Sensitivity analysis for the block Cholesky downdating problem

ABSTRACT

Some improved rigorous perturbation bounds with normwise perturbation for the block Cholesky downdating problem are first derived by combining the modified matrix-vector equation approach with the strategy for Lyapunov majorant function and the Banach fixed point theorem. Then, we investigate four distinct kinds of condition numbers, i.e. two normwise ones, and mixed and componentwise ones, for this problem, and present their explicit expressions. Furthermore, using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, we also consider the statistical estimation of these condition numbers and design two algorithms. The obtained results are illustrated by numerical examples.

KEYWORDS:

• Block Cholesky downdating problem
• rigorous perturbation bound
• normwise condition number
• mixed and componentwise condition numbers
• probabilistic spectral norm estimator
• SSCE method
• Lyapunov majorant function
• banach fixed point theorem

AMS CLASSIFICATION (2010):

• 15A23
• 15A45
• 65F35

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.

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 Nos. 11671060, 11571062, and 11771062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and the Fundamental Research Funds for the Central Universities (Grant Nos. 06112016CDJXZ238826).