Simplex--karyon algorithm of multidimensional continued fraction expansion

V. G. Zhuravlev

Informatics and Automation, Analytic number theory, Tome 299 (2017), pp. 283-303.

Résumé

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 $\mathrm{\wp }(r)$-strategy for constructing convergents is chosen, with a free objective function $\mathrm{\wp }(r)$ on the hyperplane $\mathbf{K}$.

Détail

Mots-clés : multidimensional continued fractions, best approximations, Farey sums.

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