Abstract
Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.
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References
Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. London: Springer, 2005
Chen M F. Speed of stability for birth-death processes. Front Math China, 2010, 5(3): 379–515
Chen M F. Criteria for discrete spectrum of 1D operators. Commun Math Stat, 2014, 2: 279–309
Chen M F. Efficient initials for computing the maximal eigenpair. Front Math China, 2016, 11(6): 1379–1418 See also Vol 4 in the middle of author’s homepage: http://math0.bnu.edu.cn/~chenmf A package based on the paper is available on CRAN now (by X. J. Mao). One may check it through the link: https://github.com/mxjki/PowerfulMaxEigenpair A Matlab package is also available, see the author’s homepage above The authors’ papers cited in this article can be found from Vols 1–4 in the middle of the homepage above
Chen M F. The charming leading eigenpair. Adv Math (China), 2017, 46(4): 281–297
Chen M F. Global algorithms for maximal eigenpair. Front Math China, 2017, 12(5): 1023–1043
Chen M F. Trilogy on computing maximal eigenpair. In: Yue W, Li Q L, Jin S, Ma Z, eds. Queueing Theory and Network Applications (QTNA 2017). Lecture Notes in Comput Sci, Vol 10591. Cham: Springer, 2017, 312–329
Chen M F. Hermitizable, isospectral complex matrices or differential operators. Front Math China, 2018, 13(6): 1267–1311
Chen M F, Zhang X. Isospectral operators. Commun Math Stat, 2014, 2: 17–32
Cipra B A. The best of the 20th century: Editors name top 10 algorithms. SIAM News, 2000, 33(4): 1–2
Frolov A V, Voevodin V V, Teplov A. Thomas algorithm, pointwise version. https://algowiki-project.org/en/Thomas_algorithm,_pointwise_version
From Wikipedia. Tridiagonal matrix algorithm. https://en.wikipedia.org/wiki/Tri-diagonal_matrix_algorithm
Golub G H, van der Vorst H A. Eigenvalue computation in the 20th century. J Comput Appl Math, 2000, 123(1–2): 35–65
Householder A S. Unitary triangularization of a nonsymmetric matrix. J Assoc Comput Mach, 1958, 5: 339–342
Moler C. Llewellyn Thomas. https://en.wikipedia.org/wiki/Llewellyn+Thomas 1996
Shukuzawa O, Suzuki T, Yokota I. Real tridiagonalization of Hermitian matrices by modified Householder transformation. Proc Japan Acad Ser A, 1996, 72: 102–103
Stewart G W. The decompositional approach to matrix computation. IEEE Comput Sci Eng, 2000, 2(1): 50–59
Tang T, Yang J. Computing the maximal eigenpairs of large size tridiagonal matrices with O(1) number of iterations. Numer Math Theory Methods Appl, 2018, 11(4): 877–894
van der Vorst H A, Golub G H. 150 Years old and still alive: eigenproblems. In: Duff I S, Watson G A, eds. The State of the Art in Numerical Analysis. Oxford: Oxford Univ Press, 1997, 93–119
Acknowledgements
The authors thank MS Zhou-Jing Wang for providing a program in MatLab on the Householder transformation for Hermitian matrix. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11771046), the Project from the Ministry of Education in China, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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Chen, MF., Li, YS. Development of powerful algorithm for maximal eigenpair. Front. Math. China 14, 493–519 (2019). https://doi.org/10.1007/s11464-019-0769-5
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DOI: https://doi.org/10.1007/s11464-019-0769-5