A1 Refereed original research article in a scientific journal
Landen transformations applied to approximation
Authors: Kargar, Rahim; Rainio, Oona; Vuorinen, Matti
Publisher: Yokohama Publications
Publication year: 2024
Journal: Pure and Applied Functional Analysis
Journal name in source: Pure and Applied Functional Analysis
Volume: 9
Issue: 2
First page : 503
Last page: 516
ISSN: 2189-3756
Self-archived copy’s web address: https://arxiv.org/pdf/2212.09336(external)
Preprint address: https://arxiv.org/abs/2212.09336(external)
Abstract
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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