An improved iterative optimization technique for the leftmost eigenpairs of large symmetric matrices

An accelerated optimization technique combined with a stepwise deflation procedure is presented for the efficient evaluation of the p (p ≤ 20) leftmost eigenvalues and eigenvectors of finite element symmetric positive definite (p.d.) matrices of very large size. The optimization is performed on the Rayleigh quotient of the deflated matrices by the aid of a conjugate gradient (CG) scheme effectively preconditioned with the incomplete Cholesky factorization. No "a priori" estimate of acceleration parameters is required. Numerical experiments on large arbitrarily sparse problems taken from the engineering finite elements (f.e.) practice show a very fast convergence rate for any value of p within the explored interval and particularly so for the minimal eigenpair. In this case the number of iterations needed to achieve an accurate solution may be as much as 2 orders of magnitude smaller than the problem size. Several results concerning the spectral behavior of the CG preconditioning matrices are also given and discussed. © 1988.