Padé and Gregory error estimates for the logarithm of block triangular matrices (Q2491896)

New estimates for the absolute 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 are presented. These bounds exploit the block triangular structure of \(T\) and improve the existing estimates that treat the matrix as a whole. The 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 to reducing the number of successive square roots of \(T\) needed in the inverse scaling and squaring procedure for the matrix logarithm. Numerical examples are given.