A Pade Approximate Linearization for Solving the Quadratic Eigenvalue Problem with Low- Rank Damping

The low-rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low-rank damping property, we propose a Padé Approximate Linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n+ lm, which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach, where n is the dimension of the quadratic eigenvalue problem, l and m are the rank of the damping matrix and the order of a Padé approximant, respectively. Numerical examples show that by exploiting the low-rank damping property, the PAL algorithm runs 33 – 47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems.