Modified Quaternion Newton Methods

  • Fernando Miranda
  • M. Irene Falcão
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8579)


We revisit the quaternion Newton method for computing roots of a class of quaternion valued functions and propose modified algorithms for finding multiple roots of simple polynomials. We illustrate the performance of these new methods by presenting several numerical experiments.


Quaternion Analysis Newton methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus: Theory and applications of slice hyperholomorphic functions. Progress in Mathematics, vol. 289. Birkhäuser/Springer Basel AG, Basel (2011)CrossRefGoogle Scholar
  2. 2.
    Cullen, C.G.: An integral theorem for analytic intrinsic functions on quaternions. Duke Math. J. 32, 139–148 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    De Leo, S., Ducati, G., Leonardi, V.: Zeros of unilateral quaternionic polynomials. Electron. J. Linear Algebra 15, 297–313 (2006)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Eilenberg, S., Niven, I.: The “fundamental theorem of algebra” for quaternions. Bull. Amer. Math. Soc. 50, 246–248 (1944)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Falcão, M.I.: Newton method in quaternion context. Appl. Math. Comput. (accepted for publication)Google Scholar
  6. 6.
    Falcão, M.I., Miranda, F.: Quaternions: A Mathematica package for quaternionic analysis. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2011, Part III. LNCS, vol. 6784, pp. 200–214. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Fueter, R.: Die funktionentheorie der differetialgleichungen Δ u = 0 und ΔΔ u = 0 mit vier reellen variablen. Comm. Math. Helv. (7), 307–330 (1934-1935)Google Scholar
  8. 8.
    Galántai, A., Hegedűs, C.J.: A study of accelerated Newton methods for multiple polynomial roots. Numer. Algorithms 54(2), 219–243 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Gentili, G., Struppa, D.C.: A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris 342(10), 741–744 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216(1), 279–301 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gentili, G., Struppa, D.C., Vlacci, F.: The fundamental theorem of algebra for Hamilton and Cayley numbers. Math. Z. 259(4), 895–902 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic functions in the plane and n-dimensional space. Birkhäuser Verlag, Basel (2008)Google Scholar
  13. 13.
    Janovská, D., Opfer, G.: Computing quaternionic roots by Newton’s method. Electron. Trans. Numer. Anal. 26, 82–102 (2007)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Janovská, D., Opfer, G.: The classification and the computation of the zeros of quaternionic, two-sided polynomials. Numer. Math. 115(1), 81–100 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Janovská, D., Opfer, G.: A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal. 48(1), 244–256 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Jia, Z., Cheng, X., Zhao, M.: A new method for roots of monic quaternionic quadratic polynomial. Comput. Math. Appl. 58(9), 1852–1858 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kalantari, B.: Algorithms for quaternion polynomial root-finding. J. Complexity 29(3-4), 302–322 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Niven, I.: Equations in quaternions. Amer. Math. Monthly 48, 654–661 (1941)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Ostrowski, A.M.: Solution of equations and systems of equations. Pure and Applied Mathematics, vol. IX. Academic Press, New York (1960)zbMATHGoogle Scholar
  20. 20.
    Özban, A.Y.: Some new variants of Newton’s method. Appl. Math. Lett. 17(6), 677–682 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Pogorui, A., Shapiro, M.: On the structure of the set of zeros of quaternionic polynomials. Complex Var. Theory Appl. 49(6), 379–389 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Rinehart, R.F.: Elements of a theory of intrinsic functions on algebras. Duke Math. J. 27, 1–19 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Serôdio, R., Siu, L.S.: Zeros of quaternion polynomials. Appl. Math. Lett. 14(2), 237–239 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Sprößig, W.: On operators and elementary functions in Clifford analysis. Journal for Analysis and its Applications 18(2), 349–360 (1999)MathSciNetGoogle Scholar
  25. 25.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Inc., Englewood Cliffs (1964)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Fernando Miranda
    • 1
  • M. Irene Falcão
    • 1
  1. 1.Departamento de Matemática e Aplicações and Centro de MatemáticaUniversidade do MinhoPortugal

Personalised recommendations