Sensitivity analysis for the generalized Cholesky factorization

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.