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 )(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. © 2005 IMACS. Published by Elsevier B.V. All rights reserved.