Abstract
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 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). Moré 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 a new approach using a sandwiched LTL T-LBL T factorization, with T tridiagonal, that guarantees a modification cost of at most O(n 2).
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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.
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Fang, Hr., O’Leary, D.P. Modified Cholesky algorithms: a catalog with new approaches. Math. Program. 115, 319–349 (2008). https://doi.org/10.1007/s10107-007-0177-6
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DOI: https://doi.org/10.1007/s10107-007-0177-6