Fasi, Massimiliano and Higham, Nicholas J. (2017) Multiprecision Algorithms for Computing the Matrix Logarithm. [MIMS Preprint]
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Abstract
Two algorithms are developed for computing the matrix logarithm in floating point arithmetic of any specified precision. The backward errorbased approach used in the state of the art inverse scaling and squaring algorithms does not conveniently extend to a multiprecision environment, so instead we choose algorithmic parameters based on a forward error bound. We derive a new forward error bound for Pad\'{e} approximants that for highly nonnormal matrices can be much smaller than the classical bound of Kenney and Laub. One of our algorithms exploits a Schur decomposition while the other is transformationfree and uses only the computational kernels of matrix multiplication and the solution of multiple righthand side linear systems. For double precision computations the algorithms are competitive with the state of the art algorithm of AlMohy, Higham, and Relton implemented in \texttt{logm} in MATLAB\@. They are intended for computing environments providing multiprecision floating point arithmetic, such as Julia, MATLAB via the Symbolic Math Toolbox or the Multiprecision Computing Toolbox, or Python with the mpmath or SymPy packages. We show experimentally that the algorithms behave in a forward stable manner over a wide range of precisions, unlike existing alternatives.
Item Type:  MIMS Preprint 

Uncontrolled Keywords:  multiprecision arithmetic, matrix logarithm, principal logarithm, inverse scaling and squaring method, Fr\'{e}chet derivative, Pad\'{e} approximation, Taylor approximation, forward error analysis, 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 Oct 2017 
Last Modified:  08 Nov 2017 18:18 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2587 
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Multiprecision Algorithms for Computing the Matrix Logarithm. (deposited 11 May 2017)
 Multiprecision Algorithms for Computing the Matrix Logarithm. (deposited 25 Oct 2017) [Currently Displayed]
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