A new hybrid method for finding an eigenpairs of a symmetric quadratic eigenvalue problem in an interval

The symmetric quadratic eigenvalue problem (lambda M-2 + lambda C + K)u = 0, where M, C, and K are given n x n matrices and (lambda, u) is an eigenpair, arises in a wide variety of practical applications, including vibration, acoustic, and noise control analysis[5]. In the most practical application, the problem is often of a very large dimension. Unfortunately because of the nonlinearity, the problem is extremely hard to solve numerically, and the state-of-art computational techniques, such as the Jacobi-Davidson method, are capable of computing only a few extremal eigenvalues and eigenvectors [2,3]. Fortunately, there are engineering applications that require only some of the eigenvalues lying within an interval. In this paper, a new hybrid method combining a Parametrized Newton-type method described in [1] with the Jacobi-Davidson method [2] is proposed to compute an eigenpair of quadratic pencil within an interval. The experimental results show that this method is much faster than the Jacobi-Davidson method. The results of this paper generalize those of an earlier work on parameterized Newton's Algorithm for finding an eigenpair of a symmetric matrix [1].