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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.05811 |
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| _version_ | 1866913684761083904 |
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| author | Mandal, Nitai Chakraborty, Gorachand |
| author_facet | Mandal, Nitai Chakraborty, Gorachand |
| contents | We study the dynamics of Stirling's iterative root-finding method $St_f(z)$ for rational and polynomial functions. It is seen that the Scaling theorem is not satisfied by Stirling's iterative root-finding method. We prove that for a rational function $R(z)$ with simple zeroes, the zeroes are the superattracting fixed points of $St_{R}(z)$ and all the extraneous fixed points of $St_{R}(z)$ are rationally indifferent. For a polynomial $p(z)$ with simple zeroes, we show that the Julia set of $St_p(z)$ is connected. Also, the symmetry of the dynamical plane and free critical orbits of Stirling's iterative method for quadratic unicritical polynomials are discussed. The dynamics of this root-finding method applied to Möbius map is investigated here. We have shown that the possible number of Herman rings of this method for Möbius map is at most $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05811 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the dynamics of Stirling's iterative root-finding method for rational functions Mandal, Nitai Chakraborty, Gorachand Complex Variables Dynamical Systems We study the dynamics of Stirling's iterative root-finding method $St_f(z)$ for rational and polynomial functions. It is seen that the Scaling theorem is not satisfied by Stirling's iterative root-finding method. We prove that for a rational function $R(z)$ with simple zeroes, the zeroes are the superattracting fixed points of $St_{R}(z)$ and all the extraneous fixed points of $St_{R}(z)$ are rationally indifferent. For a polynomial $p(z)$ with simple zeroes, we show that the Julia set of $St_p(z)$ is connected. Also, the symmetry of the dynamical plane and free critical orbits of Stirling's iterative method for quadratic unicritical polynomials are discussed. The dynamics of this root-finding method applied to Möbius map is investigated here. We have shown that the possible number of Herman rings of this method for Möbius map is at most $2$. |
| title | On the dynamics of Stirling's iterative root-finding method for rational functions |
| topic | Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2502.05811 |