THE STRUCTURED CONDITION NUMBER OF A DIFFERENTIABLE MAP BETWEEN MATRIX MANIFOLDS, WITH APPLICATIONS
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Dosyalar
Tarih
2019
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Siam Publications
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
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.
Açıklama
Noferini, Vanni/0000-0002-1775-041X; Tisseur, Francoise/0000-0002-1011-2570
Anahtar Kelimeler
matrix function, Frechet derivative, condition number, bilinear form, sesquilinear form, structured matrices, structured condition number, automorphism group, Lie algebra, Jordan algebra, polar decomposition
Kaynak
Siam Journal On Matrix Analysis And Applications
WoS Q Değeri
Q2
Scopus Q Değeri
Q1
Cilt
40
Sayı
2
