Preconditioned Eigensolvers for Large-scale Nonlinear Hermitian Eigenproblems with Variational Characterizations. I. Conjugate Gradient Methods

Preconditioned conjugate gradient (PCG) methods have been widely used for computing a few extreme eigenvalues of large-scale linear Hermitian eigenproblems. In this paper, we study PCG methods to compute extreme eigenvalues of nonlinear Hermitian eigenproblems of the form T (λ)v = 0 that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG method, including its global and asymptotic convergence. We propose several variants of single-vector and block PCG methods with deflation for computing multiple eigenvalues, and compare them in arithmetic and memory cost. Variable indefinite preconditioning is shown to be effective to accelerate convergence when some desired eigenvalues are not close to the lowest or highest one. Efficiency of these algorithms is illustrated by numerical experiments. AMS subject classifications. 65F15, 65F10, 65F50, 15A18, 15A22.