Geometric Mean Algorithms Based on Harmonic and Arithmetic Iterations

The geometric mean of a series of positive numbers a 1,…,a n is defined as the nth root of its product: \(\sqrt[n]{a_1\cdots a_n}\). Generalizing this concept to positive definite matrices is not straightforward due to the noncommutativity. Based on a list of desired properties –the ALM-list– initially some recursive algorithms were proposed. Unfortunately, these algorithms were typically quite time consuming. A novel optimization based approach leads to the concept of the Karcher mean and is nowadays favored as the generalization towards matrices of the geometric mean.