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 generalized symmetric eigenproblems arising from the finite element integration of elliptic PDEs. The accelerated conjugate gradient method is used to minimize successive Rayleigh quotients defined in deflated subspaces of decreasing size. The pointwise Lanczos scheme is employed in combination with both the Cholesky factorization of the stiffness matrix and the preconditioned conjugate gradient method for evaluating the recursive Lanczos vectors. The three algorithms are applied to five sample problems of varying size up to almost 5000. The numerical results show that the Lanczos approach with Cholesky triangularization is generally faster (up to a factor of 5) for small to moderately large matrices, while the optimization method is superior for large problems in terms of both storage requirement and CPU time. In the large case, the Lanczos-Cholesky scheme may be very expensive to run even on modern quite powerful computers.