Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number

Al-Mohy, Awad H. and Higham, Nicholas J. and Relton, Samuel D. (2012) Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number. [MIMS Preprint]

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Abstract

The most popular method for computing the matrix logarithm is the inverse scaling and squaring method, which is the basis of the recent algorithm of [A. H. Al-Mohy and N. J. Higham, \emph{Improved inverse scaling and squaring algorithms for the matrix logarithm}, SIAM J. Sci.\ Comput., 34 (2012), pp.~C152--C169]. We show that by differentiating the latter algorithm a backward stable algorithm for computing the Fr\'echet derivative of the matrix logarithm is obtained. This algorithm requires complex arithmetic, but we also develop a version that uses only real arithmetic when $A$ is real; as a special case we obtain a new algorithm for computing the logarithm of a real matrix in real arithmetic. We show experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the Fr\'echet derivative. We also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	matrix logarithm, principal logarithm, inverse scaling and squaring method, Fr\'{e}chet derivative, condition number, Pad\'{e} approximation, backward error analysis, matrix exponential, matrix square root, MATLAB, logm.
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Nick Higham
Date Deposited:	25 Jul 2012
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1852

Available Versions of this Item

Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number. (deposited 25 Jul 2012) [Currently Displayed]
- Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number. (deposited 13 Dec 2012)
  - Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number. (deposited 16 May 2013)
    - Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number. (deposited 06 Aug 2013)

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