Perturbation Analysis of the Cholesky Downdating and QR Updating Problems

Given upper triangular matrices $R, G$ and column vectors $x, f$ such that $R^{T}R-xx^T$ and $(R+G)^{T}(R+G)-(x+f)(x+f)^T$ are positive definite, let $U$ and $U+T$ be the corresponding Cholesky factors. In this paper, upper bounds on $\|T\|$ in terms of $\|G\|$ and $\|f\|$ and upper bounds on $\|T\|/\|U\|$ in terms of $\|G\|/\|R\|$ and $\|f\|/\|x\|$ are given, and the first order perturbation expansions of $\|T\|$ and $\|T\|/\|U\|$ are derived. Moreover, a perturbation analysis of the QR updating problem is also given.