Stopping criteria for the Ando-Li-Mathias and Bini-Meini-Poloni geometric means

We provide an upper bound for the number of iterations necessary to achieve a desired level of accuracy for the Ando-Li-Mathias [Linear Algebra Appl. 385 (2004) 305-334] and Bini-Meini-Poloni [Math. Comput. 79 (2010) 437-452] symmetrization procedures for computing the geometric mean of n positive definite matrices, where accuracy is measured by the spectral norm and the Thompson metric on the convex cone of positive definite matrices. It is shown that the upper bound for the number of iterations depends only on the diameter of the set of n matrices and the desired convergence tolerance. A striking result is that the upper bound decreases as n increases on any bounded region of positive definite matrices.