Geodesic
Simplex--karyon algorithm of multidimensional continued fraction expansion
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 283-303

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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 $\mathbf K$.
Keywords: multidimensional continued fractions, best approximations, Farey sums. 
@article{TM_2017_299_a16,
     author = {V. G. Zhuravlev},
     title = {Simplex--karyon algorithm of multidimensional continued fraction expansion},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {283--303},
     publisher = {mathdoc},
     volume = {299},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_299_a16/}
}
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V. G. Zhuravlev. Simplex--karyon algorithm of multidimensional continued fraction expansion. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 283-303. http://geodesic.mathdoc.fr/item/TM_2017_299_a16/