On Convergence of the Inexact Rayleigh Quotient Iterationwith MINRES
Speaker: Prof. JIA ZhongxiaoDepartment of Mathematical SciencesTsinghua University 29 Jul 2010 (Thursday) 10:30 - 11:30 J215 Department of Mathematics

Abstract

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 methodconverges at least quadratically under a new condition, called the uniform positiveness condition, that may allow inner tolerance $\xi_k\geq 1$ at outer iteration $k$ and can be considerably weaker than the condition $\xi_k\leq\xi<1$ with $\xi$ a constant not near one commonly used in literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditionedMINRES method 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 more refined analysis and establish a number of new convergence results. Let $\|r_k\|$ be the residual norm of approximating eigenpair at outer iteration $k$. Then all the available cubic and quadratic convergence results require $\xi_k=O(\|r_k\|)$ and $\xi_k\leq\xi$ with a fixed $\xi$ not nearone, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that $\xi_k\leq\xi$ with aconstant $\xi<1$ not near one, $\xi_k=1-O(\|r_k\|)$ and $\xi_k=1-O(\|r_k\|^2)$, respectively. Therefore, 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.

Biography

Zhongxiao Jia
Professor of Mathematics
Department of Mathematical Sciences
Tsinghua University

Services
Editor of Journal of Numerical Methods and Computer Applications;.
Standing member of Chinese Society on Industrial and Applied Mathematics and Chinese Society on Computational Mathematics

Honors and Awards

1. One of six recipients of The Sixth Leslie Fox Prize in Numerical Analysis , 1993 (Best research paper prize in numerical analysis), awarded by the Institute of Mathematics and Its Applications(IMA) once every two years.
2. One of the 49 excellent young (under 50 years old) mathematicians selected by the Section of the Mathematics and Physics of the National Natural Science Foundation of China.
3. Award grant for Excellent Young Scholars of the Ministry of Education, China, 1999-2001.
4. An expert awarded by the State Council of China, 1999.
5. The National Hundred and Thousand Talent Plan Project', 1999, one of the highest awards honoring Chinese scholars under 45.
6. A member of 'The Hundred Talent Plan Project', Tsinghua University, 2001.