The Effects of Rounding Error on an Algorithm for Downdating a Cholesky Factorization

Let the positive definite matrix A have a Cholesky factorizationA = R^TR. For a given vector xsuppose that à =A - xx^T has a Cholesky factorization à = $R˜$^T$R˜$.This paper considers an algorithm for computing $R˜$from R and x and an extension for removing a row from the QR factorization of a regression problem. It is shown that the algorithm is stable in the presence of rounding errors. However, it is also shown that the matrix $R˜$can be a very ill-conditioned function of R and x.