A modified Cholesky algorithm based on a symmetric indefinite factorization

Given a symmetric and not necessarily positive definite matrix A, a modified Cholesky algorithm computes a Cholesky factorization P(A + E)PT = RT R, where P is a permutation matrix and E is a perturbation chosen to make A + E positive definite. The aims include producing a small-normed E and making A + E reasonably well conditioned. Modified Cholesky factorizations are widely used in optimization. We propose a new modified Cholesky algorithm based on a symmetric indefinite factorization computed using a new pivoting strategy of Ashcraft, Grimes, and Lewis. We analyze the effectiveness of the algorithm, both in theory and practice, showing that the algorithm is competitive with the existing algorithms of Gill, Murray, and Wright and Schnabel and Eskow. Attractive features of the new algorithm include easy-to-interpret inequalities that explain the extent to which it satisfies its design goals, and the fact that it can be implemented in terms of existing software.