The complex step approximation to the higher order Fréchet derivatives of a matrix function

Abstract

The k th Fréchet derivative of a matrix function f is a multilinear operator from a cartesian product of k subsets of the space $\mathbb {C}^{n\times n}$ into itself. We show that the k th Fréchet derivative of a real-valued matrix function f at a real matrix A in real direction matrices E₁, E₂, $\dots$, 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 that half and seven-eighths of the computational cost can be saved for the complex step and mixed derivative, respectively. When f has an algorithm that computes its action on a vector, the computational cost drops down significantly as the dimension of the problem and k increase.