Computing the Karcher mean of symmetric positive definite matrices

We present and analyze an iterative method for approximating the Karcher mean of a set of $n\times n$ positive definite matrices $A_i$, $i=1\text{,}\dots \text{,}k$, defined as the unique positive definite solution of the matrix equation ${\sum }_{i=1}^k\log (A_i^{-1}X)=0$.