Evaluating Padé Approximants of the Matrix Logarithm

The inverse scaling and squaring method for evaluating the logarithm of a matrix takes repeated square roots to bring the matrix close to the identity, computes a Padé approximant, and then scales back. We analyze several methods for evaluating the Padé approximant, including Horner’s method (used in some existing codes), suitably customized versions of the Paterson– Stockmeyer method and Van Loan’s variant, and methods based on continued fraction and partial fraction expansions. The computational cost, storage, and numerical accuracy of the methods are compared. We find the partial fraction method to be the best method overall and illustrate the benefits it brings to a transformation-free form of the inverse scaling and squaring method recently proposed by Cheng, Higham, Kenney, and Laub [SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1112– 1125]. We comment briefly on how the analysis carries over to the matrix exponential.