An accelerated cyclic-reduction-based solvent method for solving quadratic eigenvalue problem of gyroscopic systems

Abstract

The quadratic eigenvalue problem (QEP) (λ2M+λG+K)x=0, with MT=M being positive definite, KT=K being negative definite and GT=−G, is associated with gyroscopic systems. In Guo (2004), a cyclic-reduction-based solvent (CRS) method was proposed to compute all eigenvalues of the above mentioned QEP. Firstly, the problem is converted to find a suitable solvent of the quadratic matrix equation (QME) MX2+GX+K=0. Then using a Cayley transformation and a proper substitution, the QME is transformed into the nonlinear matrix equation (NME) Z+ATZ−1A=Q with A=M+K+G and Q=2(M−K). The problem finally can be solved by applying the CR method to obtain the maximal symmetric positive definite solution of the NME as long as the QEP has no eigenvalues on the imaginary axis or for some cases where the QEP has eigenvalues on the imaginary axis. However, when all eigenvalues of the QEP are far away from or near the origin, the Cayley transformation seems not to be the best one and the convergence rate of the CRS method proposed in Guo (2004) might be further improved. In this paper, inspired by using a doubling algorithm to solve the QME, we use a Möbius transformation instead of the Cayley transformation to present an accelerated CRS (ACRS) method for solving the QEP of gyroscopic systems. In addition, we discuss the selection strategies of optimal parameter for the ACRS method. Numerical results demonstrate the efficiency of our method.