Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions

Simplex-module algorithm ($\mathcal{SM}$-algorithm) for expansion of algebraic numbers $\alpha=(\alpha_1,\ldots,\alpha_d)$ in multidimensional continued fractions is offered. The method is based on 1) minimal rational simplices $\mathbf s$, where $\alpha\in\mathbf s$, and 2) Pisot matrices $P_\alpha$ for which $\widehat \alpha=(\alpha_1,\ldots,\alpha_d,1)$ is eigenvector. A multi-dimensional generalization of the Lagrange theorem is proved.