Implementation of the Schnabel and Eskow Modified Cholesky Factorization in the Context of a Truncated-Newton Optimization Method (Classic Reprint) - Tapa dura

Sinopsis

Excerpt from Implementation of the Schnabel and Eskow Modified Cholesky Factorization in the Context of a Truncated-Newton Optimization Method

Two important questions in these modified factorizations are the following: 1) how do we perform the mcf in a numerically-stable manner without knowing the complete eigenvalue distribution of M a priori? And 2) how do we choose the appropriate numerical values for the entries of E? The first issue is important because the Cholesky factors may be ill-conditioned and may not exist for an indefinite system; thus, the standard process must be amended appropriately. Furthermore, the algorithm should be formulated so as to keep the computational cost as close as possible to that of the standard Cholesky factorization. In regard to the second question of constructing E, we must generate a clever recipe for choosing the increments ci, j n, so as to balance: 1) on one extreme, selecting large modifications that guarantee positive definiteness but perturb the original matrix excessively; with 2) on the other extreme, choosing small, just sufficient.

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