Preconditioning Saddle-Point Systems with Applications in Optimization
SIAM Journal on Scientific Computing
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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).