A New Modified Cholesky Factorization

The modified Cholesky factorization of Gill and Murray plays an important role in optimization algorithms. Given a symmetric but not necessarily positive definite matrix A, it computes a Cholesky factorization ofA +E, where E= if A is safely positive definite, and E is a diagonal matrix chosen to make A +E positive definite otherwise. The factorization costs only a small multiple of n 2 operations more than the standard Cholesky factorization. We present a new algorithm that has tnese same properties, but for which the theoretical bound on II E II is substantially smaller. It is based upon two new techniques, the use of Gerschgorin bounds in selecting the elements of E, and a new way of moiiitoring positive definiteness. In extensive computational tests on indefinite matrices, the new factorization virtually always produces smaller values of I I E I I than the existing method, without impairing the conditioning of A +E. In some cases the improvements are substantial. The new factorization may prove useful in optimization algorithms.