Computation of the matrix exponential and its derivatives by scaling and squaring

In this paper, a simple method based on the scaling and squaring technique for the evaluation of the matrix exponential and its derivatives is presented. A more general formulation with non-constant first derivatives is considered here. Both higher order and mixed derivatives are investigated. The method is based directly on the property of the exponential function and does not require the use of perturbation formulae for eigenvalues and eigenvectors. The approach provides a simple and direct algorithm for the computation of the matrix exponential and its derivatives regardless of degeneracy in the spectral decomposition of the matrix argument. If the derivatives are taken with respect to the entries of the matrix argument, the first and second linearizations can be obtained directly