Simplex-modular algorithm for the decomposition of algebraic numbers into multidimensional continued fractions

We consider a simplex-modular algorithm ($\mathcal{SM}$-algorithm) for the decomposition of algebraic numbers $\alpha=(\alpha_1,\ldots,\alpha_d)$ into multidimensional periodic continued fractions. The $\mathcal{SM}$-algorithm is based on the following: 1) the minimal rational simplexes $\mathbf{s}$ that contain the point $\alpha$; 2) integer unimodular Pisot matrices $P_{\alpha}$ for which $\widehat{\alpha}=(\alpha_1,\ldots,\alpha_d,1)$ is an eigenvector.
The $\mathcal{SM}$-algorithm is a flexible algorithm. This algorithm gives the best approximation of order $1/Q^{1+\varepsilon}_{a}$, where $Q_a$ $(a=0,1,2,\ldots)$ are denominators of the convergents and $\varepsilon>0$ depends on the settings of the $\mathcal{SM}$-algorithm.