On the Stability of Cholesky Factorization for Symmetric Quasidefinite Systems

Sparse linear equations Kd r are considered, where K is a specially structured symmetric indefinite matrix that arises in numerical optimization and elsewhere. Under certain conditions, K is quasidefinite. The Cholesky factorization PKP T LDL T is then known to exist for any permutation P, even though D is indefinite. Quasidefinite matrices have been used successfully by Vanderbei within barrier methods for linear and quadratic programming. An advantage is that for a sequence of K's, P may be chosen once and for all to optimize the sparsity of L, as in the positive-definite case. A preliminary stability analysis is developed here. It is observed that a quasidefinite matrix is closely related to an unsymmetric positive-definite matrix, for which an LDM T factorization exists. Using the Golub and Van Loan analysis of the latter, conditions are derived under which Cholesky factorization is stable for quasidefinite systems. Some numerical results confirm the predictions.