where [G.sub.NBMP] denotes the recursively defined NBMP mean of K?1 matrices with the geometric mean of two matrices (2.1) as its base.
the results for the ALM and NBMP algorithm are respectively
In fact, in [11, 25] it is shown that the ALM and NBMP mean are two instances of an entire class of means, all satisfying the required properties but with possibly different results.
For the ALM and NBMP mean, these parameters become respectively (1,1, ...
For the ALM and NBMP algorithms, the mean of two matrices is by definition known to be the analytical geometric mean, since they are recursively defined starting with this analytical expression.
While the ALM mean is proven to converge linearly  and the NBMP mean superlinearly of order 3 , both have rapidly increasing computational time as the number of matrices increases.
In Figure 3.3b, the intrinsic distances (2.2) between the results are shown, and it is clear that the ALM and NBMP mean are more similar to each other than to the CHEAP mean, especially as the condition number of the matrices increases.
Hence, this test is only meaningful for the ALM and NBMP mean, of which we show the results in Figure 3.3c.