New deterministic approaches to the least square mean

In this paper we present new deterministic approximations to the least square mean, also called geometric mean or barycenter of a finite collection of positive definite matrices. Let A1,A2,…,Am be any elements of Md(C)+, where the set Md(C)+ is the open cone in the real vector space of selfadjoint matrices H(n). We consider a sequence of blocks of m matrices, that is,(A1,…,Am,A1,…,Am,A1,…Am,…). We take a permutation on every block and then take the usual inductive mean of that new sequence. The main result of this work is that the inductive mean of this block permutation sequence approximate the least square mean on Md(C)+. This generalizes a Theorem obtain by Holbrook. Even more, we have an estimate for the rate of convergence.