Séminaire du 7 novembre 05, Régis Dupont, LIX, École polytechnique.
Arithmetic-Geometric Means of complex numbers and evaluation of modular
functions
We describe the deep connection, already studied by Gauss, between
the arithmetic-geometric mean (AGM) of complex numbers and modular forms
known as theta constants. We then show how this can be used to devise an
algorithm for the numerical evaluation of modular functions (such as the
elliptic function j) with a quasi-optimal time complexity. As a byproduct, we
also obtain an algorithm for the numerical evaluation of complex logarithms,
again with quasi-optimal time complexity.
Virginie Collette
Last modified: Mon May 23 18:32:54 CEST 2005