Multiprecision Algorithms for Computing the Matrix Logarithm

Fasi, Massimiliano and Higham, Nicholas J. (2017) Multiprecision Algorithms for Computing the Matrix Logarithm. [MIMS Preprint]

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Two algorithms are developed for computing the matrix logarithm in floating point arithmetic of any specified precision. The backward error-based 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 transformation-free and uses only the computational kernels of matrix multiplication and the solution of multiple right-hand side linear systems. For double precision computations the algorithms are competitive with the state of the art algorithm of Al-Mohy, 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: 11 May 2017
Last Modified: 08 Nov 2017 18:18

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