Simplex–karyon algorithm of multidimensional continued fraction expansion

A simplex–karyon algorithm for expanding real numbers $\alpha =(\alpha _1,\dots ,\alpha _d)$ in multidimensional continued fractions is considered. The algorithm is based on a $(d+1)$-dimensional superspace $\mathbf S$ with embedded hyperplanes: a karyon hyperplane $\mathbf K$ and a Farey hyperplane $\mathbf F$. The approximation of numbers $\alpha $ by continued fractions is performed on the hyperplane $\mathbf F$, and the degree of approximation is controlled on the hyperplane $\mathbf K$. A local $\wp (r)$-strategy for constructing convergents is chosen, with a free objective function $\wp (r)$ on the hyperplane $\mathbf K$.