A COMPARISON OF LANCZOS AND OPTIMIZATION METHODS IN THE PARTIAL SOLUTION OF SPARSE SYMMETRICAL EIGENPROBLEMS

In the present paper, we analyse the computational performance of the Lanczos method and a recent optimization technique for the calculation of the p (p less than or equal to 40) leftmost eigenpairs of generali zed symmetric eigenproblems arising from the finite element integratio n of elliptic PDEs. The accelerated conjugate gradient method is used to minimize successive Rayleigh quotients defined in deflated subspace s of decreasing size. The pointwise Lanczos scheme is employed in comb ination with both the Cholesky factorization of the stiffness matrix a nd the preconditioned conjugate gradient method for evaluating the rec ursive Lanczos vectors. The three algorithms are applied to five sampl e problems of varying size up to almost 5000. The numerical results sh ow that the Lanczos approach with Cholesky triangularization is genera lly faster (up to a factor of 5) for small to moderately large matrice s, while the optimization method is superior for large problems in ter ms of both storage requirement and CPU time. In the large case, the La nczos-Cholesky scheme may be very expensive to run even on modern quit e powerful computers.