The Structured Condition Number of a Differentiable Map Between Matrix Manifolds, with Applications

Arslan, Bahar and Noferini, Vanni and Tisseur, Francoise (2017) The Structured Condition Number of a Differentiable Map Between Matrix Manifolds, with Applications. [MIMS Preprint]

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Abstract

We study the structured condition number of differentiable maps between smooth matrix manifolds, developing a theoretical framework that extends previous results for vector subspaces to any smooth manifold. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Fran├žoise Tisseur
Date Deposited: 25 Sep 2017
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2579

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