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    The complex step approximation to the higher order Frechet derivatives of a matrix function
    (Springer, 2021) Al-Mohy, Awad H.; Arslan, Bahar
    Thekth Frechet derivative of a matrix functionfis a multilinear operator from a cartesian product ofksubsets of the spaceDOUBLE-STRUCK CAPITAL C-nxn into itself. We show that thekth Frechet derivative of a real-valued matrix functionfat a real matrixAin real direction matrices E-1, E-2, horizontal ellipsis, E-k can be computed using the complex step approximation. We exploit the algorithm of Higham and Relton (SIAM J. Matrix Anal. Appl.35(3):1019-1037,2014) with the complex step approximation and mixed derivative of complex step and central finite difference scheme. Comparing with their approach, our cost analysis and numerical experiment reveal thathalfandseven-eighthsof the computational cost can be saved for the complex step and mixed derivative, respectively. Whenfhas an algorithm that computes its action on a vector, the computational cost drops down significantly as the dimension of the problem andkincrease.
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    Structure Preserving Algorithm for the Logarithm of Symplectic Matrices
    (Murat TOSUN, 2021) Arslan, Bahar
    The current algorithms use either the full form or the Schur decomposition of the matrix in the inverse scaling and squaring method to compute the matrix logarithm. The inverse scaling and squaring method consists of two main calculations: taking a square root and evaluating the Padé approximants. In this work, we suggest using the structure preserving iteration as an alternative to Denman-Beavers iteration for taking a square root. Numerical experiments show that while using the structure preserving square root iteration in the inverse scaling and squaring method preserves the Hamiltonian structure of matrix logarithm, Denman-Beavers iteration and Schur decomposition cause a structure loss.
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    THE STRUCTURED CONDITION NUMBER OF A DIFFERENTIABLE MAP BETWEEN MATRIX MANIFOLDS, WITH APPLICATIONS
    (Siam Publications, 2019) Arslan, Bahar; Noferini, Vanni; Tisseur, Francoise
    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.