  Eigensolvers involving complex moments can determine all the eigenvalues in a
given region in the complex plane and the corresponding eigenvectors of a
regular linear matrix pencil. The complex moment acts as a filter for
extracting eigencomponents of interest from random vectors or matrices. This
study extends a projection method for regular eigenproblems to the singular
nonsquare case, thus replacing the standard matrix inverse in the resolvent
with the pseudoinverse. The extended method involves complex moments given by
the contour integrals of generalized resolvents associated with nonsquare
matrices. We establish conditions such that the method gives all finite
eigenvalues in a prescribed region in the complex plane. In numerical
computations, the contour integrals are approximated using numerical
quadratures. The primary cost lies 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 this method. The new method is more robust and efficient than
previous methods, and based on experimental results, it is conjectured to be
more efficient in parallelized settings. Notably, the proposed method does not
fail in cases involving pairs of extremely close eigenvalues, and it overcomes
the issue of problem size.
