Mathematics of Computation

Author(s): Dario A. Bini; Beatrice Meini; Federico Poloni.
Journal: Math. Comp. 79 (2010), 437-452.
MSC (2000): Primary 65F30; Secondary 15A48, 47A64
Posted: June 19, 2009
Retrieve article in: PDF

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.

1.: T. Ando, Chi-Kwong Li, and Roy Mathias, Geometric means, Linear Algebra Appl. 385 (2004), 305-334. MR 2063358 (2005f:47049)
2.: Rajendra Bhatia, Positive definite matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007. MR 2284176 (2007k:15005)
3.: Rajendra Bhatia and John Holbrook, Noncommutative geometric means, Math. Intelligencer 28 (2006), no. 1, 32-39. MR 2202893 (2007g:47023)
4.: -, Riemannian geometry and matrix geometric means, Linear Algebra Appl. 413 (2006), no. 2-3, 594-618. MR 2198952 (2007c:15030)
5.: R. F. S. Hearmon, The elastic constants of piezoelectric crystals, J. Appl. Phys. 3 (1952), 120-123.
6.: Nicholas J. Higham, The Matrix Computation Toolbox, http://www.ma.man.ac.uk/~higham/mctoolbox.
7.: Bruno Iannazzo and Beatrice Meini, Palindromic matrix polynomials and their relationships with certain functions of matrices, Tech. Report, Dipartimento di Matematica, Università di Pisa, 2009.
8.: Yongdo Lim, On Ando-Li-Mathias geometric mean equations, Linear Algebra Appl. 428 (2008), no. 8-9, 1767-1777. MR 2398117
9.: Maher Moakher, On the averaging of symmetric positive-definite tensors, J. Elasticity 82 (2006), no. 3, 273-296. MR 2231065 (2007a:74007)

Retrieve articles in Mathematics of Computation with MSC (2000): 65F30, 15A48, 47A64

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: 10.1090/S0025-5718-09-02261-3
PII: S 0025-5718(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
Posted: June 19, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS