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 Mat. Zametki, 1997, Volume 61, Issue 3, Pages 339–348 (Mi mz1508)

Comparison of various generalizations of continued fractions

A. D. Bruno, V. I. Parusnikov

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/mzm1508

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English version:
Mathematical Notes, 1997, 61:3, 278–286

Bibliographic databases:

UDC: 511.36+514.172.45
Revised: 10.10.1996

Citation: A. D. Bruno, V. I. Parusnikov, “Comparison of various generalizations of continued fractions”, Mat. Zametki, 61:3 (1997), 339–348; Math. Notes, 61:3 (1997), 278–286

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bruno, A, “Algorithms of the local nonlinear analysis”, Nonlinear Analysis-Theory Methods & Applications, 30:7 (1997), 4683
2. Bruno, AD, “Newton polyhedra and power transformations”, Mathematics and Computers in Simulation, 45:5–6 (1998), 429
3. V. I. Parusnikov, “Klein polyhedra for the fourth extremal cubic form”, Math. Notes, 67:1 (2000), 87–102
4. Bruno A., “Algorithms of the Asymptotic Nonlinear Analysis”, Direct and Inverse Problems of Mathematical Physics, International Society for Analysis, Applications and Computation, 5, eds. Gilbert R., Kajiwara J., Xu Y., Springer, 2000, 1–20
5. Bruno, AD, “Power expansions of solutions to a single algebraic or differential equation”, Doklady Mathematics, 64:2 (2001), 160
6. Pustyl'nikov, LD, “Infinite-dimensional generalized continued fractions, distribution of quadratic residues and non-residues, and ergodic theory”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 5:4 (2002), 555
7. L. D. Pustyl'nikov, “Generalized continued fractions and ergodic theory”, Russian Math. Surveys, 58:1 (2003), 109–159
8. V. I. Parusnikov, “Klein polyhedra for three extremal cubic forms”, Math. Notes, 77:4 (2005), 523–538
9. Bruno, AD, “Structure of best diophantine approximations”, Doklady Mathematics, 71:3 (2005), 396
10. V. N. Berestovskii, Yu. G. Nikonorov, “Continued Fractions, the Group $\mathrm{GL}(2,\mathbb Z)$, and Pisot Numbers”, Siberian Adv. Math., 17:4 (2007), 268–290
11. Karpenkov, ON, “Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions”, Monatshefte fur Mathematik, 152:3 (2007), 217
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13. V. I. Parusnikov, “Chetyrekhmernoe obobschenie tsepnykh drobei”, Preprinty IPM im. M. V. Keldysha, 2011, 078, 16 pp.
14. A. D. Bryuno, “Universalnoe obobschenie algoritma tsepnoi drobi”, Chebyshevskii sb., 16:2 (2015), 35–65
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