Generalized complex step approximation to estimate the first and second order Fr\'{e}chet derivative of matrix functions

Abstract

Applications of Fr\'{e}chet 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 Fr\'{e}chet derivative. We provide numerical results for the approximation to the first and the second order Fr\'{e}chet derivative of the matrix exponential and matrix square root.