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Global aspects of the continuous and discrete Newton method: A case study

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Abstract

Newton's method has recently become one of the paradigms in the revival of Julia set theory and complex dynamical systems. This paper, to a large extent experimental in nature, investigates Newton's method for some particular model problems as a real dynamical system of several simultaneous equations guided by the Julia set theory.

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References

  1. E. Allgower and K. Georg, Simplicial and continuation methods for approximating fixed points and solutions to systems of equations, SIAM Review 22 (1985), 28–85

    Google Scholar 

  2. P. Barna, Über das Newtonsche Verfahrung zur Annäherung von Wurzeln algebraischer Gleichungen, Publ. Math. Debrecen 2 (1951), 50–63

    Google Scholar 

  3. P. Barna, Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen, I. Publ. Math. Debrecen 3 (1953), 109–118, II. ibid. 4 (1956), 384–397, III. ibid. 8 (1961), 193–207

    Google Scholar 

  4. P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc., 11 (1984), 85–141

    Google Scholar 

  5. F. Branin, A widely convergent method for finding multiple solutions of simultaneous nonlinear equations, IBM J. Res. Devlop. (1972), 504–522

  6. H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144

    Google Scholar 

  7. A. Cayley, Desiderata and suggestions. No. 3.-the Newton-Fourier imaginary problem, Amer. J. Math. 2 (1879), 97

    Google Scholar 

  8. A. Cayley, Application of the Newton-Fourier method to an imaginary root of an equation, Quart. J. Pure Appl. Math. 16 (1879), 179–185

    Google Scholar 

  9. P. Collet and P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Basel, Boston, 1980

    Google Scholar 

  10. J. Curry, L. Garnett and D. Sullivan, On the iteration of a rational function: computer experiments with Newton's method, Comm. Math. Phys. 91 (1983), 267–277

    Google Scholar 

  11. R. L. Devaney, Julia sets bifurcation diagrams for exponential maps, Bull. Amer. Math. Soc. 11 (1984), 167–171

    Google Scholar 

  12. R. L. Devaney, Exploding Julia sets, Proc. Conf. Chaotic Dynamics, Georgia Tech., 1985

  13. K. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, Cambridge, 1985

    Google Scholar 

  14. P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math., France 47 (1919), 161–127; 48 (1920), 33–94, 208–314

    Google Scholar 

  15. C. Grebogi, E. Ott and J. Yorke, Crises, sudden changes in chaotic atrractors and transient chaos, pp. 181–200 in: Order in Chaos, D. Campbell and H. Rose, editors, North-Holland, Amsterdam, 1984

    Google Scholar 

  16. F. v. Haeseler, Über sofortige Attraktionsgebiete superattraktiver Zyklen, Dissertation, University of Bremen, 1985

  17. F. v. Haeseler, H.-O. Peitgen, Newton's Method and Complex Dynamical systems, Acta. Appl. Mach. 13 (1988), this issue

  18. P. Hartmann, Ordinary Differential Equations, J. Wiley, New York, 1964

    Google Scholar 

  19. E. Hille, Analytic Function Theory, vol. 2, Ginn and Co., Boston, 1962

    Google Scholar 

  20. M. Hirsch and S. Smale, Algorthims for solving f(x)=0, Comm. Pure Appl. Math. 32 (1979), 281–312

    Google Scholar 

  21. H. Jongen, P. Jonker and F. Twilt, The continuous Newton method for meromorphic functions, Springer Lecture Notes in Math. 810 (1980), 181–239

    Google Scholar 

  22. G. Julia, Memoire sur l'itération des fonction rationelles, J. Math. Pures et Appl. 81 (1918), 47–235

    Google Scholar 

  23. R. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982

    Google Scholar 

  24. R. Mane, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sc. Ec. Norm. Sup. 161 (1983), 193–217

    Google Scholar 

  25. S. W. McDonald, C. Grebogi, E. Ott and J. Yorke, Fractal basin boundaries, University of Maryland, preprint, 1985

  26. H.-O. Peitgen and P. Richter, The Beauty of Fractals, Springer-Verlag, Berlin, New York, 1986

    Google Scholar 

  27. H.-O. Peitgen, D. Saupe, and F. v. Haeseler, Cayley's problem and Julia sets, Math. Intell. 6 (1984), 11–20, Newton's method and Julia sets, GMD-Studien Nr. 97, 1985

    Google Scholar 

  28. M. Prüfer, Turbulence in multistep methods for initial value problems, SIAM J. Appl. Math. 45 (1985), 32–69

    Google Scholar 

  29. P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mtn. J. Math. 3 (1973), 162–202

    Google Scholar 

  30. D. Saupe, Discrete versus continuous Newton's method: A case study, Acta. Appl. Math. 13 (1988), this issue

  31. S. Smale, A convergent process of price adjustment and global Newton methods, J. Math. Econ. 3 (1976), 107–120

    Google Scholar 

  32. S. Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. 4 (19815), 1–36

    Google Scholar 

  33. D. Sullivan, Quasi conformal homeomorphisms and dynamics, I, II, III, preprints, 1982–1983

  34. S. Ushiki, H.-O. Peitgen, and F. v. Haeseler, Hyperbolic components of rational fractions λ(z+1/z), Proc. Conf. Theory of Dynamical Systems and Applications to Nonlinear Problems, H. Kawakami, ed., World Scientific, Singapore, 1984

    Google Scholar 

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Peitgen, H.O., Prüfer, M. & Schmitt, K. Global aspects of the continuous and discrete Newton method: A case study. Acta Appl Math 13, 123–202 (1988). https://doi.org/10.1007/BF00047504

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