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

TL;DR

This paper analyzes the convergence of the inexact Rayleigh quotient iteration when combined with the Lanczos method for solving linear systems, establishing new conditions for quadratic and linear convergence and guiding practical implementation.

Contribution

The paper provides new convergence results for inexact RQI with Lanczos, including conditions for quadratic and linear convergence, and extends the theory to preconditioned Lanczos, improving practical algorithms.

Findings

Quadratic convergence when residuals satisfy $\xi_{k+1}\leq \xi$ with $\xi extgreater 1$

Linear convergence when residuals are bounded by a small multiple of the inverse residual norm

Practical criteria for controlling residuals to achieve desired convergence rates

Abstract

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.