An improved schur-pad́e algorithm for fractional powers of a matrix and their fŕechet derivatives

The Schur-Pad́e algorithm [N. J. Higham and L. Lin, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1056-1078] computes arbitrary real powers A t of a matrix A ∈ Cn×n using the building blocks of Schur decomposition, matrix square roots, and Pad́e approximants. We improve the algorithm by basing the underlying error analysis on the quantities ||(I - A)k||1/k, for several small k, instead of ||I -A||. We extend the algorithm so that it computes along with At one or more Fŕechet derivatives, with reuse of information when more than one Fŕechet derivative is required, as is the case in condition number estimation. We also derive a version of the extended algorithm that works entirely in real arithmetic when the data is real. Our numerical experiments show the new algorithms to be superior in accuracy to, and often faster than, the original Schur-Pad́e algorithm for computing matrix powers and more accurate than several alternative methods for computing the Fŕechet derivative. They also show that reliable estimates of the condition number of At are obtained by combining the algorithms with a matrix norm estimator. Copyright © 2013 by SIAM.