Reliable Computing

, Volume 6, Issue 4, pp 459–470 | Cite as

On Factorization of Analytic Functions and Its Verification

  • Tetsuya Sakurai
  • Hiroshi Sugiura
Article

Abstract

An interval method for finding a polynomial factor of an analytic function f(z) is proposed. By using a Samelson-like method recursively, we obtain a sequence of polynomials that converges to a factor p*(z) of f(z) if an initial approximate factor p(z) is sufficiently close to p*(z). This method includes some well known iterative formulae, and has a close relation to a rational approximation. According to this factoring method, a fixed point relation for p*(z) is derived. Based on this relation, we obtain a polynomial with complex interval coefficients that includes p*(z).

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References

  1. 1.
    Bairstow, L.: The Solution of Algebraic Equations with Numerical Coefficients in the Case Where Several Pairs of Complex Roots Exist, Advisory Committee for Aeronautics, 1914–1915, pp. 239-252.Google Scholar
  2. 2.
    Bauer, F. L. and Samelson, K.: Polynomkerne und iterationsverfahren, Math. Z. 67 (1957), pp. 93-98.Google Scholar
  3. 3.
    Bini, D. A., Gemignani, L., and Meini, B.: Factorization of Analytic Functions by Means of Koenig's Theorem and Toeplitz Computations, 1998 (preprint).Google Scholar
  4. 4.
    Carstensen, C. and Sakurai, T.: Simultaneous Factorization of a Polynomial by Rational Approximation, J. Comput. Appl. Math. 61 (1995), pp. 165-178.Google Scholar
  5. 5.
    Delves, L. M. and Lyness, J. N.: A Numerical Method for Locating the Zeros of an Analytic Function, Math. Comp. 21 (1967), pp. 543-560.Google Scholar
  6. 6.
    Durand, E.: Solutions Numériques des Équations Algébriques, Masson, Paris, 1960.Google Scholar
  7. 7.
    Grau, A. A.: The Simultaneous Newton Improvement of a Complete Set of Approximate Factors of a Polynomial, SIAM J. Numer. Anal. 8 (1971), pp. 425-438.Google Scholar
  8. 8.
    Householder, A. S.: The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964.Google Scholar
  9. 9.
    Hribernig, V. and Stetter, H. J.: Detection and Validation of Clusters of Polynomial Zeros, J. Symbolic Computation 24 (1997), pp. 667-681.Google Scholar
  10. 10.
    Jenkins, M. A. and Traub, J. F.: A Three-Stage Variable-Shift Iteration for Polynomial Zeros and Its Relation to Generalized Rayleigh Iteration, Numer. Math. 14 (1970), pp. 252-263.Google Scholar
  11. 11.
    Kirrinnis, P.: Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z], J. Complexity 14(3) (1998), pp. 378-444.Google Scholar
  12. 12.
    Kravanja, P., Sakurai, T., and Van Barel, M.: A Method for Finding Clusters of Zeros of Analytic Function, in: Leuven, K. U., Dept. Computer Science Report, TW 280, 1998.Google Scholar
  13. 13.
    Li, T. Y.: On Locating All Zeros of an Analytic Function within a Bounded Domain by a Revised Delves/Lyness Method, SIAM J. Numer. Anal. 20 (1983), pp. 865-871.Google Scholar
  14. 14.
    Petković, M., Herceg, D., and Ilić, S.: Point Estimation Theory and Its Applications, Institute of Mathematics, Movi Sad, 1997.Google Scholar
  15. 15.
    Rump, S. M.: INTLAB—INTerval LABoratory, http://www.ti3.tu-harburg.de/~rump/intlab/index.html.Google Scholar
  16. 16.
    Sakurai, T., Torii, T., Ohsako, N., and Sugiura, H.: A Method for Finding Clusters of Zeros of Analytic Function, in: Proc. ICIAM'95, Hamburg, 1996, pp. 515-516.Google Scholar
  17. 17.
    Sonoda, S., Sakurai, T., Sugiura, H., and Torii, T.: Numerical Factorization of Polynomial by the Divide and Conquer Method, Trans. Japan SIAM 1 (1991), pp. 277-290 (in Japanese).Google Scholar
  18. 18.
    Stewart, G. W.: On a Companion Operator for Analytic Functions, Numer. Math. 18 (1971), pp. 26-43.Google Scholar
  19. 19.
    Stewart, G. W.: On Samelson's Iteration for Factoring Polynomials, Numer. Math. 15 (1970), pp. 306-314.Google Scholar
  20. 20.
    Torii, T. and Sakurai, T.: Global Method for the Poles of Analytic Function by Rational Interpolant on the Unit Circle, World Sci. Ser. Appl. Anal. 2 (1993), pp. 389-398.Google Scholar
  21. 21.
    Torii, T., Sakurai, T., and Sugiura, H.: An Application of Sunzi's Theorem for Solving Algebraic Equations, in: Proc. 1st China-Japan Seminar on Numerical Mathematics, 1993, pp. 155-167.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Tetsuya Sakurai
    • 1
  • Hiroshi Sugiura
    • 2
  1. 1.Institute of Information Sciences and ElectronicsUniversity of TsukubaTsukubaJapan
  2. 2.Dept. of Information EngineeringNagoya UniversityNagoyaJapan

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