The Structured Condition Number of a Differentiable Map between Matrix Manifolds, with Applications
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Date
2019
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Language
en
Pages
26
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SIAM Journal on Matrix Analysis and Applications, Volume 40, issue 2, pp. 774-799
Abstract
We study the structured condition number of differentiable maps between smooth matrix manifolds, extending previous results to maps that are only R-differentiable for complex manifolds. 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.Description
Keywords
matrix function, Frechet derivative, condition number, bilinear form, sesquilinear form, structured matrices, structured condition number, automorphism group, Lie algebra, Jordan algebra, polar decomposition, DECOMPOSITION
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Citation
Arslan, B, Noferini, V & Tisseur, F 2019, ' The Structured Condition Number of a Differentiable Map between Matrix Manifolds, with Applications ', SIAM Journal on Matrix Analysis and Applications, vol. 40, no. 2, pp. 774-799 . https://doi.org/10.1137/17M1148943