Solution to large symmetric eigenproblems by an accelerated conjugate gradient technique

The computation of the smallest eigenvalues and eigenvectors of large numerical problems is a very important task in a number of engineering applications. The eigensolution to finite element or finite difference linear models provides the shape of the normal modes of vibration and the corresponding natural frequencies of mechanical, structural and hydrodynamical systems. In the present paper the leftmost eigenpairs of large sparse symmetric positive definite matrices are assessed by an efficient numerical technique which combines a deflation procedure together with an optimization approach wherein the Rayleigh quotient is minimized by an accelerated conjugate gradient scheme. The acceleration is achieved by the aid of a preconditioning matrix given by the incomplete Cholesky factorization of the discretized model. The results from finite element matrices show that the p (with p equal to 10\uf715) smallest eigenvalues and eigenvectors are evaluated by the iterative deflating method after a number of iterations which turns out to be some orders of magnitude smaller than the problem size N. Several numerical experiments emphasize the promising features of the proposed approach