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Differentiation of matrix functionals using triangular factorization


Authors: F. R. de Hoog, R. S. Anderssen and M. A. Lukas
Journal: Math. Comp. 80 (2011), 1585-1600
MSC (2010): Primary 15A24; Secondary 15A15, 40C05, 65F30
DOI: https://doi.org/10.1090/S0025-5718-2011-02451-8
Published electronically: January 6, 2011
MathSciNet review: 2785469
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Abstract: In various applications, it is necessary to differentiate a matrix functional $ w({\bf A}({\bf x}))$, where $ {\bf A}({\bf x})$ is a matrix depending on a parameter vector $ {\bf x}$. Usually, the functional itself can be readily computed from a triangular factorization of $ {\bf A}({\bf x})$. This paper develops several methods that also use the triangular factorization to efficiently evaluate the first and second derivatives of the functional. Both the full and sparse matrix situations are considered. There are similarities between these methods and algorithmic differentiation. However, the methodology developed here is explicit, leading to new algorithms. It is shown how the methods apply to several applications where the functional is a log determinant, including spline smoothing, covariance selection and restricted maximum likelihood.


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Additional Information

F. R. de Hoog
Affiliation: CSIRO Mathematics, Informatics and Statistics, GPO Box 664, Canberra, ACT 2601, Australia
Email: Frank.deHoog@csiro.au

R. S. Anderssen
Affiliation: CSIRO Mathematics, Informatics and Statistics, GPO Box 664, Canberra, ACT 2601, Australia
Email: Bob.Anderssen@csiro.au

M. A. Lukas
Affiliation: Mathematics and Statistics, Murdoch University, South Street, Murdoch WA 6150, Australia
Email: M.Lukas@murdoch.edu.au

DOI: https://doi.org/10.1090/S0025-5718-2011-02451-8
Keywords: Differentiation matrix functionals, triangular factorization, algorithmic differentiation, log-det relationships, robust generalized cross validation, smoothing splines, REML, covariance selection
Received by editor(s): May 5, 2009
Published electronically: January 6, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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