A note on computing matrix geometric means

作者:Bini Dario Andrea*; Iannazzo Bruno
来源:Advances in Computational Mathematics, 2011, 35(2-4): 175-192.
DOI:10.1007/s10444-010-9165-0

摘要

A new definition is introduced for the matrix geometric mean of a set of k positive definite nxn matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires O(n (3) k (2)) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando et al. (Linear Algebra Appl 385:305-334, 2004) have complexity O(n (3) k!2 (k) ). The 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 ten properties stated by Ando, Li and Mathias; a counterexample shows that monotonicity is not fulfilled.

  • 出版日期2011-11