A SCHUR-FRECHET ALGORITHM FOR COMPUTING THE LOGARITHM AND EXPONENTIALOF A MATRIX

The Schur-Frechet method of evaluating matrix functions consists of pu tting the matrix in upper triangular form, computing the scalar functi on values along the main diagonal, and then using the Frechet derivati ve of the function to evaluate the upper diagonals. This approach requ ires a reliable method of computing the Frechet derivative. For the lo garithm this can be done by using repeated square roots and a hyperbol ic tangent form of the logarithmic Frechet derivative. Pade approximat ions of the hyperbolic tangent lead to a Schur-Frechet algorithm for t he logarithm that avoids problems associated with the standard ''inver se scaling and squaring'' method. Inverting the order of evaluation in the logarithmic Frechet derivative gives a method of evaluating the d erivative of the exponential. The resulting Schur-Frechet algorithm fo r the exponential gives superior results compared to standard methods on a set of test problems from the literature.