Some reflections on Newton’s method

  • F. Twilt
Part of the Mathematics and Its Applications book series (MAIA, volume 81)


An infinitesimal version (Newton flow) of Newton’s iteration for finding zeros of rational functions on the complex plane is introduced. Structural stability aspects are discussed, including a characterization and classification of structurally stable Newton flows in terms of certain plane graphs. Possible generalizations to other classes of functions are indicated and several open problems are posed.


Newton flow structural stability rational function 

1991 Mathematics Subject Classification

05C10 30C15 34D30 49M15 


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  1. [I-1]
    J. Stoer and R. Bulirsch: Introduction to numerical analysis Springer, New York (1980).Google Scholar
  2. [I-2]
    D.G. Luenberger: Introduction to linear and nonlinear programming Addison-Wesley, London (1973).zbMATHGoogle Scholar
  3. [I-3]
    G.W. Gear: Numerical initial value problems in ordinary differential equations Prentice-Hall, New York (1971).zbMATHGoogle Scholar
  4. [I-4]
    E. van Groesen: A kaleidoscopic excursion into numerical calculations of differential equations This issue (1992).Google Scholar
  5. [I-5]
    H.Th. Jongen, P. Jonker and F. Twilt: Nonlinear optimization in R npart II, Methoden und Verfahren der Mathematischen Physik32 Peter Lang, Frankfurt a M., Bern, New York (1986).Google Scholar
  6. [I-6]
    A.I. Markushevich: Theory of functions of a complex variable Vol. II, Englewood Cliffs, Prentice Hall (1965).Google Scholar
  7. [I-7]
    S. Lefschetz: Differential equations: Geometric theory Interscience Publ.Google Scholar
  8. [I-8]
    A.A. Andronov, E.L. Leontovich, I.I. Gordon and A.G. Maier: Theory of bifurcations of dynamical systems on a plane Wiley, New York (1973).Google Scholar
  9. [I-9]
    J.W. Milnor: Topology from a differential viewpoint Univ. Press of Virginia (1965).Google Scholar
  10. [I-10]
    P.J. Giblin: Graphs surfaces and homology ,Wiley, New York (1977).zbMATHGoogle Scholar
  11. [I-11]
    O.L. Mangasarian: Nonlinear programming Mc. Graw Hill, New York (1986).Google Scholar
  12. [I-12]
    L. Mirsky: Transversal theory Acad. Press, New York (1971).zbMATHGoogle Scholar
  13. [I-13]
    A.I. Markushevich: Theory of functions of a complex variable Vol.III, Englewood Cliffs, Prentice Hall (1967).zbMATHGoogle Scholar
  14. [A-1]
    E. Hairer and G. Wanner: Solving ordinary differential equations Vol. 2 , Stiff and differential-algebraic problems Springer Series in Computational Mathematics14 Springer, Berlin (1991).Google Scholar
  15. [A-2]
    D. Braess: Ueber die Einzugbereiche der Nullstellen von Polynomen beim Newton- Verfahren Num. Math.29 pp. 123–132 (1977).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [A-3]
    F.H. Branin: A widely convergent method for finding multiple solutions of simultaneous nonlinear equations IBM J. Res. Develop., pp. 504–522 (1972).Google Scholar
  17. [A-4]
    I. Diener: On the global convergence of pathfollowing methods to determine all solutions to a system of nonlinear equations Math. Prog.39 pp. 181–188 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  18. [A-5]
    I. Diener: Trajectory nets connecting all critical points of a smooth function Math. Prog.36 pp. 340–353 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  19. [A-6]
    C.B. Garcia and F.J. Gould: Relation between several pathfollowing algorithms and local and global Newton methods SIAM Review22 no. 3, pp. 263–274 (1980).MathSciNetzbMATHCrossRefGoogle Scholar
  20. [A-7]
    M.W. Hirsch and S. Smale: Algorithms for solving f ( x ) = 0, Comm. Pure Appl. Math.32 , pp. 281–312, (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  21. [A-8]
    H.Th. Jongen, P. Jonker and F. Twilt: The continuous Newton method for meromorphic functions In: Geometric approaches to differential equations (R. Martini, ed.), Lect. Notes in Math.,810 Springer, pp. 181–239 (1980).CrossRefGoogle Scholar
  22. [A-9]
    H.Th. Jongen, P. Jonker and F. Twilt: The continuous desingularized Newton method for meromorphic functions Acta Appl. Math.13 pp. 81–121 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  23. [A-10]
    H.Th. Jongen, P. Jonker and F. Twilt: On the classification of plane graphs representing structurally stable rational Newton flows Journal of Comb. Theor. Series B,15 -2, pp. 256–270 (1991).MathSciNetCrossRefGoogle Scholar
  24. [A-11]
    H.B. Keller: Global homotopies and Newton methods In: Recent advances in numerical analysis (C. de Boor, G.H. Golub, eds.), Acad. Press, pp. 73–74 (1978).Google Scholar
  25. [A-12]
    M. Shub, D. Tischler and R.F. Williams: The Newtonian graph of a complex polynomial SIAM J. Math. Anal.,19 -1, pp. 246–256 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  26. [A-13]
    S. Smale: A convergent process of price adjustment and global Newton methods J. Math. Economics3 pp. 107–120 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  27. [A-14]
    S. Smale: On the efficiency of algorithms of analysis Bull. Am. Math. Soc.,13 -2, pp. 87–121 (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  28. [A-15]
    M.C. Peixoto: Structural stability on two dimensional manifolds Topology1 pp. 101–120 (1962).MathSciNetzbMATHCrossRefGoogle Scholar
  29. [A-16]
    M.C. Peixoto: On the classification of flows on 2-manifolds In: Dynamical Systems (M.M. Peixoto, ed.), Acad. Press, New York, pp. 389–419 (1973).Google Scholar
  30. [A-17]
    F. Harary, G. Prins and W.T. Tutte: The number of plane trees Indag. Math.26 (1964).Google Scholar
  31. [A-18]
    G.F. Helminck, M. Streng and F. Twilt: The qualitative behaviour of Newton flows for Weierstrass P-functions In preparation.Google Scholar
  32. [A-19]
    F. Twilt, P. Jonker and M. Streng: Gradient Newton flows for complex polynomials In: Proc. Fifth French-German Conference on Optimization, Lect. Notes in Math.1405 (S. Dolecki, ed.), pp. 177–190 (1989).Google Scholar
  33. [A-20]
    W. Wesselink: De continue Newton methode voor functies van de vorm R 1( z ) exp( R 2( z )) met R 1( z ) en R 2( z ) rationaal D-report, Dept. of Appl. Math., Univ. of Twente (1991).Google Scholar
  34. [A-21]
    H.G. Meier: Diskrete und kontinuierliche Newton Systeme im Komplexen Ph.D. thesis, RWTH Aachen (1991)zbMATHGoogle Scholar
  35. [A-22]
    W. Bergweiler et al.: Newton’s method for meromorphic functions Preprint31 Lehrstuhl C für Mathematik, RWTH Aachen (1991).Google Scholar
  36. [A-23]
    H.Th. Jongen, P. Jonker and F. Twilt: A note on Branin’s method for finding the critical points of smooth mappings In: Parametric optimization and related topics, (J. Guddat, H.Th. Jongen, B. Kummer and F. Nozicka, eds.), Akad. Verlag, Berlin (1987).Google Scholar
  37. [A-24]
    M. de Graaf: The continuous Newton method in three dimensions D—report, Dept. of Appl. Math., Univ. of Twente (1990).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • F. Twilt
    • 1
  1. 1.Faculty of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

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