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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Differentiation of matrix functionals using triangular factorization
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by F. R. de Hoog, R. S. Anderssen and M. A. Lukas PDF
Math. Comp. 80 (2011), 1585-1600 Request permission


In various applications, it is necessary to differentiate a matrix functional $w(\textbf {A}(\textbf {x}))$, where $\textbf {A}(\textbf {x})$ is a matrix depending on a parameter vector $\textbf {x}$. Usually, the functional itself can be readily computed from a triangular factorization of $\textbf {A}(\textbf {x})$. 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:
  • R. S. Anderssen
  • Affiliation: CSIRO Mathematics, Informatics and Statistics, GPO Box 664, Canberra, ACT 2601, Australia
  • Email:
  • M. A. Lukas
  • Affiliation: Mathematics and Statistics, Murdoch University, South Street, Murdoch WA 6150, Australia
  • Email:
  • Received by editor(s): May 5, 2009
  • Published electronically: January 6, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1585-1600
  • MSC (2010): Primary 15A24; Secondary 15A15, 40C05, 65F30
  • DOI:
  • MathSciNet review: 2785469