Algorithms associated with arithmetic, geometric and harmonic means and integrable systems

Gauss’ algorithm for arithmetic–geometric mean (AGM) can be regarded as a discrete-time integrable dynamical system having an elliptic theta function solution and a conserved quantity. In this paper we consider algorithms associated with arithmetic, geometric and harmonic means from a viewpoint of integrable systems. First, a max-plus limit and its inverse limit of the AGM algorithm are discussed. These mean operations are shown to be connected to each other by the max-plus limit. Secondly, continous-time dynamical systems associated with the arithmetic–harmonic mean (AHM) algorithm are found. Thirdly, it is shown that the AHM algorithm in indefinite case has a chaotic dynamics and is a generator of numbers which obey the Cauchy distribution. Finally, an extension of the AHM algorithm to the space of positive-definite symmetric matrices is considered.