We study preconditioned eigensolvers for large-scale nonlinear Hermitian
eigenproblems of the form
that admit the min-max
principle of eigenvalues. For the computation of a small number of
extreme eigenvalues, we propose variants of preconditioned conjugate
gradient (PCG) methods. These algorithms are natural extensions of
PCG-like methods for linear Hermitian eigenproblems. To compute a large
number of eigenvalues, we explore a new solver called the preconditioned
locally minimal residual (PLMR) method. With stable preconditioners that
enhance eigenvalues in desired parts of the spectrum, this algorithm only
requires deflation of a small set of recently converged eigenvalues, and
its arithmetic and CPU time cost increase linearly with the number of
desired eigenvalues. Numerical experiments illustrate the efficiency of
these algorithms.