交替投影法求解对称随机逆特征值问题
Alternating Method for Symmetric Stochastic Inverse Eigenvalue Problems
摘要: 本文主要研究对称随机矩阵的逆特征值问题。通过将该问题转化为求两个集合交点的可行性问题，提出用交替投影法进行求解。因为其中一个集合不是凸集，关于凸可行性问题的收敛性结果不能用来分析算法的收敛性。对于算法的收敛性，本文在已有关于两个黎曼流形的交替投影算法收敛性的研究结果上，建立了交替投影算法在一定条件下的线性收敛性。最后数值例子也表明了算法的有效性。
Abstract: The symmetric stochastic inverse eigenvalue problem (StIEP) is considered. By reformulating it into a feasibility problem of two sets, the alternating method is proposed for solving it. Since one of the sets is not convex, the convergence analysis technique for the convex cases does not work. For the convergence of the algorithm, this paper establishes the linear convergence of the alternating projection algorithm under certain conditions on the existing results on the convergence of the alternating projection algorithms for two Riemannian manifolds. At last, some numerical experiments are provided to illustrate the efficiency of the method.
文章引用：党婵娟, 王湘美. 交替投影法求解对称随机逆特征值问题[J]. 应用数学进展, 2021, 10(6): 2206-2216. https://doi.org/10.12677/AAM.2021.106230

参考文献

[1] Kolmogorov, A.N. (1937) Markov Chains with Countably Many Possible States. Bull. University. Moscow, 3, 1-16.
[2] Suleimanova, K.R. (1949) Stochastic Matrices with Real Eigenvalues. Soviet Mathematics—Doklady, 66, 343-345.
[3] Loewy, R. and London, D.A. (1978) Note on an Inverse Problem for Nonnegative Matrices. Linear and Multilinear Algebra, 6, 83-90. [Google Scholar] [CrossRef] 
[4] Johnson, C. (1981) Row Stochastic Matrices Similar to Doubly Stochastic Matrices. Linear and Multilinear Algebra, 66, 113-130. [Google Scholar] [CrossRef] 
[5] Laffey, T.J. and Meehan, E. (1999) A Characterization of Trace Zero Nonnegative 5×5 Matrices. Linear Algebra and Its Applications, 302, 295-302. [Google Scholar] [CrossRef] 
[6] Chu, M.T. and Golub, G.H. (2002) Structured Inverse Eigenvalue Problems. Acta Numerica, 11, 1-71. [Google Scholar] [CrossRef] 
[7] Barcilon, V. (1990) A Two-Dimensional Inverse Eigenvalue Problem. Journal of Inverse and Ill-Posed Problems, 6, 11-20. [Google Scholar] [CrossRef] 
[8] Egleston, P.D., Lenker, T.D. and Narayan, S.K. (2004) The Nonnegative Inverse Eigenvalue Problem. Linear Algebraandits Applications, 379, 475-490. [Google Scholar] [CrossRef] 
[9] Soules, G.W. (2008) Constructing Symmetric Nonnegative Matrices. Linear and Multilinear Algebra, 13, 241-251. [Google Scholar] [CrossRef] 
[10] Robert, R. (1996) An Inequality for Nonnegative Matrices and the Inverse Eigenvalue Problem. Linear and Multilinear Algebra, 41, 367-375. [Google Scholar] [CrossRef] 
[11] Lin, M.M. (2015) An Algorithm for Constructing Nonnegative Matrices with Prescribed Real Eigenvalues. Applied Mathematicsand Computation, 31, 582-590. [Google Scholar] [CrossRef] 
[12] Combettes, P.L. and Trussell, H.J. (1990) Method of Successive Projections for Finding a Common Point of Sets in Metric Spaces. Journal of Optimization Theory and Applications, 67, 487-507. [Google Scholar] [CrossRef] 
[13] Morel, P. (1976) Des algorithmes pour le problème inverse des valeurs propres. Linear Algebra and Its Applications, 13, 251-273. [Google Scholar] [CrossRef] 
[14] Chu, M.T. and Chen, X.Z. (1996) On the Least Squares Solution of Inverse Eigenvalue Problems. Siam Journalon Numerical Analysis, 33, 2417-2430. [Google Scholar] [CrossRef] 
[15] Perfect, H. (1955) Methods of Constructing Certain Stochastic Matrices. Duke Mathematical Journal, 22, 305-311. [Google Scholar] [CrossRef] 
[16] Chu, M.T. (1998) Inverse Eigenvalue Problems. Siam Review, 40, 1-39. [Google Scholar] [CrossRef] 
[17] Xu, M. (2010) On Computing Minimal Realizable Spectral Radii of Non-Negative Matrices. Numerical Linear Algebra with Applications, 12, 77-86. [Google Scholar] [CrossRef] 
[18] Chu, M.T. and Driessel, K.R. (1991) Constructing Symmetric Nonnegative Matrices with Prescribed Eigenvalues by Differential Equations. SIAM Journal on Numerical Analysis, 22, 1372-1387. [Google Scholar] [CrossRef] 
[19] Joseph, K.T. (2012) Inverse Eigenvalue Problem in Structural Design. AIAA Journal, 30, 2890-2896. [Google Scholar] [CrossRef] 
[20] Dawson, C.B. and Cha, P.D. (2019) A Sensitivity-Based Approach to Solving the Inverse Eigenvalue Problem for Linear Structures Carrying Lumped Attachments. Materials Scienceand Engineering R-Reports, 120, 537-566. [Google Scholar] [CrossRef] 
[21] Chu, M.T. and Guo, Q.Y. (1998) A Numerical Method for the Inverse Stochastic Spectrum Problem. SIAM Journal on Matrix Analysis and Applications, 19, 1027-1039. [Google Scholar] [CrossRef] 
[22] Orsi, R. (2006) Numerical Methods for Solving Inverse Eigenvalue Problems for Nonnegative Matrices. SIAM Journal on Matrix Analysis and Applications, 28, 190-212. [Google Scholar] [CrossRef] 
[23] Wu, S.J. and Lin, M.M. (2014) Numerical Methods for Solving Nonnegative Inverse Singular Value Problems with Prescribed Structure. Inverse Problems, 30, 55-69. [Google Scholar] [CrossRef] 
[24] Bai, Z.J., Serra-Capizzano, S. and Zhi, Z. (2012) Nonnegative Inverse Eigenvalue Problems with Partial Eigendata. Bit Numerical Mathematics, 120, 387-431. [Google Scholar] [CrossRef] 
[25] Yao, T.T., Bai, Z.J. and Zhao, Z. (2018) A Riemannian Variant of the Fletcher-Reeves Conjugate Gradient Method for Stochastic Inverse Eigenvalue Problems with Partial Eigendata. Numerical Linear AlgebraWithApplications, 26, 1-19. [Google Scholar] [CrossRef] 
[26] Lewis, A.S. (2008) Alternating Projections on Manifolds. Informs Journal on Computing, 33, 216-234. [Google Scholar] [CrossRef] 
[27] 陈维桓, 李兴校. 黎曼几何引论[M]. 北京: 北京大学出版社, 2002.
[28] Carmo, M.P. (1992) Riemannian Geometry. Birkhauser, Boston. [Google Scholar] [CrossRef] 
[29] Sakai, T. (1996) Riemannian Geometry. American Mathematical Society, Providence. [Google Scholar] [CrossRef] 
[30] Bauschke, H.H. and Borwein, J.M. (1993) On the Convergence of von Neumann’s Alternating Projection Algorithm for Two Sets. Set-Valued Analysis, 1, 185-212. [Google Scholar] [CrossRef] 

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