Geodesic
Comparison of various generalizations of continued fractions
Matematičeskie zametki, Tome 61 (1997) no. 3, pp. 339-348.

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We use the Euler, Jacobi, Poincaré, and Brun matrix algorithms as well as two new algorithms to evaluate the continued fraction expansions of two vectors $L$ related to two Davenport cubic forms $g_1$ and $g_2$. The Klein polyhedra of $g_1$ and $g_2$ were calculated in another paper. Here the integer convergents $P_k$ given by the cited algorithms are considered with respect to the Klein polyhedra. We also study the periods of these expansions. It turns out that only the Jacobi and Bryuno algorithms can be regarded as satisfactory. 
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A. D. Bruno; V. I. Parusnikov. Comparison of various generalizations of continued fractions. Matematičeskie zametki, Tome 61 (1997) no. 3, pp. 339-348. http://geodesic.mathdoc.fr/item/MZM_1997_61_3_a2/

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