Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

Efficient computation of extreme eigenvalues of large-scale linear Hermitian eigenproblems can be achieved by preconditioned conjugate gradient (PCG) methods. In this paper, we study PCG methods for computing 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 eigenvalue. The efficiency of variants of PCG is illustrated by numerical experiments. Overall, the locally optimal block preconditioned conjugate gradient (LOBPCG) is the most efficient method, as in the linear setting.