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
Article

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.

Keywords

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Benner, R. Byers, V. Mehrmann and H. Xu, A unified deflating subspace approach for classes of polynomial and rational matrix equations, Preprint SFB393/00-05, Zentrum für Technomathematik, Universität Bremen, Bremen, Germany (January 2000). Google Scholar
  2. [2]
    R. Bhatia, Matrix Analysis (Springer, New York, 1997). Google Scholar
  3. [3]
    D.A. Bini, L. Gemignani and B. Meini, Computations with infinite Toeplitz matrices and polynomials, Linear Algebra Appl. 343/344 (2002) 21–61. Google Scholar
  4. [4]
    D.A. Bini, L. Gemignani and B. Meini, Solving certain matrix equations by means of Toeplitz computations: Algorithms and applications, in: Fast Algorithms for Structured Matrices: Theory and Applications, ed. V. Olshevsky, Contemporary Mathematics, Vol. 323 (Amer. Math. Soc., Providence, RI, 2003) pp. 151–167. Google Scholar
  5. [5]
    D.A. Bini and B. Meini, Improved cyclic reduction for solving queueing problems, Numer. Algorithms 15(1) (1997) 57–74. Google Scholar
  6. [6]
    D.A. Bini and B. Meini, Non-skip-free M/G/1-type Markov chains and Laurent matrix power series, Linear Algebra Appl. 386 (2004) 187–206. Google Scholar
  7. [7]
    D.A. Bini and V.Y. Pan, Polynomial and Matrix Computations, Vol. 1: Fundamental Algorithms (Birkhäuser, Boston, MA, 1994). Google Scholar
  8. [8]
    A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices (Springer, New York, 1999). Google Scholar
  9. [9]
    B.L. Buzbee, G.H. Golub and C.W. Nielson, On direct methods for solving Poisson’s equations, SIAM J. Numer. Anal. 7(4) (1970) 627–656. Google Scholar
  10. [10]
    S.H. Cheng, N.J. Higham, C.S. Kenney and A.J. Laub, Approximating the logarithm of a matrix to specified accuracy, SIAM J. Matrix Anal. Appl. 22(4) (2001) 1112–1125. Google Scholar
  11. [11]
    P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, London, 1984). Google Scholar
  12. [12]
    M.A. Hasan, A.A. Hasan and K.B. Ejaz, Computation of matrix nth roots and the matrix sector function, in: Proc. of the 40th IEEE Conf. on Decision and Control, Orlando, FL (2001) pp. 4057–4062. Google Scholar
  13. [13]
    M.A. Hasan, J.A.K. Hasan and L. Scharenroich, New integral representations and algorithms for computing nth roots and the matrix sector function of nonsingular complex matrices, in: Proc. of the 39th IEEE Conf. on Decision and Control, Sydney, Australia (2000) pp. 4247–4252. Google Scholar
  14. [14]
    N.J. Higham, The Matrix Computation Toolbox, http://www.ma.man.ac.uk/~higham/mctoolbox.
  15. [15]
    N.J. Higham, Newton’s method for the matrix square root, Math. Comp. 46(174) (1986) 537–549. Google Scholar
  16. [16]
    N.J. Higham, The matrix sign decomposition and its relation to the polar decomposition, Linear Algebra Appl. 212/213 (1994) 3–20. Google Scholar
  17. [17]
    N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed. (SIAM, Philadelphia, PA, 2002). Google Scholar
  18. [18]
    W.D. Hoskins and D.J. Walton, A faster, more stable method for computing the pth roots of positive definite matrices, Linear Algebra Appl. 26 (1979) 139–163. Google Scholar
  19. [19]
    C.S. Kenney and A.J. Laub, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl. 10(2) (1989) 191–209. Google Scholar
  20. [20]
    Ç.K. Koç and B. Bakkaloğlu, Halley’s method for the matrix sector function, IEEE Trans. Automat. Control 40(5) (1995) 944–949. Google Scholar
  21. [21]
    M.L. Mehta, Matrix Theory: Selected Topics and Useful Results, 2nd ed. (Hindustan Publishing, Delhi, 1989). Google Scholar
  22. [22]
    B. Meini, The matrix square root from a new functional perspective: Theoretical results and computational issues, Technical Report 1455, Dipartimento di Matematica, Università di Pisa (2003), to appear in SIAM J. Matrix Anal. Appl. Google Scholar
  23. [23]
    M.A. Ostrowski, Recherches sur la méthode de Graeffe et les zeros des polynômes et des séries de Laurent, Acta Math. 72 (1940) 99–257. Google Scholar
  24. [24]
    B. Philippe, An algorithm to improve nearly orthonormal sets of vectors on a vector processor, SIAM J. Algebra Discrete Methods 8(3) (1987) 396–403. Google Scholar
  25. [25]
    G. Schulz, Iterative Berechnung der reziproken Matrix, Z. Angew. Math. Mech. 13 (1933) 57–59. Google Scholar
  26. [26]
    L.-S. Shieh, Y.T. Tsay and R.E. Yates, Computation of the principal nth roots of complex matrices, IEEE Trans. Automat. Control 30(6) (1985) 606–608. Google Scholar
  27. [27]
    M.I. Smith, A Schur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl. 24(4) (2003) 971–989. Google Scholar
  28. [28]
    J.S.H. Tsai, L.S. Shieh and R.E. Yates, Fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function, Comput. Math. Appl. 15(11) (1988) 903–913. Google Scholar
  29. [29]
    Y.T. Tsay, L.S. Shieh and J.S.H. Tsai, A fast method for computing the principal nth roots of complex matrices, Linear Algebra Appl. 76 (1986) 205–221. Google Scholar

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

Personalised recommendations