THE STRUCTURED CONDITION NUMBER OF A DIFFERENTIABLE MAP BETWEEN MATRIX MANIFOLDS, WITH APPLICATIONS

Noferini, Vanni/0000-0002-1775-041X; Tisseur, Francoise/0000-0002-1011-2570 WOS:000473026800016 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. Republic of Turkey Ministry of National EducationMinistry of National Education - Turkey; European Research Council Advanced grant MATFUN [267526]; Engineering and Physical Sciences Research CouncilUK Research & Innovation (UKRI)Engineering & Physical Sciences Research Council (EPSRC) [EP/I005293]; Royal Society-Wolfson Research Merit AwardRoyal Society of London The first author's research was supported by the Republic of Turkey Ministry of National Education. The second author's work was partially supported by European Research Council Advanced grant MATFUN (267526). The third author's research was supported by Engineering and Physical Sciences Research Council grant EP/I005293 and by a Royal Society-Wolfson Research Merit Award.