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An effective matrix geometric mean satisfying the Ando-Li-Mathias properties


Authors: Dario A. Bini, Beatrice Meini and Federico Poloni
Journal: Math. Comp. 79 (2010), 437-452
MSC (2000): Primary 65F30; Secondary 15A48, 47A64
DOI: https://doi.org/10.1090/S0025-5718-09-02261-3
Published electronically: June 19, 2009
MathSciNet review: 2552234
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Abstract: We propose a new matrix geometric mean satisfying the ten properties given by Ando, Li and Mathias [Linear Alg. Appl. 2004]. This mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by Ando, Li and Mathias is the limit of a sequence having order of convergence 1. This makes this new mean very easily computable. We provide a geometric interpretation and a generalization which includes as special cases our mean and the Ando-Li-Mathias mean.


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Additional Information

Dario A. Bini
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy
Email: bini@dm.unipi.it

Beatrice Meini
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy
Email: meini@dm.unipi.it

Federico Poloni
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 6, 56126 Pisa, Italy
Email: poloni@sns.it

DOI: https://doi.org/10.1090/S0025-5718-09-02261-3
Keywords: Matrix geometric mean, geometric mean, positive definite matrix
Received by editor(s): December 22, 2008
Received by editor(s) in revised form: January 26, 2009
Published electronically: June 19, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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