Sensitivity analysis for the generalized Cholesky block downdating problem

Abstract

In this article, first we will present the new rigorous perturbation bounds with normwise perturbation for the generalized Cholesky block downdating problem by combining the unified matrix–vector equation approach with the method of Lyapunov majorant functions and the Banach fixed point theorem. Then, we will derive the explicit expressions for the six distinct kinds of condition numbers, i.e. four normwise ones, mixed and componentwise ones. Furthermore, with the help of probabilistic spectral norm estimator and the small-sample statistical condition estimation method, these condition numbers can be estimated with high accuracy. At the end, the obtained results are illuminated by the numerical examples.

Keywords:

• Generalized Cholesky block downdating problem
• rigorous perturbation bound
• condition numbers
• probabilistic spectral norm estimator
• SSCE method
• Lyapunov majorant function
• banach fixed point theorem

AMS classification (2010):

• 15A23
• 15A45
• 65F35

Acknowledgements

The authors would like to thanks the editor and the two anonymous referees for their helpful comments for improving the manuscript. They would also like to acknowledge Prof. Michiel E. Hochstenbach and Prof. Hanyu Li for providing the MATLAB program of probabilistic spectral norm estimator.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the National Natural Science Foundation of China (Grant Numbers 11671060, 11571062, and 11771062).