A simple proof of the exponential convergence of the modified Jacobi-Perron algorithm

Abstract

It has recently been shown in Ito et al that the modied JacobiPerron algorithm is strongly convergent in the sense of Brentjes almost everywhere with exponential rate Their proof relies on very complicated computations In this paper we will show that the original paper of Podsypanin

on the modied Jacobi-Perron algorithm almost contains a proof of this convergence with exponential rate The only ingredients missing in that paper are some ergodic-theoretical facts about the transformation generating the approximations This leads to a very simple proof of the beforementioned exponential convergence in the modied JacobiPerron algorithm