A modified Cholesky factorization algorithm introduced originally by Gill a
nd Murray and refined by Gill, Murray, and Wright is used extensively in op
timization algorithms. Since its introduction in 1990, a different modified
Cholesky factorization of Schnabel and Eskow has also gained widespread us
age. Compared with the Gill-Murray-Wright algorithm, the Schnabel-Eskow alg
orithm has a smaller a priori bound on the perturbation, added to ensure po
sitive definiteness, and some computational advantages, especially for larg
e problems. Users of the Schnabel-Eskow algorithm, however, have reported c
ases from two different contexts where it makes a far larger modification t
o 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 diffic
ulties with the method, without harming its theoretical properties or its c
omputational behavior in any other cases. In new computational tests, the m
odifications to the original matrix made by the new algorithm appear virtua
lly always to be smaller than those made by the Gill-Murray-Wright algorith
m, sometimes by significant amounts. The perturbed matrix is allowed to be
more ill-conditioned with the new algorithm, but this seems to be appropria
te in the known contexts where the underlying problem is ill-conditioned.