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2013.21: The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential

2013.21: Mary Aprahamian and Nicholas J. Higham (2014) The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential. SIAM Journal on Matrix Analysis and Applications, 35 (1). pp. 88-109. ISSN 1095-7162

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DOI: 10.1137/130920137

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:Article
Additional Information:

Uncontrolled Keywords:matrix unwinding function, unwinding number, matrix logarithm, matrix power, matrix exponential, argument reduction
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2013.21
Deposited By:Nick Higham
Deposited On:01 February 2014

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