Numerical Algorithms

, Volume 39, Issue 4, pp 349–378 | Cite as

Algorithms for the matrix pth root

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


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.


matrix pth root matrix sign function Wiener–Hopf factorization Newton’s method Graeffe iteration cyclic reduction Laurent polynomial 


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Copyright information

© Springer 2005

Authors and Affiliations

  • Dario A. Bini
    • 1
  • Nicholas J. Higham
    • 2
  • Beatrice Meini
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Department of MathematicsUniversity of ManchesterManchesterEngland

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