## 2013.1: An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives

2013.1:
Nicholas J. Higham and Lijing Lin
(2013)
*An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives.*

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## Abstract

The Schur--PadÃƒÂ© algorithm [N. J. Higham and L. Lin, A Schur--PadÃƒÂ© 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 PadÃƒÂ© 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 FrÃƒÂ©chet derivatives, with reuse of information when more than one FrÃƒÂ©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--PadÃƒÂ© algorithm for computing matrix powers and more accurate than several alternative methods for computing the FrÃƒÂ©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, FrÃƒÂ©chet derivative, condition number, condition estimate, Schur decomposition, PadÃƒÂ© approximation, PadÃƒÂ© approximant, matrix logarithm, matrix exponential, MATLAB |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2013.1 |

Deposited By: | Dr Lijing Lin |

Deposited On: | 03 May 2013 |

### Available Versions of this Item

- An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives (deposited 18 September 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--PadÃ© Algorithm for Fractional Powers of a Matrix and their FrÃ©chet Derivatives (deposited 16 January 2013)

- An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives (deposited 03 May 2013)

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