Frontiers of Mathematics in China >

2021 , Vol. 16 >Issue 2: 345 - 379

DOI: https://doi.org/10.1007/s11464-021-0909-6

RESEARCH ARTICLE

Computing top eigenpairs of Hermitizable matrix

  • Mu-Fa CHEN , 1,2,3 ,
  • Zhi-Gang JIA 1,4 ,
  • Hong-Kui PANG 1,4
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  • 1. Research Institute of Mathematical Science, Jiangsu Normal University, Xuzhou 221116, China
  • 2. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • 3. Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China
  • 4. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received date: 26 Jul 2020

Accepted date: 04 Dec 2020

Published date: 15 Apr 2021

Copyright

2021 Higher Education Press
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Abstract

The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which only little is known up to now. This is different from some mature algorithms, that are clearly limited only to medium-sized matrix for calculating full spectrum. It is hoped that a combination of this paper with the earlier works, to be seen soon, may provide some effective algorithms for computing the spectrum in practice, especially for matrix mechanics.

Key words： Hermitizable; Householder transformation; birth-death matrix; isospectral matrices; top eigenpairs; algorithm

Cite this article

Mu-Fa CHEN , Zhi-Gang JIA , Hong-Kui PANG . Computing top eigenpairs of Hermitizable matrix[J]. Frontiers of Mathematics in China, 2021 , 16(2) : 345 -379 . DOI: 10.1007/s11464-021-0909-6

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Cao Z H. Matrix Eigenvalue Problem. Shanghai: Shanghai Scientific & Technical Publishers, 1983 (in Chinese)

2
Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. London: Springer, 2005

3
Chen M F. Efficient initials for computing maximal eigenpair. Front Math China, 2016, 11(6): 1379–1418

DOI

4
Chen M F. Global algorithms for maximal eigenpair. Front Math China, 2017, 12(5): 1023–1043

DOI

5
Chen M F. Hermitizable, isospectral complex matrices or differential operators. Front Math China, 2018, 13(6): 1267–1311

DOI

6
Chen M F. On spectrum of Hermitizable tridiagonal matrices. Front Math China, 2020, 15(2): 285–303

DOI

7
Chen M F, Li Y S. Development of powerful algorithm for maximal eigenpair. Front Math China, 2019, 14(3): 493–519

DOI

8
Chung K L, Yan W M. The complex Householder transform. IEEE Trans Signal Process, 1997, 45(9): 2374–2376

DOI

9
Golub G H, Van Loan C F. Matrix Computations. 4th ed. Baltimore: The Johns Hopkins Univ Press, 2013

10
Householder A S. Unitary triangularization of a nonsymmetric matrix. J Assoc Comput Mach, 1958, 5: 339–342

DOI

11
Jiang E X. Symmetric Matrix Computation. Shanghai: Shanghai Scientific & Technical Publishers, 1984 (in Chinese)

12
Min C. A new understanding of the QR method. J Korean Soc Ind Appl Math, 2010, 14(1): 29–34

13
Niño A, Muñoz-Caro C, Reyes S. A concurrent object-oriented approach to the eigen- problem treatment in shared memory multicore environments. Lecture Notes in Computer Sci, Vol 6782. Cham: Springer, 2011, 630–642

DOI

14
Parlett B N. The Symmetric Eigenvalue Problem. Philadelphia: SIAM, 1998

DOI

15
Press W H, Teukolsky S A, Vetterling W T, Flannery B P. Numerical Recipes. The Art of Scientific Computing. 3rd ed. Cambridge: Cambridge Univ Press, 2007

16
Shukuzawa O, Suzuki T, Yokota I. Real tridiagonalization of Hermitian matrices by modified Householder transformation. Proc Japan Acad Ser A, 1996, 72(5): 102–103

DOI

17
Wang Z J. Householder transformation for Hermitizable matrix. Master Thesis. Beijing: Beijing Normal Univ, 2018

18
Wilkinson J H. The Algebraic Eigenvalue Problem. Oxford: Oxford Univ Press, 1965

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