Projection method for eigenvalue problems of linear nonsquare matrix pencils

Abstract

In this contribution, we extend a projection method for square matrix pencils to the singular non-square case. Eigensolvers involving complex moments for linear square matrix eigenproblems are paid much attention in the eigencomputation due to its coarse-grained parallelizability. Such solvers determine all finite eigenvalues in a prescribed region in the complex plane and the corresponding eigenvectors. For extracting eigencomponents from random vectors or matrices, a complex moment forms a filter that allows desired eigencomponents to pass through. The challenge is to extend the definition of a complex matrix moment to the non-square case. This extension naively replaces the standard matrix inverse in the resolvent with the pseudoinverse. The extended method involves complex moments associated with contour integrals of generalized resolvents associated with rectangular matrices. We give conditions such that this method gives all finite eigenvalues in a prescribed region in the complex plane and the corresponding eigenvectors. Surprisingly, almost the same algorithm as the original complex moment eigensolver works similarly on the nonsquare case. In numerical computations, the contour integrals are approximated by using numerical quadratures. The dominant cost is in the solutions of linear least squares problems that arise from quadrature points, and they can be readily parallelized in practice. Numerical experiments on large matrix pencils illustrate the method, comparing with previous methods, and show that the new method is more robust and efficient, in particular, in parallel. Further, the proposed method does not fail for the existence of many pairs of extremely close eigenvalues and breaks the previous barrier of problem size. We also discuss the limitation of the scope of problems that the proposed method can solve.