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.
Highlights
Condition numbers measure the sensitivity of a problem to perturbation in the data
The special case of matrix functions was considered by Kenney and Laub [15] and was analyzed in detail by Higham in the monograph [10, Chap
We show that the structured condition number cond\scrM (f, X) can be expressed as the norm of the differential of the restriction of f to \scrM at X
Summary
Condition numbers measure the sensitivity of a problem to perturbation in the data. Let f : \BbbF n\times n \rightar \BbbF n\times n be differentiable, where \BbbF = \BbbR or \BbbC. Our interest is in the sensitivity of f when perturbations are constrained to preserve structure. A general theory of conditioning was first developed by Rice [20]. The special case of matrix functions was considered by Kenney and Laub [15] and was analyzed in detail by Higham in the monograph [10, Chap. For a matrix X \in \BbbF n\times n such that f (X) is defined in an open neighborhood \scrD \subsete \BbbF n\times n of X, the absolute condition number of f (X) is (1.1)
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