Schröder's method and the infinity point

Víctor Galilea; José Manuel Gutiérrez; Víctor Galilea; José Manuel Gutiérrez

doi:10.3934/math.2025581

AIMS Mathematics

2025, Volume 10, Issue 6: 12919-12934. doi: 10.3934/math.2025581

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Schröder's method and the infinity point

Víctor Galilea ,
José Manuel Gutiérrez ^,

Department of Mathematics and Computer Sciences, Universidad de La Rioja, C/ Madre de Dios, 53, Logroño 26006, La Rioja, Spain

Received: 12 December 2024 Revised: 24 April 2025 Accepted: 09 May 2025 Published: 05 June 2025
MSC : 37F10, 65S05

This paper presents an initial investigation into the dynamic properties of a well-known iterative method for solving nonlinear equations: Schröder's method. We characterize the degree of the rational map induced by applying the method to polynomial equations, along with other dynamical features such as the nature of extraneous fixed points and the presence of attracting cycles. Particular attention is given to the significant dynamical differences between Schröder's method and other iterative methods, notably Newton's method, with a focus on the behavior at infinity.
- complex dynamics,
- numerical methods,
- Newton's method,
- Schröder's method,
- infinity point
Citation: Víctor Galilea, José Manuel Gutiérrez. Schröder's method and the infinity point[J]. AIMS Mathematics, 2025, 10(6): 12919-12934. doi: 10.3934/math.2025581

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Abstract

This paper presents an initial investigation into the dynamic properties of a well-known iterative method for solving nonlinear equations: Schröder's method. We characterize the degree of the rational map induced by applying the method to polynomial equations, along with other dynamical features such as the nature of extraneous fixed points and the presence of attracting cycles. Particular attention is given to the significant dynamical differences between Schröder's method and other iterative methods, notably Newton's method, with a focus on the behavior at infinity.

References

[1]	E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, (German), Mate. Ann., 2 (1870), 317–365.
[2]	G. W. Stewart, On infinitely many algorithms for solving equations, Technical Reports from UMIACS, 1993, TR-2990.
[3]	M. A. Hernández-Verón, A note on Halley's method, Extracta Mathematicae, 3 (1988), 104–106.
[4]	J. F. Traub, Iterative methods for the solution of equations, Englewood Cliffs: Prentice-Hall, 1964.
[5]	G. E. Roberts, J. Horgan-Kobelski, Newton's versus Halley's methods: a dynamical systems approach, Int. J. Bifur. Chaos, 14 (2004), 3459–3475. https://doi.org/10.1142/S0218127404011399 doi: 10.1142/S0218127404011399
[6]	J. M. Gutiérrez, J. L. Varona, Superattracting extraneous fixed points and $n$-cycles for Chebyshev's method on cubic polynomials, Qual. Theory Dyn. Syst., 19 (2020), 54. https://doi.org/10.1007/s12346-020-00390-5 doi: 10.1007/s12346-020-00390-5
[7]	D. K. R. Babajee, A. Cordero, J. R. Torregrosa, Study of iterative methods through the Cayley Quadratic Test, J. Comput. Appl. Math., 291 (2016), 358–369. https://doi.org/10.1016/j.cam.2014.09.020 doi: 10.1016/j.cam.2014.09.020
[8]	B. Campos, E. G. Villalba, P. Vindel, Dynamical and numerical analysis of classical multiple roots finding methods applied for different multiplicities, Comput. Appl. Math., 43 (2024), 230. https://doi.org/10.1007/s40314-024-02746-y doi: 10.1007/s40314-024-02746-y
[9]	P. I. Marcheva, S. I. Ivanov, Convergence and dynamics of Schröder's method for zeros of analytic functions with unknown multiplicity, Mathematics, 13 (2025), 275. https://doi.org/10.3390/math13020275 doi: 10.3390/math13020275
[10]	T. Nayak, S. Pal, The Julia sets of Chebyshev's method with small degrees, Nonlinear Dyn., 110 (2022), 803–819. https://doi.org/10.1007/s11071-022-07648-4 doi: 10.1007/s11071-022-07648-4
[11]	J. M. Gutiérrez, V. Galilea, Two dynamic remarks on the chebyshev-halley family ofiterative methods for solving nonlinear equations, Axioms, 12 (2023), 1114. https://doi.org/10.3390/axioms12121114 doi: 10.3390/axioms12121114
[12]	K. Kneisl, Julia sets for the super-Newton method, Cauchy's method and Halley's method, Chaos, 11 (2001), 359–370. https://doi.org/10.1063/1.1368137 doi: 10.1063/1.1368137
[13]	E. R. Vrscay, W. J. Gilbert, Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions, Numer. Math., 52 (1988), 1–16. https://doi.org/10.1007/BF01401018 doi: 10.1007/BF01401018
[14]	V. Galilea, J. M. Gutiérrez, A characterization of the dynamics of Schröder's method for polynomials with two roots, Fractal Fract., 5 (2021), 25. https://doi.org/10.3390/fractalfract5010025 doi: 10.3390/fractalfract5010025
[15]	S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Scientia Series A: Mathematical Sciences, 10 (2004), 3–35.
[16]	E. R. Vrscay, Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: a computer assisted study, Math. Comp., 46 (1986), 151–169. https://doi.org/10.1090/s0025-5718-1986-0815837-5 doi: 10.1090/s0025-5718-1986-0815837-5
[17]	A. F. Beardon, Iteration of rational functions: complex analytic dynamical systems, New York: Springer, 1991.
[18]	J. Hubbard, D. Schleicher, S. Sutherland, How to find all roots of complex polynomials by Newton's method, Invent. Math., 146 (2001), 1–33. https://doi.org/10.1007/s002220100149 doi: 10.1007/s002220100149

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