This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algori...This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.展开更多
Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In th...Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.展开更多
基金Acknowledgements The main results of the paper have been reported at Anhui Normal University, Jiangsu Normal University, the International Workshop on SDEs and Numerical Methods at Shanghai Normal University, Workshop on Markov Processes and Their Applications at Hunan University of Arts and Science, and Workshop of Probability Theory with Applications at University of Macao. The author acknowledges Professors Dong-Jin Zhu, Wan-Ding Ding, Ying-Chao Xie, Xue-Rong Mao, Xiang-Qun Yang, Xu-Yan Xiang, Jie Xiong, Li-Hu Xu, and their teams for very warm hospitality and financial support. The author also thanks Ms. Yue-Shuang Li for her assistance in computing large matrices. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003）, the ＂985＂ project from the Ministry of Education in China, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.
基金supported in part by NSF grants DMS-0611548 and OCI-0749217 and DOE grant DE-FC02-06ER25794supported in part by NSF of China under the contract number 10871049 and Shanghai Down project 200601.
文摘Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.
