A new definition is introduced for the matrix geometric mean of a set of k positive definite $n\times n$ matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires $O(n^3k^2)$ arithmetic operations per step whereas the methods based on the symmetrization technique of Ando, Li and Mathias [Linear Algebra Appl., 385 (2004), pp. 305–334] have complexity $O(n^3k!2^k)$. he new mean is obtained from the properties of the centroid of a triangle rephrased in terms of geodesics in a suitable Riemannian geometry on the set of positive definite matrices. It satisfies most part of the 10 properties stated by Ando, Li and Mathias; a counterexample shows that monotonicity is not fulfilled.
|Autori:||BINI D; IANNAZZO B.|
|Titolo:||A note on computing matrix geometric means|
|Anno del prodotto:||2011|
|Digital Object Identifier (DOI):||10.1007/s10444 -010-9165-0|
|Appare nelle tipologie:||1.1 Articolo in rivista|