multivariate approaches).

Practical issues, complexity and memory
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The recursive algorithms (\[def:Gammarm\]) and (\[def:Gammamedaver\]) require each $O(d^2)$ elementary operations at each update. With the additional online estimation given in (\[algo:vectp\]) of the $q$ eigenvectors associated to the $q$ largest eigenvalues, $O(qd^2)$ additional operations are required. The orthogonalization procedure only requires $O(q^2d)$ elementary operations.

Note that the use of classical Newton-Raphson algorithms for estimating the MCM (see [@FFC2012]) can not be envisaged for high dimensional data since the computation or the approximation of the Hessian matrix would require $O(d^4)$ elementary operations. The well known and fast Weiszfeld’s algorithm requires $O(nd^2)$ elementary operations for each sample with size $n$. However, the estimation cannot be updated automatically if the data arrive sequentially. Another drawback compared to the recursive algorithms studied in this paper is that all the data must be stored in memory, which is of order $O(nd^2)$ elements whereas the recursive technique require an amount of memory of order $O(d^2)$.

The performances of the recursive algorithms depend on the values of tuning parameters $c_\gamma$, $c_m$ and $\alpha$. The value of parameter $\alpha$ is often chosen to be $\alpha=2/3$ or $\alpha=3/4$. Previous empirical studies (see [@CCZ11] and [@CardCC10]) have shown that, thanks to 