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Landen transformations applied to approximation
Tekijät: Kargar, Rahim; Rainio, Oona; Vuorinen, Matti
Kustantaja: Yokohama Publications
Julkaisuvuosi: 2024
Journal: Pure and Applied Functional Analysis
Tietokannassa oleva lehden nimi: Pure and Applied Functional Analysis
Vuosikerta: 9
Numero: 2
Aloitussivu: 503
Lopetussivu: 516
ISSN: 2189-3756
Rinnakkaistallenteen osoite: https://arxiv.org/pdf/2212.09336
Preprintin osoite: https://arxiv.org/abs/2212.09336
Tiivistelmä
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 φk(r) which gives a majorant for |f(x)| when f: B2 → B2, f(0) = 0, is a quasiconformal mapping of the unit disk B2. It turns out that the approximation method is very simple: Five steps of Landen iteration are enough to achieve machine precision.
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 φk(r) which gives a majorant for |f(x)| when f: B2 → B2, f(0) = 0, is a quasiconformal mapping of the unit disk B2. It turns out that the approximation method is very simple: Five steps of Landen iteration are enough to achieve machine precision.
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