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.
A note on computing matrix geometric means
BINI, DARIO ANDREA;
2011-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
