Algorithms for the matrix pth root

Dario A. Bini; Nicholas J. Higham; Beatrice Meini

doi:10.1007/s11075-004-6709-8

Algorithms for the matrix pth root

Dario A. Bini, Nicholas J. Higham, Beatrice Meini

Research output: Contribution to journal › Article › peer-review

Abstract

New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener-Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newton's method for the inverse pth root. Preliminary computational experiments are presented to compare the methods. © Springer 2005.

Original language	English
Pages (from-to)	349-378
Number of pages	29
Journal	Numerical Algorithms
Volume	39
Issue number	4
DOIs	https://doi.org/10.1007/s11075-004-6709-8
Publication status	Published - Aug 2005

Keywords

Cyclic reduction
Graeffe iteration
Laurent polynomial
Matrix pth root
Matrix sign function
Newton's method
Wiener-Hopf factorization

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10.1007/s11075-004-6709-8

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title = "Algorithms for the matrix pth root",

abstract = "New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener-Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newton's method for the inverse pth root. Preliminary computational experiments are presented to compare the methods. {\textcopyright} Springer 2005.",

keywords = "Cyclic reduction, Graeffe iteration, Laurent polynomial, Matrix pth root, Matrix sign function, Newton's method, Wiener-Hopf factorization",

author = "Bini, {Dario A.} and Higham, {Nicholas J.} and Beatrice Meini",

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AU - Bini, Dario A.

AU - Higham, Nicholas J.

AU - Meini, Beatrice

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N2 - New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener-Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newton's method for the inverse pth root. Preliminary computational experiments are presented to compare the methods. © Springer 2005.

AB - New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener-Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newton's method for the inverse pth root. Preliminary computational experiments are presented to compare the methods. © Springer 2005.

KW - Cyclic reduction

KW - Graeffe iteration

KW - Laurent polynomial

KW - Matrix pth root

KW - Matrix sign function

KW - Newton's method

KW - Wiener-Hopf factorization

U2 - 10.1007/s11075-004-6709-8

DO - 10.1007/s11075-004-6709-8

M3 - Article

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VL - 39

SP - 349

EP - 378

JO - Numerical Algorithms

JF - Numerical Algorithms

IS - 4

ER -

Algorithms for the matrix pth root

Abstract

Keywords

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