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

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)]f(x)=log[(1+x)/(1−x)] and partial sums of Gregoryʹs series. These bounds show that if the norm of all diagonal blocks of the Cayley-transform B=(T−I)−1(T+I)B=(T−I)(T+I)−1 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.