Convergence Analysis of Inexact Rayleigh Quotient Iteration

We consider the computation of the smallest eigenvalue and associated eigenvector of a Hermitian positive definite pencil. Rayleigh quotient iteration (RQI) is known to converge cubically, and we first analyze how this convergence is affected when the arising linear systems are solved only approximately. We introduce a special measure of the relative error made in the solution of these systems and derive a sharp bound on the convergence factor of the eigenpair in a function of this quantity. This analysis holds independently of the way the linear systems are solved and applies to any type of error. For instance, it applies to rounding errors as well. We next consider the Jacobi–Davidson method. It acts as an inexact RQI method in which the use of iterative solvers is made easier because the arising linear systems involve a projected matrix that is better conditioned than the shifted matrix arising in classical RQI. We show that our general convergence result straightforwardly applies in this context and permits us to trace the convergence of the eigenpair in a function of the number of inner iterations performed at each step. On this basis, we also compare this method with some form of inexact inverse iteration, as recently analyzed by Neymeyr and Knyazev.