A Novel Deflation Technique for Solving Quadratic Eigenvalue Problems

In this paper we propose numerical algorithms for solving large-scale quadratic eigenvalue problems for which a set of eigenvalues closest to a fixed target and the associated eigenvectors are of interest. The desired eigenvalues are usually with smallest modulo in the spectrum. The algorithm based on the quadratic Jacobi-Davidson (QJD) algorithm is proposed to find the first smallest eigenvalue closest to the target. To find the sucessive eigenvalues closest to the target, we propose a novel explicit non-equivalence low-ank deflation technique. The technique transforms the smallest eigenvalue to infinity, while all other eigenvalues remain unchanged. Thus, the original second smallest eigenvalue becomes the smallest of the new quadratic eigenvalue problem, which can then be solved by the QJD algorithm. To compare with locking and restarting quadratic eigensolver [14], our numerical experience shows that the QJD method combined with our explicit non-equivalence deflation is robust and efficient.