Data Structures and Algorithms

   

The Backward Differentiation of the Bordering Algorithm for an Indefinite Cholesky Factorization

Authors: Stephen P. Smith

The bordering method of the Cholesky decomposition is backward differentiated to derive a method of calculating first derivatives. The result is backward differentiated again and an algorithm for calculating second derivatives results. Applying backward differentiation twice also generates an algorithm for conducting forward differentiation. The differentiation methods utilize three main modules: a generalization of forward substitution for calculating the forward derivatives; a generalization of backward substitution for calculating the backward derivatives; and an additional module involved with the calculation of second derivatives. Separating the methods into three modules lends itself to optimization where software can be developed for special cases that are suitable for sparse matrix manipulation, vector processing and/or blocking strategies that utilize matrix partitions. Surprisingly, the same derivative algorithms fashioned for the Cholesky decomposition of a positive definite matrix can be used again for matrices that are indefinite. The only differences are very minor adjustments involving an initialization step that leads into backward differentiation and a finalization step that follows forward differentiation.

Comments: 19 Pages.

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Submission history

[v1] 2017-07-17 17:13:10 (removed)
[v2] 2017-07-19 13:54:31 (removed)
[v3] 2017-07-20 12:52:57 (removed)
[v4] 2017-08-13 18:09:20

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