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:=A^{2^{-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 Padé approximant to the matrix exponential at 0. This equation is solved by a substitution technique in the style of those developed in (Fasi & Iannazzo, Elect. Trans. Num. Anal., 53 (2020), pp. 500--521). The new method is tailored to the special structure of the diagonal Padé approximants to the exponential, and in terms of computational cost is more efficient than the state-of-the-art inverse scaling and squaring algorithm