For linear square matrix eigenproblems, eigensolvers involving complex moments can determine all the eigenvalues in a given region in the complex plane and the corresponding eigenvectors . A complex moment plays the role of a filter and extracts eigencomponents of interest from random vectors or matrices. This study extends a projection method for regular matrix pencils to the singular nonsquare case . Extending the definition of a complex moment matrix to the nonsquare case has posed a challenge. This extension replaces the standard matrix inverse in the resolvent with the pseudoinverse. The extended method involves complex moments given by  contour integrals of generalized resolvents associated with nonsquare matrices. We give 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 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. As compared to previous methods, the new method is found to be more robust and efficient and is conjectured by experiment results to be more efficient in parallelized settings. Notably, the proposed method does not fail in cases involving pairs of extremely close eigenvalues ; it overcomes the issue of problem size.
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.
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