The dual inverse scaling and squaring algorithm for the matrix logarithm

The inverse scaling and squaring algorithm computes the logarithm of a square matrix A by evaluating a rational approximant to the logarithm at the matrix B:=A2-s for a suitable choice of s. We introduce a dual approach and approximate the logarithm of B by solving the rational equation r(X)=B, where r is a diagonal PadE approximant to the matrix exponential at 0. This equation is solved by a substitution technique in the style of those developed by Fasi & Iannazzo (2020, Substitution algorithms for rational matrix equations. Elect. Trans. Num. Anal., 53, 500-521). The new method is tailored to the special structure of the diagonal PadE approximants to the exponential and in terms of computational cost is more efficient than the state-of-The-Art inverse scaling and squaring algorithm.