Landen transformations applied to approximation
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909206352756736 |
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| author | Kargar, Rahim Rainio, Oona Vuorinen, Matti |
| author_facet | Kargar, Rahim Rainio, Oona Vuorinen, Matti |
| contents | We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as inverses of such quotients. In particular, we consider the distortion function $φ_K(r)$ which gives a majorant for $|f(x)|$ when $f: \mathbb{B}^2 \to \mathbb{B}^2, f(0)=0,$ is a quasiconformal mapping of the unit disk $\mathbb{B}^2.$ It turns out that the approximation method is very simple: five steps of Landen iteration is enough to achieve machine precision. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09336 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Landen transformations applied to approximation Kargar, Rahim Rainio, Oona Vuorinen, Matti Complex Variables Classical Analysis and ODEs Primary: 30C62, Secondary: 33C75 We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as inverses of such quotients. In particular, we consider the distortion function $φ_K(r)$ which gives a majorant for $|f(x)|$ when $f: \mathbb{B}^2 \to \mathbb{B}^2, f(0)=0,$ is a quasiconformal mapping of the unit disk $\mathbb{B}^2.$ It turns out that the approximation method is very simple: five steps of Landen iteration is enough to achieve machine precision. |
| title | Landen transformations applied to approximation |
| topic | Complex Variables Classical Analysis and ODEs Primary: 30C62, Secondary: 33C75 |
| url | https://arxiv.org/abs/2212.09336 |