最近亏损矩阵的可信验证
The Verification of the Nearest Defective Matrix
DOI: 10.12677/AAM.2020.911228, PDF,    科研立项经费支持
作者: 张春磊, 李 喆*：长春理工大学理学院，吉林 长春
关键词: 特征值；亏损矩阵；验证算法；Eigenvalue； Defective Matrix； Verification
摘要: 给定具有n个互异特征值的阶矩阵Ã，以及其近似特征值λ，本文提出的验证算法可计算给定矩阵Ã的微小摄动区间矩阵A，以及给定实数λ的微小摄动区间∧Λ，使得在区间矩阵A中存在一个实亏损矩阵。
Abstract: Given an  matrix Ã having n distinct eigenvalues and its approximate eigenvalue λ, we propose a verification algorithm for constructing an interval matrix A near to Ã and an interval ∧Λ near to λ. The computed interval matrix A is guaranteed to contain a real defective matrix .
文章引用：张春磊, 李喆. 最近亏损矩阵的可信验证[J]. 应用数学进展, 2020, 9(11): 1974-1984. https://doi.org/10.12677/AAM.2020.911228

参考文献

[1] Kublanovskaya, V.N. (1966) On a Method of Solving the Complete Eigenvalue Problem for a Degenerate Matrix. USSR Computational Mathematics and Mathematical Physics, 6, 1-14. [Google Scholar] [CrossRef] 
[2] Ruhe, A. (1970) An Algorithm for Numerical Determination of the Structure of a General Matrix. BIT Numerical Mathematics, 10, 196-216. [Google Scholar] [CrossRef] 
[3] Sridhar, B. and Jordan, D. (1973) An Algorithm for Calculation of the Jordan Canonical form of a Matrix. Computers & Electrical Engineering, 1, 239-254. [Google Scholar] [CrossRef] 
[4] Kågström, B. and Ruhe, A. (1980) Algorithm 560: JNF an Algorithm for Numerical Computation of the Jordan Normal Form of a Complex Matrix. ACM Transactions on Mathematical Software, 6, 437-443. [Google Scholar] [CrossRef] 
[5] Demmel, J.W. (1986) Computing Stable Eigendecompositions of Matrices. Linear Algebra and Its Applications, 79, 163-193. [Google Scholar] [CrossRef] 
[6] Chatelin, F. (1986) Ill Conditioned Eigenproblems. North-Holland Mathematics Studies, 127, 267-282. [Google Scholar] [CrossRef] 
[7] Wilkinson, J.H. (1965) The Algebraic Eigenvalue Problem. Clarendon Press, Oxford.
[8] Wilkinson, J.H. (1972) Notes on Matrices with a Very Ill-conditioned Eigenproblem. Numerische Mathematik, 19, 175-178. [Google Scholar] [CrossRef] 
[9] Wilkinson, J.H. (1984) Sensitivity of Eigenvalues. Utilitas Mathematica, 25, 5-76.
[10] Wilkinson, J.H. (1984) On Neighbouring Matrices with Quadratic Elementary Divisors. Numerische Mathematik, 44, 1-21. [Google Scholar] [CrossRef] 
[11] Wilkinson, J.H. (1986) Sensitivity of Eigenvalues, II. Utilitas Mathematica, 30, 243-286.
[12] Malyshev, A.N. (1999) A Formula for the 2-Norm Distance from a Matrix to the Set of Matrices with Multiple Eigenvalues. Numerische Mathematik, 83, 443-454. [Google Scholar] [CrossRef] 
[13] Lippert, R. A. and Edelman, A. (1999) The Computation and Sensitivity of Double Eigenvalues. In: Chen, Z., Li, Y., Micchelli, C.A. and Xu, Y., Eds., Advances in Computational Mathematics: Proceedings of the Guangzhou International Symposium, Dekker, New York, 353-393.
[14] Alam, R. and Bora, S. (2005) On Sensitivity of Eigenvalues and Eigendecompositions of Matrices. Linear Algebra and Its Applications, 396, 273-301. [Google Scholar] [CrossRef] 
[15] Alam, R., Bora, S., Byers, R. and Michael, L.O. (2011) Characterization and Construction of the Nearest Defective Matrix via Coalescence of Pseudospectral Components. Linear Algebra and Its Applications, 435, 494-513. [Google Scholar] [CrossRef] 
[16] Akinola, R.O., Freitag, M.A. and Spence, A. (2014) The Calculation of the Distance to a Nearby Defective Matrix. Numerical Linear Algebra with Applications, 21, 403-414. [Google Scholar] [CrossRef] 
[17] Rump, S.M. (1999) INTLAB—Interval Laboratory. Tibor Csendes. Developments in Reliable Computing. Kluwer Academic Publishers, Dordrecht, 77-104. [Google Scholar] [CrossRef] 
[18] Golub, G.H. and Charles, F. (1996) Matrix Computations. Johns Hopkins University Press, Baltimore.
[19] Rump, S.M. (2010) Verification Methods: Rigorous Results Using Floating-Point Arithmetic. Acta Numerica, 19, 287-449. [Google Scholar] [CrossRef] 

为你推荐