Stewart, G. W.In a recent paper, Chang and Paige have shown that the usual perturbation bounds for Cholesky factors can systematically overestimate the errors. In this note we sharpen their results and extend them to the factors of the LU decomposition. The results are based on a new formula for the first order terms of the error in the factors. (Also cross-referenced as UMIACS-TR-95-93)en-USTechnical Report