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We consider the solution of large-scale nonlinear algebraic Hermitianeigenproblems of the form $T(\lambda)v=0$ that admit a variationalcharacterization of eigenvalues. These problems arise in a variety ofapplications and are generalizations of linear Hermitian eigenproblems$Av\!=\!\lambda Bv$. In this paper, we propose a Preconditioned Locally MinimalResidual (PLMR) method for efficiently computing interior eigenvalues ofproblems of this type. We discuss the development of search subspaces,preconditioning, and eigenpair extraction procedure based on the refinedRayleigh-Ritz projection. Extension to the block methods is presented, and amoving-window style soft deflation is described. Numerical experimentsdemonstrate that PLMR methods provide a rapid and robust convergence towardsinterior eigenvalues. The approach is also shown to be efficient and reliablefor computing a large number of extreme eigenvalues, dramatically outperformingstandard preconditioned conjugate gradient methods.

Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. II. Interior eigenvalues