The effect of rounding error on an algorithm for downdating a Cholesky factorization

Let the positive definite matrix A have a Cholesky factorization A = RTR. For a given vector x suppose that A = A — xxT has a Cholesky factorization A = R~TR. 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. 1