Modified Gram–Schmidt-based methods for block downdating the Cholesky factorization

The hyperbolic modified Gram–Schmidt (HMGS) method is proposed for block downdating the Cholesky factorization. The method might be unsatisfactory due to rounding errors. A modified version based on the MGS process is presented and is shown to be mixed stable. Numerical tests show that the new method has the same numerical properties as the generalized LINPACK-type algorithm, and can work better than the Householder-based algorithm given by Bojanczyk and Steinhardt (1991) [9].