Padé and Gregory error estimates for the logarithm of block triangular matrices

In this paper we give bounds for the error arising in the approximation of the logarithm of a block triangular matrix T by Padé approximants of the function f(x)=log[(1+x)/(1−x)] and partial sums of Gregory's series. These bounds show that if the norm of all diagonal blocks of the Cayley-transform B=(T−I)−1(T+I) is sufficiently close to zero, then both approximation methods are accurate. This will contribute for reducing the number of successive square roots of T needed in the inverse scaling and squaring procedure for the matrix logarithm.