2011.83: Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm
2011.83: Awad H. AlMohy and Nicholas J. Higham (2011) Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm.
There is a more recent version of this eprint available. Click here to view it.
Full text available as:
PDF  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader 381 Kb  
Other (Zip file containing MATLAB codes) 7 Kb 
Abstract
A popular method for computing the matrix logarithm is the inverse scaling and squaring method, which essentially carries out the steps of the scaling and squaring method for the matrix exponential in reverse order. Here we make several improvements to the method, putting its development on a par with our recent version [\emph{SIAM J. Matrix Anal.\ Appl.}, 31 (2009), pp.\ 970989] of the scaling and squaring method for the exponential. In particular, we introduce backward error analysis to replace the previous forward error analysis; obtain backward error bounds in terms of the quantities $\A^p\^{1/p}$, for several small integer $p$, instead of $\A\$; and use special techniques to compute the argument of the Pad\'e approximant more accurately. We derive one algorithm that employs a Schur decomposition, and thereby works with triangular matrices, and another that requires only matrix multiplications and the solution of multiple righthand side linear systems. Numerical experiments show the new algorithms to be generally faster and more accurate than their existing counterparts and suggest that the Schurbased method is the method of choice for computing the matrix logarithm.
Item Type:  MIMS Preprint 

Additional Information: 

Uncontrolled Keywords:  matrix logarithm, inverse scaling and squaring method, matrix exponential, backward error analysis, Pad\'e approximation, matrix square root, MATLAB, \texttt{logm} 
Subjects:  MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis 
MIMS number:  2011.83 
Deposited By:  Nick Higham 
Deposited On:  19 October 2011 
Available Versions of this Item
 Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm (deposited 25 July 2012)
 Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm (deposited 20 April 2012)
 Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm (deposited 19 October 2011) [Currently Displayed]
Download Statistics: last 4 weeks
Repository Staff Only: edit this item