Al-Mohy, Awad H.Arslan, Bahar2021-03-202021-03-2020211017-13981572-9265http://doi.org/10.1007/s11075-020-00998-3https://hdl.handle.net/20.500.12885/376Thekth 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.eninfo:eu-repo/semantics/closedAccessMatrix functionFrechet derivativeHigher order Frechet derivativeComplex step approximationAction of matrix functionsArticle10.1007/s11075-020-00998-387310611074WOS:000564517600001Q1Q2