Parallel preconditioned conjugate gradient optimization of the Rayleigh quotient for the solution of sparse eigenproblems

A parallel algorithm based on the multidimensional minimization of the Rayleigh quotient is proposed to evaluate the leftmost eigenpairs of the generalized symmetric positive definite eigenproblem. The minimization is performed via a conjugate gradient-like procedure accelerated by a factorized approximate inverse preconditioner (FSAI) and by a number of block preconditioners. The resulting code obtains a high level of parallel efficiency and proves to be comparable with the PARPACK package on a set of large matrices arising from various discretizations of PDEs of elliptic/parabolic type.