  In this paper we presents 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.
