For the Hermitian inexact Rayleigh quotient iteration(RQI),we consider the local convergence of the in exact RQI with the Lanczos method for the linear systems involved.Some attractive properties are derived for the r...For the Hermitian inexact Rayleigh quotient iteration(RQI),we consider the local convergence of the in exact RQI with the Lanczos method for the linear systems involved.Some attractive properties are derived for the residual,whose norm is ξk,of the linear system obtained by the Lanczos method at outer iteration k+1.Based on them,we make a refned analysis and establish new local convergence results.It is proved that(i) the inexact RQI with Lanczos converges quadratically provided that ξk≤ξ with a constant ξ1 and (ii) the method converges linearly provided that ξk is bounded by some multiple of1/||rk|| with rkthe residual norm of the approximate eigenpair at outer iteration k.The results are fundamentally diferent from the existing ones that always require ξk<1,and they have implications on efective implementations of the method.Based on the new theory,we can design practical criteria to control ξkto achieve quadratic convergence and implement the method more efectively than ever before.Numerical experiments confrm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.展开更多
基金supported by National Basic Research Program of China(Grant No.2011CB302400)National Natural Science Foundation of China(Grant No.11071140)
文摘For the Hermitian inexact Rayleigh quotient iteration(RQI),we consider the local convergence of the in exact RQI with the Lanczos method for the linear systems involved.Some attractive properties are derived for the residual,whose norm is ξk,of the linear system obtained by the Lanczos method at outer iteration k+1.Based on them,we make a refned analysis and establish new local convergence results.It is proved that(i) the inexact RQI with Lanczos converges quadratically provided that ξk≤ξ with a constant ξ1 and (ii) the method converges linearly provided that ξk is bounded by some multiple of1/||rk|| with rkthe residual norm of the approximate eigenpair at outer iteration k.The results are fundamentally diferent from the existing ones that always require ξk<1,and they have implications on efective implementations of the method.Based on the new theory,we can design practical criteria to control ξkto achieve quadratic convergence and implement the method more efectively than ever before.Numerical experiments confrm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.
