An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives

Higham, Nicholas J. and Lin, Lijing (2013) An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives. SIAM. J. Matrix Anal. & Appl., 34 (3). pp. 1341-1360. ISSN 1095-7162

Official URL: http://epubs.siam.org/doi/pdf/10.1137/130906118

Abstract

The Schur--PadÃ�Æ�Ã�Â© algorithm [N. J. Higham and L. Lin, A Schur--PadÃ�Æ�Ã�Â© algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. Appl., 32(3):1056--1078, 2011] computes arbitrary real powers $A^t$ of a matrix $A\in\mathbb{C}^{n\times n}$ using the building blocks of Schur decomposition, matrix square roots, and PadÃ�Æ�Ã�Â© 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 $A^t$ one or more FrÃ�Æ�Ã�Â©chet derivatives, with reuse of information when more than one FrÃ�Æ�Ã�Â©chet 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--PadÃ�Æ�Ã�Â© algorithm for computing matrix powers and more accurate than several alternative methods for computing the FrÃ�Æ�Ã�Â©chet derivative. They also show that reliable estimates of the condition number of $A^t$ are obtained by combining the algorithms with a matrix norm estimator.

Item Type: Article matrix power, fractional power, matrix root, FrÃ�Æ�Ã�Â©chet derivative, condition number, condition estimate, Schur decomposition, PadÃ�Æ�Ã�Â© approximation, PadÃ�Æ�Ã�Â© approximant, matrix logarithm, matrix exponential, MATLAB MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Dr Lijing Lin 18 Sep 2013 20 Oct 2017 14:13 http://eprints.maths.manchester.ac.uk/id/eprint/2021