Modified Cholesky algorithms: a catalog with new approaches

Given an n × n symmetric possibly indefinite matrix A, a modified Cholesky algorithm computes a factorization of the positive definite matrix A + E , where E is a correction matrix. Since the factorization is often used to compute a Newton-like downhill search direction for an optimization problem, the goals are to compute the modification without much additional cost and to keep A + E wellconditioned and close to A. Gill, Murray and Wright introduced a stable algorithm, with a bound of ‖E‖2 = O(n2). An algorithm of Schnabel and Eskow further guarantees ‖E‖2 = O(n). We present variants that also ensure ‖E‖2 = O(n). Moré and Sorensen and Cheng and Higham used the block L BLT factorization with blocks of order 1 or 2. Algorithms in this class have a worst-case cost O(n3) higher than the standard Cholesky factorization. We present a new approach using a sandwiched LT LT -L BLT factorization, with T tridiagonal, that guarantees a modification cost of at most O(n2). H.-r. Fang’s work was supported by National Science Foundation Grant CCF 0514213. D. P. O’Leary’s work was supported by National Science Foundation Grant CCF 0514213 and Department of Energy Grant DEFG0204ER25655. H.-r. Fang Department of Computer Science and Engineering, University of Minnesota, 200 Union Street, Minneapolis, MN 55455, USA D. P. O’Leary (B) Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, A.V. Williams Building, College Park, Maryland, MD 20742, USA e-mail: [email protected]