Computing the Karcher mean of symmetric positive definite matrices (Q1940322)

This paper deals with a new iteration procedure for computing the Karcher mean of \(k\) symmetric positive definite \(n \times n\) matrices. It is based on an iteration formula expressed by a gradient descent algorithm and subsequently, on its linearization depending on a certain parameter \(\vartheta\). Its global quadratic convergence in the commutative case and its local linear convergence in the general case are proved; a criterion for determining a value of \(\vartheta\) which guarantees local convergence is provided. It is shown by numerical experiments that starting with the cheap mean the iteration converges quickly to the Karcher mean. An adaptive version of this iteration procedure is presented in [\url{http://bezout.dm.unipi.it/software/mmtoolbox/}].