0 On Convergence of the Inexact Rayleigh Quotient Iteration with the Lanczos Method Used for Solving Linear Systems∗
Zhongxiao Jia · 2012
For the Hermitian inexact Rayleigh quotient iteration (RQI), we present new general convergence results, independent of iterative solvers for inner linear systems. We prove that the method converges quadratically under a new condition, called the uniform positiveness condition. This condition is much weaker than the commonly used one for quadratic convergence that, at outer iteration k, requires the relative residual norm ξk (inner tolerance or accuracy) of the inner linear system to be smaller than one considerably and may allow ξk ≥ 1. Our focus is on the inexact RQI with the Lanczos method used for solving the linear systems. We derive some attractive properties of the residuals obtained by Lanczos. Based on these properties and the new general convergence results, we establish a number of insightful convergence results that relate accuracy of outer iterations to inner tolerance. It appears that the inexact RQI with Lanczos converges quadratically provided that ξk ≤ ξ with ξ a constant that can be bigger than one considerably, that is, the linear systems are solved with no accuracy in the sense of solving the linear systems. The results are fundamentally different from the existing quadratic convergence results and have a strong impact on effective implementations of the method. Based on the new theory, we design practical criteria to control inner tolerance to achieve quadratic convergence and implement the method much more effectively than ever before, so that much computational cost is saved. Numerical experiments support our theory and show its practical value.