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. 88109. ISSN 10957162
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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 SchurParlett method with a special reordering is proposed. It is shown that matrix argument reduction using the function $\mathrm{mod}(A) = A2\pi i\, \mathcal{U}(A)$, 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 
Available Versions of this Item
 The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential (deposited 01 February 2014) [Currently Displayed]
 The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential (deposited 25 October 2013)
 The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential (deposited 02 October 2013)
 The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential (deposited 15 May 2013)
 The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential (deposited 07 May 2013)
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