Perturbation analysis for the periodic generalized coupled Sylvester equation

ABSTRACT

In this paper, we consider the perturbation analysis for the periodic generalized coupled Sylvester (PGCS) equation. The normwise backward error for this equation is first investigated. Then, we present its normwise and componentwise perturbation bounds using the fixed-point theorem, from which the normwise and effective condition numbers are derived. Moreover, the mixed and componentwise condition numbers for the PGCS equation are also given. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the statistical condition estimation method are applied and two algorithms are devised. The obtained results are illustrated by numerical experiments.

KEYWORDS:

• Periodic generalized coupled Sylvester equation
• backward error
• perturbation bound
• fixed-point theorem
• condition number
• probabilistic spectral norm estimator
• SCE method

2010 AMS SUBJECT CLASSIFICATIONS:

• 65F35
• 15A23
• 15A45

Acknowledgments

The authors would like to thank the editor and the two anonymous referees for their helpful comments for improving the manuscript. They also would like to acknowledge Prof. Michiel E. Hochstenbach for providing the Matlab program of probabilistic spectral norm estimator. Part of this work was finished when the first author was a visiting scholar at the Department of Mathematics and Statistics of Auburn University from August 2014 to August 2015.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by the National Natural Science Foundation of China [Grant No. 11671060] and the Fundamental Research Funds for the Central Universities [Grant No. 106112015CDJXY100003].