AlMohy, Awad H. and Higham, Nicholas J. (2009) The Complex Step Approximation to the Fréchet Derivative of a Matrix Function. [MIMS Preprint]
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Abstract
We show that the Fréchet derivative of a matrix function $f$ at $A$ in the direction $E$, where $A$ and $E$ are real matrices, can be approximated by $\Im f(A+ihE)/h$ for some suitably small $h$. This approximation, requiring a single function evaluation at a complex argument, generalizes the complex step approximation known in the scalar case. The approximation is proved to be of second order in $h$ for analytic functions $f$ and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating $f$ employs complex arithmetic. The ease of implementation of the approximation, and its superiority over finite differences, make it attractive when specialized methods for evaluating the Fréchet derivative are not available, and in particular for condition number estimation when used in conjunction with a block 1norm estimation algorithm.
Item Type:  MIMS Preprint 

Additional Information:  To appear in Numerical Algorithms 
Uncontrolled Keywords:  Fr\'echet derivative, matrix function, complex step approximation, complex arithmetic, finite difference, matrix sign function, condition number estimation, block 1norm estimator, CICADA 
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:  02 Oct 2009 
Last Modified:  08 Nov 2017 18:18 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/1295 
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The Complex Step Approximation to the Fréchet Derivative of a Matrix Function. (deposited 27 Apr 2009)
 The Complex Step Approximation to the Fréchet Derivative of a Matrix Function. (deposited 02 Oct 2009) [Currently Displayed]
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