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 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.
Funding agencies:
MathWorks
Royal Society of London European Commission
Istituto Nazionale di Alta Matematica (INdAM-GNCS Project 2019)