A numerical method for quadratic eigenvalue problems of gyroscopic systems

We consider the quadratic eigenvalues problem (QEP) of gyroscopic systems (λ2M+λG+K)x=0(λ2M+λG+K)x=0, where M=M⊤,G=-G⊤M=M⊤,G=-G⊤ and K=K⊤∈Rn×nK=K⊤∈Rn×n with MM being positive definite. Guo [Numerical solution of a quadratic eigenvalue problem, Linear Algebra and its Applications   385 (2004) 391–406] showed that all eigenvalues of the QEP can be found by solving the maximal solution of a nonlinear matrix equation Z+A⊤Z-1A=QZ+A⊤Z-1A=Q with quadratic convergence when the QEP has no eigenvalues on the imaginary axis. The convergence becomes linear or more slower (Guo, 2004) when the QEP allows purely imaginary eigenvalues having even partial multiplicities. In this paper, we consider the general case when the QEP has eigenvalues on the imaginary axis. We propose an eigenvalue shifting technique to transform the original gyroscopic system to a new gyroscopic system, which shifts purely imaginary eigenvalues to eigenvalues with nonzero real parts, while keeps other eigenpairs unchanged. This transformation ensures that the new method for computing the maximal solution of the nonlinear matrix equation converges quadratically. Numerical examples illustrate the efficiency of our method.