Aprahamian, Mary and Higham, Nicholas J. (2013) The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential. [MIMS Preprint]
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Abstract
A new matrix function corresponding to the scalar unwinding number of Corless, Hare, and Jeffrey is introduced. This matrix unwinding function, $\mathcal{U}$, is shown to be a valuable tool for deriving identities involving the matrix logarithm and fractional matrix powers, revealing, for example, the precise relation between $\log A^\alpha$ and $\alpha \log A$. The unwinding function is also shown to be closely connected with the matrix sign function. An algorithm for computing the unwinding function based on the Schur--Parlett method with a special reordering is proposed. It is shown that matrix argument reduction using the function $\mathrm{mod}(A) = A-2\pi i\, \mathcal{U}(A)$, which has eigenvalues with imaginary parts in the interval $(-\pi,\pi]$ and for which $\e^A = \e^{\mathrm{mod}(A)}$, can give significant computational savings in the evaluation of the exponential by scaling and squaring algorithms.
Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | matrix unwinding function, unwinding number, matrix logarithm, matrix power, matrix exponential, argument reduction |
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: | 07 May 2013 |
Last Modified: | 08 Nov 2017 18:18 |
URI: | https://eprints.maths.manchester.ac.uk/id/eprint/1974 |
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- The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential. (deposited 07 May 2013) [Currently Displayed]
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