Perturbation analysis for block downdating of the generalized Cholesky factorization

The generalized Cholesky factorization is a generalization of the classical Cholesky factorization and its block downdating problem means finding the downdated generalized Cholesky factorization when a matrix XXT is subtracted from the original matrix, where X is full column rank. In this paper, we consider the perturbation analysis of this problem. Some first order perturbation bounds are first obtained using the refined matrix equation approach and the matrix-vector equation approach. These results generalize the corresponding ones for the block downdating problem of the classical Cholesky factorization. Then, the rigorous perturbation bounds are also obtained using the combination of the classical and refined matrix equation approaches. Each of these bounds is composed of a small constant multiple of the first order term of the corresponding first order perturbation bound and an additional second order term.