Efficient initials for computing maximal eigenpair

Abstract

This paper introduces some efficient initials for a well-known algorithm (an inverse iteration) for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.

Keywords

Perron-Frobenius theorem / power iteration / Rayleigh quotient iteration / efficient initial / tridiagonal matrix / Q-matrix

References

[1] Chen M F. Explicit bounds of the first eigenvalue. Sci China Ser A, 2000, 43(10): 1051–1059

[2] Chen M F. Variational formulas and approximation theorems for the first eigenvalue. Sci China Ser A, 2001, 44(4): 409–418

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

[4] Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. London: Springer, 2005

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

[6] Chen M F. Criteria for discrete spectrum of 1D operators. Commun Math Stat, 2014, 2: 279–309

[7] Chen M F. Unified speed estimation of various stabilities. Chinese J Appl Probab Statist, 2016, 32(1): 1–22

[8] Chen M F, Zhang X. Isospectral operators. Commun Math Stat, 2014, 2: 17–32

[9] Chen M F, Zhang Y H. Unified representation of formulas for single birth processes. Front Math China, 2014, 9(4): 761–796

[10] Golub G H, van Loan C F. Matrix Computations. 4th ed. Baltimore: Johns Hopkins Univ Press, 2013

[11] Hua L K. Mathematical theory of global optimization on planned economy, (II) and (III). Kexue Tongbao, 1984, 13: 769–772 (in Chinese)

[12] Langville A N, Meyer C D. Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton: Princeton Univ Press, 2006

[13] Meyer C. Matrix Analysis and Applied Linear Algebra. Philadelphia: SIAM, 2000

[14] von Mises R, Pollaczek-Geiringer H. Praktische Verfahren der Gleichungsaufösung. ZAMM Z Angew Math Mech, 1929, 9: 152–164

[15] Wielandt H. Beiträge zur mathematischen Behandlung komplexer Eigenwertprobleme. Teil V: Bestimmung höherer Eigenwerte durch gebrochene Iteration. Bericht B 44/J/37, Aerodynamische Versuchsanstalt Göttingen, Germany, 1944

[16] Wilkinson J H. The Algebraic Eigenvalue Problem. Oxford: Oxford Univ Press, 1965

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg