The Matrix Means Toolbox includes the following files:
alm.m - Computes the Ando-Li-Mathias mean of k matrices [1]
cheap.m - Computes the cheap mean of k matrices [3]
dist.m - Computes the Riemannian and Euclidean distance between A and B
explog.m - Computes the explog mean of k matrices [2]
geomean.m - Computes the geometric mean of two matrice with three algorithms [6]
gradient.m - Computes the Karcher mean of k matrices using the Riemannian gradient descent [7]
nbmp.m - Computes the Nakamura/Bini-Meini-Poloni mean of k matrices [5,8]
random.m - Generates k random positive definite matrices
rich.m - Computes the Karcher mean of k matrices using a Richardson-like iteration [4]
sharp.m - Computes $A\#_t B$
Bibliography
[1] T. Ando, C.-K. Li and R. Mathias, Geometric means, Linear Algebra Appl., 385-1 (2004) pp. 305-334.
[2] V. Arsigny, P. Fillard, X. Pennec and N. Ayache, Geometric means in a novel vector space structure on symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl., 29-1 (2006/07), pp. 328-347.
[3] D. A. Bini and B. Iannazzo, A note on computing matrix geometric means, to
appear in Advances Comp. Math., 35-2/4 (2010), pp. 175-192.
[4] D. A. Bini and B. Iannazzo, Computing the Karcher mean of symmetric positive definite matrices, to appear in Linear Algebra Appl., 2012.
[5] D. A. Bini, B. Meini and F. Poloni, An effective matrix geometric mean satisfying the Ando-Li-Mathias properties, Math. Comp., 79 (2010), pp. 437-452.
[6] B. Iannazzo and B. Meini, The Palyndromic Cyclic Reduction and the Matrix Geometric Mean, Manuscript.
[7] J. H. Manton, A globally convergent numerical algorithm for computing the centre of mass on compact lie groups, Eighth International Conference on Contro, Automation, Robotics and Vision, 2004. ICARCV 2004 8th, Kunming, China.
[8] K. Nakamura, Geometric Means of Positive Operators, Kyungpook Math. J. 49 (2009), 167-181
History
mmtoolbox 1.1b2 - November 26, 2011 - Some minor adjustments to the formatting
of the files.
mmtoolbox 1.1b1 - August, 19, 2011 - rewritten all files from scratch with a uniform style, help in line,
and more; changed the square root computation to the Cholesky factor whenever possible;
changed the name of the file manton.m to gradient.m; changed the name of the file bmp.m to nbmp.m, as
soon as the reference [8] was individuated; fixed a bug in dist.m.
mmtoolbox 1.0 - July 11, 2010 - first release
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