A Revised Modified Cholesky Factorization Algorithm

A modified Cholesky factorization algorithm introduced originally by Gill and Murray and refined by Gill, Murray, and Wright is used extensively in optimization algorithms. Since its introduction in 1990, a different modified Cholesky factorization of Schnabel and Eskow has also gained widespread usage. Compared with the Gill--Murray--Wright algorithm, the Schnabel--Eskow algorithm has a smaller a priori bound on the perturbation, added to ensure positive definiteness, and some computational advantages, especially for large problems. Users of the Schnabel--Eskow algorithm, however, have reported cases from two different contexts where it makes a far larger modification to the original matrix than is necessary and than is made by the Gill--Murray--Wright method. This paper reports on a simple modification to the Schnabel--Eskow algorithm that appears to correct all the known computational difficulties with the method, without harming its theoretical properties or its computational behavior in any other cases. In new computational tests, the modifications to the original matrix made by the new algorithm appear virtually always to be smaller than those made by the Gill--Murray--Wright algorithm, sometimes by significant amounts. The perturbed matrix is allowed to be more ill-conditioned with the new algorithm, but this seems to be appropriate in the known contexts where the underlying problem is ill-conditioned.