On the optimal solution of large eigenpair problems

AbstractThe problem of approximation of an eigenpair of a large n × n matrix A is considered. We study algorithms which approximate an eigenpair of A using the partial information on A given by b, Ab, …, Ajb, j ⪡ n, i.e., by Krylov subspaces. A new algorithm called the generalized minimal residual (gmr) algorithm is analyzed. Its optimality for some classes of matrices is proved. We compare the gmr algorithm with the widely used Lanczos algorithm for symmetric matrices. The gmr and Lanczos algorithms cost essentially the same per step and they have the same stability characteristics. Since the gmr algorithm never requires more steps than the Lanczos algorithm, and sometimes uses substantially fewer steps, the gmr algorithm seems preferable. We indicate how to modify the gmr algorithm in order to approximate p eigenpairs of A. We also show some other problems which can be nearly optimally solved by gmr-type algorithms. The gmr algorithm for symmetric matrices was implemented and some numerical results are described. The detailed implementation, more numerical results, and the Fortran subroutine can be found in Kuczyński (“Implementation of the gmr Algorithm for Large Symmetric Eigenproblems,” Report, Columbia University, 1985). The Fortran subroutine is also available via anonymous FTP as “pub/gmrval” on COLUMBIA-EDU [128.59.16.1] on the Arpanet