider the quadratic eigenvalue problem (QEP) expressed in various commonly used bases, including Taylor, Newton, and Lagrange bases. We propose to investigate the backward errors of the computed eigenpairs and condition numbers of eigenvalues for QEP solved by a class of block Kronecker linearizations. To improve the backward error and condition number of the QEP expressed in a non-monomial basis, we combine the tropical scaling with the block Kronecker linearization. We then establish upper bounds for the backward error of an approximate eigenpair of the QEP relative to the backward error of an approximate eigenpair of the block Kronecker linearization with and without tropical scaling. Moreover, we get bounds for the normwise condition number of an eigenvalue of the QEP relative to that of the block Kronecker linearization. Our investigation is accompanied by adequate numerical experiments to justify our theoretical findings.