A Schur--Fréchet Algorithm for Computing the Logarithm and Exponential of a Matrix

The Schur--Frechet method of evaluating matrix functions consists of putting the matrix in upper triangular form, computing the scalar function values along the main diagonal, and then using the Frechet derivative of the function to evaluate the upper diagonals. This approach requires a reliable method of computing the Frechet derivative. For the logarithm this can be done by using repeated square roots and a hyperbolic tangent form of the logarithmic Frechet derivative. Pade approximations of the hyperbolic tangent lead to a Schur--Frechet algorithm for the logarithm that avoids problems associated with the standard "inverse scaling and squaring" method. Inverting the order of evaluation in the logarithmic Frechet derivative gives a method of evaluating the derivative of the exponential. The resulting Schur--Frechet algorithm for the exponential gives superior results compared to standard methods on a set of test problems from the literature.