Large order eigenproblems arise in a variety of applications. The long computing times and large storage requirements of these problems motivate development of fast serial and efficient parallel eigensolvers. New numerical methods will be investigated for computing all eigenvalues and eigenvectors of real unsymmetric tridiagonal matrices that arise in applications and from reduction of general real matrices. The immediate goal is to provide new methods for the tridiagonal case. Because current methods for the general eigenproblem can suffer from parallel inefficiency or numerical instability, the larger goal is to develop accurate parallel methods that might ultimately be extended to the general eigenproblem without initial reduction to tridiagonal form. The research derives from methods that have been successful for the symmetric tridiagonal eigenproblem. The work will begin with a study of parallel methods for computing eigenvectors of an unsymmetric tridiagonal matrix given its computed eigenvalues. It will continue by developing divide and conquer methods to compute both eigenvalues and eigenvectors of an unsymmetric tridiagonal matrix. It will conclude with a mechanism for roughly locating the eigenvalues of any tridiagonal matrix. The latter technique is geared toward accelerating rootfinding methods for computing eigenvalues. Serial and parallel algorithms will be designed, implemented, and tested for all approaches.
