Modified Cholesky Algorithms: A Catalog with New Approaches

Given an n by 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 well-conditioned and close to A. Gill, Murray and Wright introduced a stable algorithm, with a bound of ||E||_2=O(n^2). An algorithm of Schnabel and Eskow further guarantees ||E||_2=O(n). We present variants that also ensure ||E||_2=O(n). Mor\'{e} and Sorensen and Cheng and Higham used the block LBL^T factorization with blocks of order 1 or 2. Algorithms in this class have a worst-case cost O(n^3) higher than the standard Cholesky factorization. We present two new algorithms using an LTL^T factorization, with T tridiagonal, that guarantees a modification cost of at most O(n^2).


University of Maryland CS Technical Report CS-TR-4807, August 2006.



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