A truncated-CG style method for symmetric generalized eigenvalue problems

A numerical algorithm is proposed for computing an extreme eigenpair of a symmetric/positive-definite matrix pencil (A,B)(A,B). The leftmost or the rightmost eigenvalue can be targeted. Knowledge of (A,B)(A,B) is only required through a routine that performs matrix–vector products. The method has excellent global convergence properties and its local rate of convergence is superlinear. It is based on a constrained truncated-CG trust-region strategy to optimize the Rayleigh quotient, in the framework of a recently proposed trust-region scheme on Riemannian manifolds.