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. [MIMS Preprint]

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Abstract

The Schur--Pad�© algorithm [N. J. Higham and L. Lin, A Schur--Pad�© 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 Pad�© 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 Fr�©chet derivatives, with reuse of information when more than one Fr�©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--Pad�© algorithm for computing matrix powers and more accurate than several alternative methods for computing the Fr�©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:	MIMS Preprint
Uncontrolled Keywords:	matrix power, fractional power, matrix root, Fr�©chet derivative, condition number, condition estimate, Schur decomposition, Pad�© approximation, Pad�© approximant, matrix logarithm, matrix exponential, MATLAB
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Dr Lijing Lin
Date Deposited:	03 May 2013
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1972

Available Versions of this Item

• An Improved Schur--Padé Algorithm for Fractional Powers of a Matrix and their Fréchet Derivatives. (deposited 16 Jan 2013)
  • An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives. (deposited 03 May 2013) [Currently Displayed]
    • An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives. (deposited 18 Sep 2013)

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