# Perturbation and error analyses for block downdating of a Cholesky decomposition

## Abstract

A new perturbation result is presented for the problem of block downdating a Cholesky decomposition*X*^{ T }*X = R*^{ T }*R*. Then, a condition number for block downdating is proposed and compared to other downdating condition numbers presented in literature recently. This new condition number is shown to give a tighter bound in many cases. Using the perturbation theory, an error analysis is presented for the block downdating algorithms based on the LINPACK downdating algorithm and stabilized hyperbolic transformations. An error analysis is also given for block downdating using Corrected Seminormal Equations (CSNE), and it is shown that for ill-conditioned downdates this method gives more accurate results than the algorithms based on the LINPACK downdating algorithm or hyperbolic transformations. We classify the problems for which the CSNE downdating method produces a downdated upper triangular matrix which is comparable in accuracy to the upper triangular factor obtained from the QR decomposition by Householder transformations on the data matrix with the row block deleted.

## Key words

Block downdating Cholesky decomposition condition number error analysis perturbation theory seminormal equations## Preview

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