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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The explicit expressions of the normwise, mixed, and componentwise condition numbers and their upper bounds for the generalized Cholesky factorization are first obtained. Then, some improved rigorous perturbation bounds with normwise or componentwise perturbation in the given matrix are derived by bringing together the modified matrix-vector equation approach with the method of Lyapunov majorant function and the Banach fixed point theorem. Theoretical and experimental results show that these new bounds are always tighter than the corresponding ones in the literature. [ABSTRACT FROM AUTHOR]