The Pade method for computing the matrix exponential

We analyze the Pade method for computing the exponential of a real matrix. More precisely, we study the roundoff error introduced by the method in the general case and in three special cases: (1) normal matrices; (2) essentially nonnegative matrices (a(ij) greater than or equal to 0, i not equal j); (3) matrices A such that A = D-1 BD, with D diagonal and B essentially nonnegative. For these special matrices, it turns out that the Pade method is stable. Finally, we compare the Ward upper bound with our results and show that our bounds are generally tighter.