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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]

The explicit expressions of normwise, mixed and componentwise condition numbers for the K-weighted pseudoinverse L K † are first presented. With the intermediate result, i.e. the derivative of L K † , we can recover the explicit expressions of condition numbers for solution of least squares problem with equality constraint. Then, we investigate the statistical estimation of condition numbers of L K † using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, numerical examples are provided to illustrate these results. [ABSTRACT FROM AUTHOR]

In this paper, we give the explicit expressions for the strong and the weak rigorous multiplicative perturbation bounds for the block Cholesky downdating problem. First, we derived the strong rigorous multiplicative perturbation bounds by combining the modified matrix-vector equation approach with the strategy for Lyapunov majorant function and the Banach fixed point theorem. Then, the weak rigorous multiplicative bounds are obtained with the help of the matrix-equation approach. The obtained results are illustrated by numerical tests. [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]