2013.84: Estimating the Condition Number of the Frechet Derivative of a Matrix Function
2013.84: Nicholas J. Higham and Samuel D. Relton (2014) Estimating the Condition Number of the Frechet Derivative of a Matrix Function. SIAM Journal on Scientific Computing, 36 (6). C617C634. ISSN 10648275
This is the latest version of this eprint.
Full text available as:
PDF  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader 283 Kb 
DOI: 10.1137/130950082
Abstract
The Fr\'{e}chet derivative $L_f$ of a matrix function $f \colon \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Fr\'{e}chet derivative and derive upper and lower bounds for it that differ by at most a factor $2$. For a wide class of functions we derive an algorithm for estimating the 1norm condition number that requires $O(n^3)$ flops given $O(n^3)$ flops algorithms for evaluating $f$ and $L_f$; in practice it produces estimates correct to within a factor $6n$. Numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning.
Item Type:  Article 

Additional Information: 

Uncontrolled Keywords:  matrix function, condition number, Fr\'{e}chet derivative, Kronecker form, matrix exponential, matrix logarithm, matrix powers, matrix $p$th root, MATLAB, expm, logm, sqrtm 
Subjects:  MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis 
MIMS number:  2013.84 
Deposited By:  Nick Higham 
Deposited On:  26 November 2014 
Available Versions of this Item
 Estimating the Condition Number of the Frechet Derivative of a Matrix Function (deposited 26 November 2014) [Currently Displayed]
Download Statistics: last 4 weeks
Repository Staff Only: edit this item