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

Tutkimustuotos: Lehtiartikkelivertaisarvioitu

Tutkijat

Organisaatiot

  • Bursa Technical University
  • University of Manchester

Kuvaus

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.

Yksityiskohdat

AlkuperäiskieliEnglanti
Sivut774-799
Sivumäärä26
JulkaisuSIAM Journal on Matrix Analysis and Applications
Vuosikerta40
Numero2
TilaJulkaistu - 2019
OKM-julkaisutyyppiA1 Julkaistu artikkeli, soviteltu

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