A Landen-type method for computation of Weierstrass functions

TL;DR

This paper introduces a Landen-type transformation for Weierstrass functions applicable to general lattices, providing an efficient quadratic convergence method for computing elliptic functions, periods, and integrals from invariants.

Contribution

It develops a novel Landen-type transformation for Weierstrass functions applicable to all lattices, enabling an effective quadratic convergence algorithm for elliptic computations.

Findings

Quadratic convergence rate of the proposed method

Effective computation of Weierstrass functions and invariants

Applicable to general complex lattices

Abstract

We establish a version of the Landen's transformation for Weierstrass functions and invariants that is applicable to general lattices in complex plane. Using it we present an effective method for computing Weierstrass functions, their periods, and elliptic integral in Weierstrass form given Weierstrass invariants $g_2$ and $g_3$ of an elliptic curve. Similarly to the classical Landen's method our algorithm has quadratic rate of convergence.