On two families of iterative methods without memory
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913754555351040 |
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| author | Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc |
| author_facet | Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc |
| contents | We study two natural families of methods of order $n\ge 2$ that are useful for solving numerically one variable equations $f(x)=0.$ The first family consists on the methods that depend on $x,f(x)$ and its successive derivatives up to $f^{(n-1)}(x)$ and the second family comprises methods that depend on $x,g(x)$ until $g^{\circ n}(x),$ where $g^{\circ m}(x)=g(g^{\circ (m-1)}(x))$ and $g(x)=f(x)+x$. The first family includes the well-known Newton, Chebyshev, and Halley methods, while the second one contains the Steffensen method. Although the results for the first type of methods are well known and classical, we provide new, simple, detailed, and self-contained proofs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On two families of iterative methods without memory Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc Numerical Analysis Dynamical Systems 65H05, 58C30 We study two natural families of methods of order $n\ge 2$ that are useful for solving numerically one variable equations $f(x)=0.$ The first family consists on the methods that depend on $x,f(x)$ and its successive derivatives up to $f^{(n-1)}(x)$ and the second family comprises methods that depend on $x,g(x)$ until $g^{\circ n}(x),$ where $g^{\circ m}(x)=g(g^{\circ (m-1)}(x))$ and $g(x)=f(x)+x$. The first family includes the well-known Newton, Chebyshev, and Halley methods, while the second one contains the Steffensen method. Although the results for the first type of methods are well known and classical, we provide new, simple, detailed, and self-contained proofs. |
| title | On two families of iterative methods without memory |
| topic | Numerical Analysis Dynamical Systems 65H05, 58C30 |
| url | https://arxiv.org/abs/2503.18815 |