This project is concerned with the development of algorithms for calculating the eigensystems of general matrices by reducing them to a form which is more compact than the usual Hessenberg form and by applying the LR algorithm ot the condensed form. Preliminary tests suggest that the proposed algorithm will produce accurate results in a much shorter time than the standard QR algorithm for a dense eigensystem. For the most condensed form of a tridiagonal matrix, a divide-and-conquer algorithm will be developed for use in parallel computers. For a specialized Hamilton eigenvalue problem which requires the solution of an algebraic Riccati equation, our algorithm will be based on symplectic similarity transformations which preserve the Hamiltonian structure of the matrix.
