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Advances in Computational Mathematics

, Volume 35, Issue 2–4, pp 175–192 | Cite as

A note on computing matrix geometric means

  • Dario Andrea Bini
  • Bruno Iannazzo
Article

Abstract

A new definition is introduced for the matrix geometric mean of a set of k positive definite n×n 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.

Keywords

Matrix geometric mean Matrix function Riemannian centroid Geodesic 

Mathematics Subject Classifications

65F30 15A15 

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Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly

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