• 论文 •

### 一类特征值反问题(IEP)的基于矩阵方程的Ulm型算法

1. 云南大学数学与统计学院, 昆明 650091
• 收稿日期:2020-01-18 出版日期:2021-11-14 发布日期:2021-11-12
• 基金资助:
国家自然科学基金（11861077）资助.

Wang Yihong, Li Yaotang. A ULM-TYPE ALGORITHM BASED ON MATRIX EQUATION FOR A CLASS OF INVERSE EIGENVALUE PROBLEMS (IEP)[J]. Mathematica Numerica Sinica, 2021, 43(4): 444-456.

### A ULM-TYPE ALGORITHM BASED ON MATRIX EQUATION FOR A CLASS OF INVERSE EIGENVALUE PROBLEMS (IEP)

Wang Yihong, Li Yaotang

1. School of Mathematics and Statistics, Yunnan University, Kunming 650091, China
• Received:2020-01-18 Online:2021-11-14 Published:2021-11-12

A new algorithm for solving a class of inverse eigenvalue problems of matrices is constructed by using the Ulm method for solving operator equations. The algorithm avoids the shortcomings of solving a system of linear equations in each iteration of the algorithm in[Aishima K., A quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems, Linear Algebra and its Applications, 2018, 542:310-333], and it is proved that under the condition that the given spectrum data are different from each other, the algorithm has the quadratic convergence in the sense of root convergence. Numerical experiments show that the algorithm in this paper is better than the algorithm above when the matrix order is large.

MR(2010)主题分类:

()
 [1] Brussard P J, Glaudemants P W. Shell Model Applications in Nuclear Spectroscopy[M]. New York:Elsevier, 1977.[2] Hald O. On Discrete and numerical Sturm-Liouville Problems[D]. Ph. D dissertation. New York:Dept. Mathematics, New York University, 1970.[3] Gladwell G M L. inverse problems in vibration[J]. Appl. Mech. Rev, 1986, 39:1013-1018.[4] Parker R L, Whaler K A. Numerical methods for establishing solutions to the inverse problem of electromagnetic induction[J]. J.Geophys.Res, 1981, 86:9574-9584.[5] Joseph K T. Inverse eigenvalue problem in structural design[J]. AIAA J, 1992, 30:2890-2896.[6] Ravi M S, Rosenthal J, Wang X A. On decentralized dynamic pole placement and feedback stabilization[J]. IEEE Trans. Automat.Control, 1995, 40:1603-1614.[7] Trench W F. Numerical solution of the inverse eigenvalue problem for real symmetric Toeplitz matrices[J]. SIAM J. Sci. Comput. 1997, 18:1722-1736.[8] Aishima K. A quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems[J]. Linear Algebra Appl, 2018, 542:310-333.[9] Friedland S, Nocedal J, Overton M L. The formulation and analysis of numerical methods for inverse eigenvalue problems[J]. SIAM J. Numer. Anal, 1987, 24:634-667.[10] Chan R H, Chung H L, Xu S F. The inexact Newton-like method for inverse eigenvalue problem[J]. BIT, 2003, 43:7-20.[11] Bai Z J, Chan R H, Morini B. An inexact Cayley transform method for inverse eigenvalue problem[J]. Inverse Probl, 2004, 20:1675-1689.[12] Ezquerro J A, Hernández M A. The Ulm method under mild differentiability conditions[J]. Numer. Math, 2008, 109:193-207.[13] Ulm S. On iterative method with successive approximation of the inverse oprator[J]. Izv. Akad. Nauk. Est. SSR, 1967, 16:403-411.[14] Gutiérrez J M, Hernandez M A, Romero N. A note on a modification of Moser s method[J]. Journal of Complexity, 2008, 24:185-197.[15] Shen W P, Li C, Jin X Q. An Ulm-like method for inverse eigenvalue problems[J]. Appl. Numer. Math, 2011, 61:356-357.[16] Shen W P, Li C. An Ulm-like Cayley transform method for inverse eigenvalue problems[J]. Taiwan.J. Math, 2012, 16:367-386.[17] 徐成贤,陈志平,李乃成.近代优化方法[M].北京:科学出版社,2002.[18] Golub G H, Van Loan C F, Matrix Computations. 3th ed[M]. The Johns Hopkins University Press, Baltimore, 1996.[19] Chan R H, Xu S F, Zhou H. On the convergence rate of a quasi-Newton method for inverse eigenvalue problems[J]. SIAM J. Numer. Anal., 1999, 36(2):436-441.
 [1] 王华, 乌力吉. 垂直线性互补问题的一种光滑算法[J]. 计算数学, 2009, 31(1): 1-14.