Global algorithms for maximal eigenpair

Abstract

This paper is a continuation of our previous work [Front. Math. China, 2016, 11(6): 1379–1418] where an efficient algorithm for computing the maximal eigenpair was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces mainly two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real (with some negative off-diagonal elements) or complex matrices.

Keywords

Maximal eigenpair / shifted inverse iteration / global algorithm

References

[1] ArapostathisA, BorkarV S, KumarK S. Risk-sensitive control and an abstract Collatz–Wielandt formula.J Theoret Probab, 2016, 29(4): 1458–1484

[2] ChangK C. Nonlinear extensions of the Perron–Frobenius theorem and the Krein–Rutman theorem.J Fixed Point Theory Appl, 2014, 15: 433–457

[3] ChenM F. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed.Singapore: World Scientific, 2004

[4] ChenM F. Speed of stability for birth-death processes.Front Math China, 2010, 5(3): 379–515

[5] ChenM F. Efficient initials for computing the maximal eigenpair.Front Math China, 2016, 11(6): 1379–1418; see also volume 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://cran.r-project.org/web/packages/EfficientMaxEigenpair/index.html

[6] ChenM F. Efficient algorithm for principal eigenpair of discrete p-Laplacian.Preprint, 2017

[7] ChenM F, ZhangX. Isospectral operators.Commun Math Stat, 2014, 2: 17–32

[8] ChenR R. An extended class of time-continuous branching processes.J Appl Probab, 1997, 34(1): 14–23

[9] DonskerW D, VaradhanS R S. On a variational formula for the principal eigenvalue for operators with maximum principle.Proc Natl Acad Sci, 1975, 72(3): 780–783

[10] NoutsosD. On Perron–Frobenius property of matrices having some negative entries.Linear Algebra Appl, 2006, 412: 132–153

[11] NoutsosD. Perron Frobenius theory and some extensions.2008, http://www.pdfdrive. net/perron-frobenius-theory-and-some-extensions-e10082439.html

[12] NoutsosD, VargaR S. On the Perron–Frobenius theory for complex matrices.Linear Algebra Appl, 2012, 437: 1071–1088

[13] SheuS J. Stochastic control and principal eigenvalue.Stochastics, 1984, 11(3-4): 191–211

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2017 Higher Education Press and Springer-Verlag GmbH Germany