The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, ...The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments.展开更多
For the Hermitian inexact Rayleigh quotient iteration （RQI）, we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for t...For the Hermitian inexact Rayleigh quotient iteration （RQI）, we consider the local convergence of the inexact 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 refined analysis and establish new local convergence results. It is proved that （i） the inexact RQI with Lanezos converges quadratically provided that ξk ≤ξ with a constant ξ≥） 1 and （ii） the method converges linearly provided that ξk is bounded by some multiple of 1/‖τk‖ with ‖τk‖ the residual norm of the approximate eigenpair at outer iteration k. The results are fundamentally different from the existing ones that always require ξk 〈 1, and they have implications on effective implementations of the method. Based on the new theory, we can design practical criteria to control ξk to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.展开更多
RQI is an approach for eigenvectors of matrices. In 1974, B.N Parlett proved that it was a ''succeessful algorithm'' with cubic convergent speed for normal matrices. After then, several authors develop...RQI is an approach for eigenvectors of matrices. In 1974, B.N Parlett proved that it was a ''succeessful algorithm'' with cubic convergent speed for normal matrices. After then, several authors developed relevant theory and put this research into dynamical frame. [3] indicated that RQI failed for non-normal matrices with complex eigenvalues. In this paper, RQI for non-normal matrices with only real spectrum is analyzed. The authors proved that eigenvectors are super-attractive fixed points of RQI. The geometrical and topological behaviours of two periodic orbits are considered too. The existness of three or higher periodic orbits and their geometry are considered too. The existness of three or higher periodic orbits and their geometry are still open and of interest. It will be reported in our forthcomming paper.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 10471074, 10771116)the Doctoral Program of the Ministry of Education of China (Grant No. 20060003003)
文摘The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments.
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 inexact 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 refined analysis and establish new local convergence results. It is proved that （i） the inexact RQI with Lanezos converges quadratically provided that ξk ≤ξ with a constant ξ≥） 1 and （ii） the method converges linearly provided that ξk is bounded by some multiple of 1/‖τk‖ with ‖τk‖ the residual norm of the approximate eigenpair at outer iteration k. The results are fundamentally different from the existing ones that always require ξk 〈 1, and they have implications on effective implementations of the method. Based on the new theory, we can design practical criteria to control ξk to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.
文摘RQI is an approach for eigenvectors of matrices. In 1974, B.N Parlett proved that it was a ''succeessful algorithm'' with cubic convergent speed for normal matrices. After then, several authors developed relevant theory and put this research into dynamical frame. [3] indicated that RQI failed for non-normal matrices with complex eigenvalues. In this paper, RQI for non-normal matrices with only real spectrum is analyzed. The authors proved that eigenvectors are super-attractive fixed points of RQI. The geometrical and topological behaviours of two periodic orbits are considered too. The existness of three or higher periodic orbits and their geometry are considered too. The existness of three or higher periodic orbits and their geometry are still open and of interest. It will be reported in our forthcomming paper.
