On multistep Rayleigh quotient iterations for Hermitian eigenvalue problems

Abstract

We present a multistep Rayleigh quotient iteration, as well as its inexact variant, for computing an eigenpair of a large sparse Hermitian matrix. Theoretical analysis shows that both exact and inexact multistep Rayleigh quotient iterations converge much faster than the exact and inexact Rayleigh quotient iterations, respectively. For the inexact multistep Rayleigh quotient iteration, we use the preconditioned conjugate gradient method to solve the inner linear systems, and find that significant saving in the number of inner iteration steps can be achieved when choosing a proper preconditioner. Numerical examples demonstrate effectiveness and superiority of our methods.