Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T13:55:02.600Z Has data issue: false hasContentIssue false
  • English
  • Français

An algorithm for the total, or partial, factorization of a polynomial

Published online by Cambridge University Press:  24 October 2008

M. R. Farmer  and
G. Loizou
M. R. Farmer
Affiliation:
Department of Computer Science, Birkbeck College, London
G. Loizou
Affiliation:
Department of Computer Science, Birkbeck College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

A globally convergent algorithm is presented for the total, or partial, factorization of a polynomial. Firstly, a circle is found containing all the zeros. Secondly, a search procedure locates smaller circles, each containing a zero, and the multiplicities are then calculated. Thirdly, a simultaneous Iteration Function is used to accelerate convergence. The Iteration Function is chosen from a class of such functions derived herein to deal with the general case of multiple zeros; various properties of these functions are also discussed. Finally, sample numerical results are given which demon-strate the effectiveness of the algorithm.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 3 , November 1977 , pp. 427 - 437
Copyright
Copyright © Cambridge Philosophical Society 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Aberth, O.Iteration methods for finding all zeros of a polynomial simultaneously. Math. Comp. 27 (1973), 339–344.CrossRefGoogle Scholar
(2)Alefeld, G. & Herzberger, J.On the convergence speed of some algorithms for the simultaneous approximation of polynomial roots. Siam J. Numer. Anal. 11 (1974), 237–243.CrossRefGoogle Scholar
(3)Alefeld, G. & Herzberger, J.Über Simultanverfahren zur Bestimmung reeller Polynomwurzeln. Z. Angew. Math. Mech. 54 (1974), 413–420.CrossRefGoogle Scholar
(4)Berezin, I. S. & Zhidkov, N. P.Computing methods, vol. 11 (Oxford, Pergamon Press, 1965).Google Scholar
(5)Börsch-Supan, W.A posteriori error bounds for the zeros of polynomials. Numer. Math. 5 (1963), 380–398.CrossRefGoogle Scholar
(6)Braess, D. & Hadeler, K. P.Simultaneous inclusion of the zeros of a polynomial. Numer. Math. 21 (1973), 161–165.Google Scholar
(7)Carrano, F. M.A modified Bairstow method for multiple zeros of a polynomial. Math. Comp. 27 (1973), 781–792.CrossRefGoogle Scholar
(8)Dekker, T. J. Newton-Lagnlerre iteration. Colloq. Internat. Cnrs, no. 165. Programmation en mathématiques numériques (1968), 189–200.Google Scholar
(9)Dochev, K. & Byrnev, P.Certain modifications of Newton's method for the approximate solution of algebraic equations. Ž. Vyčisl. Mat. i Mat. Fiz. 4 (1964), 915–920.Google Scholar
(10)Dunaway, D. K.Calculation of zeros of a real polynomial through factorization using Euclid's algorithm. Siam J. Numer. Anal. 11 (1974), 1087–1104.CrossRefGoogle Scholar
(11)Dvorčuk, J.Factorization of a polynomial into quadratic factors by Newton method. Apl. Mat. 14 (1969), 54–80.Google Scholar
(12)Ehrlich, L. W.A modified Newton method for polynomials. Comm. ACM 10 (1967), 107–108.CrossRefGoogle Scholar
(13)Farmer, M. R. & Lolzou, G.A class of iteration functions for improving, simultaneously, approximations to the zeros of a polynomial. Bit 15 (1975), 250–258.CrossRefGoogle Scholar
(14)Gargantini, I. & Münzner, W. An experimental program for the simultaneous determination of all zeros of a polynomial. Ibm Research Report RZ-237 (1967).Google Scholar
(15)Gargantini, I. & Henrici, P.Circular arithmetic and the determination of polynomial zeros. Numer. Math. 18 (1972), 305–320.CrossRefGoogle Scholar
(16)Garside, G. R., Jarratt, P. & Mack, C.A new method for solving polynomial equations. Comput. J. 11 (1968), 87–90.CrossRefGoogle Scholar
(17)Grau, A. A.The simultaneous Newton improvement of a complete set of approximate factors of a polynomial. SIAM J. Numer. Anal. 8 (1971), 425–438.CrossRefGoogle Scholar
(18)Halley, E.A new, exact and easy method of finding the roots of equations generally, and that without any previous reduction. Philos. Trans. R. Soc. London A 18 136–145.CrossRefGoogle Scholar
(19)Hamilton, H. J.Roots of equations by functional iteration. Duke Math. J. 13 (1946), 113–121.CrossRefGoogle Scholar
(20)Hansen, E. & Patrick, M.A family of root finding methods. Numer. Math. 27 (1977), 257–269.CrossRefGoogle Scholar
(21)Henrici, P. & Watkins, B. O.Finding zeros of a polynomial by the Q-D algorithm. Comm. ACM 8 (1965), 570–574.CrossRefGoogle Scholar
(22)Ishiguro, M.A numerical method for extraction of multiple roots of algebraic equations. Information Processing in Japan 12 (1972), 46–50.Google Scholar
(23)Jenkins, M. A. & Traub, J. F.Principles for testing polynomial zero finding programs. ACM Trans. Math. Software 1 (1975), 26–34.CrossRefGoogle Scholar
(24)Kerner, I. O.Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen. Numer. Math. 8 (1966), 290–294.CrossRefGoogle Scholar
(25)Kiss, I.Über eine Verallgemeinerung des newtonschen Näherungsverfahrens. Z. Angew. Math. Mech. 34 (1954), 68–69.CrossRefGoogle Scholar
(26)Lagouanelle, J. L.Sur une méthode de calcul de l'ordre de multiplicité des zéros d'un polynome. C. R. Acad. Sci. Paris A 262 (1966), 626–627.Google Scholar
(27)Lehmer, D. H.A machine method for solving polynomial equations. J. Assoc. Comput. Mach. 8 (1961), 151–162.CrossRefGoogle Scholar
(28)Ortega, J. M. & Rheinboldt, W. C.Iterative solution of nonlinear equations in several variables (New York, Academic Press, 1970).Google Scholar
(29)Papaconstantinou, C.Iterative solution of the generalized eigenvalue problem. J. Inst. Math. Appl. 17 (1976), 255–260.CrossRefGoogle Scholar
(30)Papaconstantinou, C.On the numerical solution of the generalized eigenproblem. IEEE Trans. Automatic Control AC-20 (1975), 437–439.Google Scholar
(31)Petrić, J., Jovanović, M. & Stamatović, S.Algorithm for simultaneous determination of all roots of algebraic polynomial equations. Mat. Vesnik 9 (1972), 325–332.Google Scholar
(32)Prešić, M. D.Un procédé itératif pour déterminer k zéros d'un polynome. C. R. Acad. Sci. Paris A 273 (1971), 446–449.Google Scholar
(33)Rail, L. B.Convergence of the Newton process to multiple solutions. Numer. Math. 9 (1966), 23–37.CrossRefGoogle Scholar
(34)Stengl, J. E. & Isaacs, C. D.A new solution method for the determinantal equation of a matric polynomial. Proceedings ACM National Conference (1966), pp. 21–27.Google Scholar
(35)Stewart, G. W.III On Leer's method for finding the zeros of a polynomial. Math. Comp. 23 (1969), 829–835.CrossRefGoogle Scholar
(36)Tošić, D. D. & Millovanovlć, G. V.An application of Newton's method to simultaneous determination of zeros of a polynomial. Univ. Beograd. Publ. Elektrotehn. Fak. (Ser. Mat. Piz.), nos. 412–460 (1973), 175–177.Google Scholar
(37)Traub, J. F.Iterative methods for the solution of equations (Englewood Cliffs, New Jersey, Prentice Hall, 1964).Google Scholar
(38)University Of London Computer Centre. Multi-digit computations. Bulletin B5.2/3 (1975).Google Scholar
(39)Wilkinson, J. H.The algebraic eigenvalue problem (Oxford University Press, 1965).Google Scholar