Numerical Algorithms

, Volume 3, Issue 1, pp 411–418 | Cite as

Numerical factorization of a polynomial by rational Hermite interpolation

  • Tetsuya Sakurai
  • Hiroshi Sugiura
  • Tatsuo Torii
Article

Abstract

We derive a class of iterative formulae to find numerically a factor of arbitrary degree of a polynomialf(x) based on the rational Hermite interpolation. The iterative formula generates the sequence of polynomials which converge to a factor off(x). It has a high convergence order even for a factor which includes multiple zeros. Some numerical examples are also included.

Subject classification

AMS (MOS) Primary: 65D15 Secondary: 65H05 

Keywords

Rational Hermite interpolant root finding algorithm algebraic equation Euclidean algorithm numerical factorization 

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References

  1. [1]
    L. Bairstow, The solution of algebraic equations with numerical coefficients in the case where several pairs of complex roots exist, Advisory Committee for Aeronautics, Technical Report for 1914–15 (1915) pp. 239–252.Google Scholar
  2. [2]
    F. Cordellier, On the use of Kronecker's algorithm in the generalized rational interpolation problem, Numer. Algorithms 1 (1991) 401–413.Google Scholar
  3. [3]
    T.L. Freeman, A divide and conquer method for polynomial zeros, J. Comput. Appl. Math. 30 (1990) 71–79.CrossRefGoogle Scholar
  4. [4]
    R.J. McEliece and J.B. Shearer, A property of Euclid's algorithm and an application to Padé approximation, SIAM J. Appl. Math. 34 (1978) 611–615.CrossRefGoogle Scholar
  5. [5]
    A.W. Nourein, Root determination by use of Padé approximants, BIT 16 (1976) 291–297.CrossRefGoogle Scholar
  6. [6]
    T. Sakurai, H. Sugiura and T. Torii, An iterative method for algebraic equation by Padé approximation, Computing 46 (1991) 131–141.Google Scholar
  7. [7]
    T. Sakurai, H. Sugiura and T. Torii, A high order iterative formula for simultaneous determination of zeros of a polynomial, J. Comput. Appl. Math. 38 (1991) 387–397.CrossRefMathSciNetGoogle Scholar
  8. [8]
    S. Sonoda, T. Sakurai, H. Sugiura and T. Torii, Numerical factorization of a polynomial by the divide and conquer method, Trans. Japan SIAM 1 (1991) 277–290 (in Japanese).Google Scholar
  9. [9]
    G.W. Stewart, Some iterations for factoring a polynomial, Numer. Math. 13 (1969) 458–471.Google Scholar

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1992

Authors and Affiliations

  • Tetsuya Sakurai
    • 1
  • Hiroshi Sugiura
    • 1
  • Tatsuo Torii
    • 1
  1. 1.Department of Information EngineeringNagoya UniversityNagoyaJapan

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