Yazar "Al-Mohy, Awad H." seçeneğine göre listele

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    Generalized Complex Step Approximation to Estimate the First and Second Order Frechet Derivative of Matrix Functions
    (Univ Nis, Fac Sci Math, 2022) Arslan, Bahar; Al-Mohy, Awad H.
    Applications of Frechet derivative emerge in the sensitivity analysis of matrix functions. Our work extends the generalized complex step approximation using the complex computation f (A + e(i theta)hE) as a tool to matrix case, and combines it with finite difference formula to estimate the Frechet derivative. We provide numerical results for the approximation to the first and the second order Frechet derivative of the matrix exponential and matrix square root.
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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.