Bivariate Newton-Raphson method and toroidal attraction basins

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• 作者：Luis Javier Hernández Paricio
• 关键词：Roots of polynomial equations ; Bivariate Newthon ; Raphson method ; Discrete semi ; flow ; Basin of attraction ; Bivariate polynomials ; Intersection of algebraic curves ; Toroidal fractals ; Toroidal basins of attraction ; 65H04 ; 65S05 ; 68W25
• 刊名：Numerical Algorithms
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：71
• 期：2
• 页码：349-381
• 全文大小：2,458 KB
• 参考文献：1.Abate, M.: Discrete homomorphic local dynamical systems, in Holomorphic Dynamical Systems. Lect. Notes Math., 1–55 (2010)
  2.Altman, M.: A generalization of Newton’s method. Bull. Acad. Polon. Sci. 3, 189–193 (1955)MathSciNet MATH
  3.Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer (2008)
  4.Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Scientia, Ser. A Math. Sci. 10, 3–35 (2004)MathSciNet MATH
  5.Bartle, R.G.: Newton’s method in Banach spaces. Proc. Amer. Math. Soc. 6, 827–831 (1955)MathSciNet MATH
  6.Beardon, A.F.: Iteration of Rational Functions, Graduate Texts in Mathematics. Springer-Verlag, New York (1991)CrossRef
  7.Chun, C., Neta, B., Kozdon, J., Scott, M.: Choosing weight functions in iterative methods for simple roots. Appl. Math. Comput. 227, 788–800 (2014)CrossRef MathSciNet
  8.Eskin, A., Mirzakhani, M.: Invariant and stationary measures for the SL2(R) action on moduli space, preprint 2013; arXiv:1302.​3320
  9.Eskin, A., Rzakhani, M.M., Mohammadi, A.: Isolation, equidistribution, and orbit closures for the SL2(R) action on moduli space, preprint, 2013; preprint, 2013; arXiv:1305.​3015
  10.García-Calcines, J.M., Hernández, L.J., Rivas, M.T.: Limit and end functors of dynamical systems via exterior spaces. Bull. Belg. Math. Soc. Simon Stevin 20, 937–957 (2013)MathSciNet MATH
  11.García-Calcines, J.M., Hernández, L.J., Rivas, M.T.: A completion construction for continuous dynamical systems. Topology Methods Nonlinear Analytical 44(2), 497–526 (2014)
  12.Gutiérrez, J.M., Hernández-Paricio, L.J., Marañón-Grandes, M.: Influence of the multiplicity of the roots on the basins of attraction of Newton’s method. Numer. Algorithm. 66(3), 431–455 (2014)CrossRef MATH
  13.Hernández, L.J., Marañón, M., Rivas, M.T.: Plotting basins of end points of rational maps with Sage. Tbilisi Math. J. 5(2), 71–99 (2012)MATH
  14.Hildebrandt, T.H., Graves, L.M.: Implicit functions and their differentials in general analysis. Trans. Amer. Math. Sot. 29, 127–153 (1927)CrossRef MathSciNet MATH
  15.Kantorovic, L.V.: On Newton’s method for functional equations. Dokl. Akad. Nauk SSSR 59, 1237–1240 (1948)MathSciNet MATH
  16.Kantorovic, L.V.: Functional analysis and applied mathematics. Uspehi Mat. Nauk (N.S.) 3(28), 89–185 (1948). No. (6)MathSciNet
  17.Kantorovic, L.V.: On Newton’s method. Trudy Mat. Inst. Steklov. 28, 104–144 (1949)MathSciNet
  18.Lewis, O.: Gereralized Julia sets: An extension of Cayley’s problem, Doctoral thesis, Harvey Mudd College (Claremont, California), Department of Mathematics (2005)
  19.McClure, M.: Newton’s method for complex polynomials. Math. Educ. Res. 11(2) (2006)
  20.Neta, B., Chun, C., Scott, M.: Basins of attraction for several methods to find simple roots of nonlinear equations. Appl. Math. Comput. 218, 10548–10556 (2012)CrossRef MathSciNet MATH
  21.Neta, B., Chun, C., Scott, M.: On the development of iterative methods for multiple roots. Appl. Math. Comput. 224, 358–361 (2013)CrossRef MathSciNet
  22.Ortega, J.M., Rheinbold, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM (2000)
  23.Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1960)MATH
  24.Peitgen, H.O., Jürgens, H., Saupe, D.: Cayley’s problem and Julia sets. Math. Intell. 6(2), 11–20 (1984)CrossRef MATH
  25.Schröder, J.: Über das Newtonsche Verfahren. Arch. Rat. Mech. Anal. 1, 154–180 (1957)CrossRef MATH
  26.Shaw, W.T.: Complex Analysis with Mathematica. Cambridge University Press (2006)
  27.Stein, M.L.: Sufficient conditions for the convergence of Newton’s method in complex Banach spaces. Proc. Amer. Math. Soc. 3, 858–863 (1952)MathSciNet MATH
  28.Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall (1964)
  29.Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24(1), 37–46 (2002)CrossRef MathSciNet MATH
  30.Yang, W.: Symmetries in the Julia sets of Newton’s method for multiple roots. Appl. Math. Comput. 217, 2490–2494 (2010)CrossRef MathSciNet MATH
  31.Wang, X., Chang, P.: Research on fractal structure of generalized M-J sets utilized Lyapunov exponents and periodic scanning techniques. Appl. Math. Comput. 175(2), 1007–1025 (2006)CrossRef MathSciNet MATH
  32.Wang, X.-Y., Chang, P.-J., Ni-ni, G.: Additive perturbed generalized Mandelbrot-Julia sets. Appl. Math. Comput. 189(1), 754–765 (2007)CrossRef MathSciNet MATH
  33.Xingyuan, W., Ge, F.: The quasi-sine Fibonacci hyperbolic dynamic system. Fractals 18(1), 45–51 (2010)CrossRef MathSciNet
  34.Wang, X., Jia, R., Sun, Y.: The Generalized Julia Set Perturbed by Composing Additive and Multiplicative Noises. Discrete Dyn. Nat. Soc. 2009 (2009). Article ID 781976, 18 pages
  35.Xingyuan, W., Ruihong, J., Zhenfeng, Z.: The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative. Appl. Math. Comput. 210(1), 107–118 (2009)CrossRef MathSciNet MATH
  36.Xingyuan, W., Bo, L.: Julia sets of the Schröder iteration functions of a class of one-parameter polynomials with high degree. J. Appl. Math. Comput. 178(2), 461–473 (2006)CrossRef MathSciNet MATH
  37.Wang, X., Luo, C.: Generalized Julia sets from a non-analytic complex mapping. Appl. Math. Comput. 181(1), 113–122 (2006)CrossRef MathSciNet MATH
  38.Xingyuan, W., Li, Y.-K., Sun, Y.-Y., Song, J.-M., Ge, F.-D.: Julia sets of Newton’s method for a class of complex-exponential function F(z)=P(z)e Q(z). Nonlinear Dynam. 62(4), 955–966 (2010)CrossRef MathSciNet
  39.Xingyuan, W., Wei, L., Xuejing, Y.: Research on brownian movement based on generalized mandelbrot-julia sets from a class complex mapping system. Modern Phys. Lett. B 21(20), 1321–1341 (2007)CrossRef MATH
  40.Xing-Yuan, W., Qing-Ye, M.: Study on the physical meaning for generalized Mandelbrot-Julia sets based on the Langevin problem. Acta Phys. Sin. 53(2), 388–395 (2004)MATH
  41.Xingyuan, W., Qijiang, S.: The generalized Mandelbort-Julia sets from a class of complex exponential map. Appl. Math. Comput. 181(2), 816–825 (2006)CrossRef MathSciNet MATH
  42.Wang, Xing-Y., Song, Wen-J.: The generalized M-J sets for bicomplex numbers. Nonlinear Dynam. 72(1-2), 17–26 (2013)CrossRef MathSciNet MATH
  43.Wang, Xing-Y., Song, Wen-J.: Hyperdimensional generalized M-J sets in hypercomplex number space. Nonlinear Dynam. 73(1-2), 843–852 (2013)CrossRef MathSciNet
  44.Wang, X.-Y., Sun, Y.-Y.: The general quaternionic M-J sets on the mapping image. Comput. Math. Appl. 53(11), 1718–1732 (2007)CrossRef MathSciNet MATH
  45.Xingyuan, W., Wang, T.: Julia sets of generalized Newton’s method. Fractals 15(4), 323–336 (2007)CrossRef MathSciNet MATH
  46.Wang, X., Wang, Z., Lang, Y., Zhang, Z.: Noise perturbed generalized Mandelbrot sets. J. Math. Anal. Appl. 347(1), 179–187 (2008)CrossRef MathSciNet MATH
  47.Xingyuan, Wang, Wei, Liu: The Julia set of Newton’s method for multiple root. Appl. Math. Comput. 172(1), 101–110 (2006)CrossRef MathSciNet MATH
  48.Xingyuan, W., Xuejing, Y.: Julia sets for the standard Newton’s method, Halley’s method, and Schröder’s method. Appl. Math. Comput. 189(2), 1186–1195 (2007)CrossRef MathSciNet MATH
  49.Xingyuan, W., Xuejing, Y.: Julia set of the Newton transformation for solving some complex exponential equation. Fractals 17(2), 197–204 (2009)CrossRef MathSciNet MATH
  
• 作者单位：Luis Javier Hernández Paricio (1)
  
  1. Department of Mathematics and Computer Science, University of La Rioja, 2 Luis de Ulloa St., 26004, Logroño, Spain
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

When the numerical Newton-Raphson method is applied to find the intersections of two algebraic curves (that is, the roots of a pair of bivariate polynomials), some difficulties appear when the value of a denominator of the corresponding bivariate rational functions is zero. In this paper we give a solution to these problems by using adequate homogeneous coordinates and extending the domain of the iteration function. The iteration of a map given by a pair of bivariate rational maps is analyzed by taking a canonical extension, which is defined on the product of two copies of an (real or complex) augmented projective line. This method gives a global description of the basins of attraction of fixed points of an iteration. In particular, the attraction basins of fixed points associated to the intersection points of the two algebraic curves is obtained. As a consequence of our techniques, we are able to plot the attraction basins of a real root either in a torus (considered as a compactification of the real plane) or in an open square, which is homeomorphic to the global real plane containing the two algebraic curves. Keywords Roots of polynomial equations Bivariate Newthon-Raphson method Discrete semi-flow Basin of attraction Bivariate polynomials Intersection of algebraic curves Toroidal fractals Toroidal basins of attraction