A Backward Stable Algorithm for Quadratic Eigenvalue Problems

Quadratic eigenvalue problems (QEPs) appear in almost all vibration analysis of systems, such as buildings, circuits, acoustic structures, and so on. Conventional numerical method for QEPs is to linearize a QEP as a doublly-sized generalized eigenvalue problem (GEP), then call a backward stable algorithm to solve the GEP, for example, the QZ for dense GEP, and at last recover approximated eigenpairs of original QEP from those of the GEP. However, the growth factor of the condition number in linearization, that is the ratio of the condition numbers between the QEP and the linearized GEP, may be much greater than 1, the growth factor of the backward error in recovery, that is the ratio of the backward errors between the recovered approximated eigenpairs of the QEP and the ones of the GEP, may also much greater than 1. To improve these growth factors, one needs to use a scaling before linearizations, carefully choose the linearizations, and properly recover the approximated eigenpairs. The FLV scaling by Fan, Lin and van Dooren can effectively improve the growth factors for not heavily damped QEP, the tropical scaling by Gaubert and Sharify can effectively improve the growth factors for heavily damped QEP with well-conditioned matrices. In this talk, we give an algorithm for solving the complete solution of a QEP, and prove that the growth factors of condition numbers and backward errors are of order 1, and in turn, the algorithm is backward stable for all QEPs, no matter the QEP is heavily damped or not, no matter the matrices in QEP are wellor ill-conditioned.