On Convergence of the Inexact Rayleigh Quotient Iteration with the Lanczos Method Used for Solving Linear Systems

For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has established new local general convergence results, independent of iterative solvers for inner linear systems. The theory shows that the method locally converges quadratically under a new condition, called the uniform positiveness condition. In this paper we first consider the local convergence of the inexact RQI with the unpreconditioned Lanczos method for the linear systems. Some attractive properties are derived for the residuals, whose norms are $\xi_{k+1}$'s, of the linear systems obtained by the Lanczos method. Based on them and the new general convergence results, we make a refined analysis and establish new local convergence results. It is proved that the inexact RQI with Lanczos converges quadratically provided that $\xi_{k+1}\leq\xi$ with a constant $\xi\geq 1$. The method is guaranteed to converge linearly provided that $\xi_{k+1}$ is bounded by a small multiple of the reciprocal of the residual norm $\|r_k\|$ of the current approximate eigenpair. The results are fundamentally different from the existing convergence results that always require $\xi_{k+1}<1$, and they have a strong impact on effective implementations of the method. We extend the new theory to the inexact RQI with a tuned preconditioned Lanczos for the linear systems. Based on the new theory, we can design practical criteria to control $\xi_{k+1}$ to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory.Comment: 20 pages, 8 figures. arXiv admin note: text overlap with arXiv:0906.223

Harmonic and Refined Harmonic Shift-Invert Residual Arnoldi and Jacobi--Davidson Methods for Interior Eigenvalue Problems

This paper concerns the harmonic shift-invert residual Arnoldi (HSIRA) and Jacobi--Davidson (HJD) methods as well as their refined variants RHSIRA and RHJD for the interior eigenvalue problem. Each method needs to solve an inner linear system to expand the subspace successively. When the linear systems are solved only approximately, we are led to the inexact methods. We prove that the inexact HSIRA, RHSIRA, HJD and RHJD methods mimic their exact counterparts well when the inner linear systems are solved with only low or modest accuracy. We show that (i) the exact HSIRA and HJD expand subspaces better than the exact SIRA and JD and (ii) the exact RHSIRA and RHJD expand subspaces better than the exact HSIRA and HJD. Based on the theory, we design stopping criteria for inner solves. To be practical, we present restarted HSIRA, HJD, RHSIRA and RHJD algorithms. Numerical results demonstrate that these algorithms are much more efficient than the restarted standard SIRA and JD algorithms and furthermore the refined harmonic algorithms outperform the harmonic ones very substantially.Comment: 15 pages, 4 figure