Evaluating Pade approximants of the matrix logarithm

The inverse scaling and squaring method for evaluating the logarithm of a m atrix takes repeated square roots to bring the matrix close to the identity , computes a Pade approximant, and then scales back. We analyze several met hods for evaluating the Pade approximant, including Horner's method (used i n some existing codes), suitably customized versions of the Paterson Stockm eyer method and Van Loan's variant, and methods based on continued fraction and partial fraction expansions. The computational cost, storage, and nume rical accuracy of the methods are compared. We nd the partial fraction meth od to be the best method overall and illustrate the bene ts it brings to a transformation-free form of the inverse scaling and squaring method recentl y 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.