Effective preconditioning for the simultaneous evaluation of the smallest eigenpairs of sparse matrices

A preconditioned simultaneous iteration method is described for the solution of the generalized eigenproblem Ax=lBx, where A an B are are real symmetric positive definite matrices. The procedure relies on the optimization of the Rayleigh quotient over a subspace of orthogonal vectors by a conjugate gradient technique effectively preconditioned with the pointwise incomplete Cholesky factorization. The numerical experiments show that, while the simultaneous conjugate gradient scheme fails to converge, the accelerated iterations yield accurate results in a number of steps which is much smaller than N.