This project will investigate new eigenvalue algorithms based upon the idea of homotopy continuation. These consist of deforming a simple matrix into the matrix in question, while following the easily found eigenpairs of the simple matrix to the eigenpairs being sought. The issues to be studied include: - Further exploration of a recently developed continuation algorithm for eigenvalue problems of symmetric triadiagonal matrices. - Continuation algorithms for eigenvalue problems of large sparse matrices. - Parallelization and vectorization of such algorithms for a variety of advanced architectures. - Continuation algorithms for eigenvalue problems of nonsymmetric matrices. The emphasis will be on the development of efficient computational algorithms and their numerical implementation. Hopefully, the work will lead to software packages that can be applied to practical eigenvalue problems, especially those of large, sparse matrices.
