# 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. (2013) Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number. SIAM J. Sci. Comput., 35 (4). C394 -C410. ISSN 1095-7197

Official URL: http://epubs.siam.org/doi/abs/10.1137/120885991

## 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]. For real matrices we develop a version of the latter algorithm that works entirely in real arithmetic and is twice as fast as and more accurate than the original algorithm. We show that by differentiating the algorithms we obtain backward stable algorithms for computing the Fr\'echet derivative. We demonstrate experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the Fr\'echet derivative and we also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm.

Item Type: Article 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. MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Nick Higham 06 Aug 2013 20 Oct 2017 14:13 http://eprints.maths.manchester.ac.uk/id/eprint/2015