On convergence of the inexact Rayleigh quotient iteration with MINRES

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm ξk≥1ξk≥1 of the inner linear system at outer iteration k+1k+1 and can be considerably weaker than the condition ξk≤ξ<1ξk≤ξ<1 with ξξ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖rk‖‖rk‖ be the residual norm of approximating eigenpair at outer iteration kk. Then all the available cubic and quadratic convergence results require ξk=O(‖rk‖)ξk=O(‖rk‖) and ξk≤ξξk≤ξ with a fixed ξξ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that ξk≤ξξk≤ξ with a constant ξ<1ξ<1 not near one, ξk=1−O(‖rk‖)ξk=1−O(‖rk‖) and ξk=1−O(‖rk‖2)ξk=1−O(‖rk‖2), respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.