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