Perturbation Analysis of the Cholesky Downdating and QR Updating Problems

Abstract

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.