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

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Matematicheskie Zametki, 1997, Volume 61, Issue 3, Pages 339–348
DOI: https://doi.org/10.4213/mzm1508 (Mi mzm1508)

This article is cited in 26 scientific papers (total in 27 papers)

Comparison of various generalizations of continued fractions

A. D. Bruno, V. I. Parusnikov

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

Full-text PDF (226 kB) Citations (27)

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

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.

Received: 14.11.1995
Revised: 10.10.1996

English version:
Mathematical Notes, 1997, Volume 61, Issue 3, Pages 278–286
DOI: https://doi.org/10.1007/BF02355409

Bibliographic databases:

UDC: 511.36+514.172.45

Language: Russian

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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\transl

\jour Math. Notes

\yr 1997

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Linking options:

https://www.mathnet.ru/eng/mzm1508

https://doi.org/10.4213/mzm1508

https://www.mathnet.ru/eng/mzm/v61/i3/p339

This publication is cited in the following 27 articles:

Hanka Řada, Štěpán Starosta, Vítězslav Kala, “Periodicity of general multidimensional continued fractions using repetend matrix form”, Expositiones Mathematicae, 42:3 (2024), 125571
Yu. A. Basalov, “O russkoi nauchnoi shkole diofantovykh priblizhenii”, Chebyshevskii sb., 21:1 (2020), 388–403
A. A. Lodkin, “Klein Sail and Diophantine Approximation of a Vector”, J Math Sci, 247:5 (2020), 680
A. A. Lodkin, “Parus Kleina i diofantovy priblizheniya vektora”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye i algoritmicheskie metody. XXX, Zap. nauchn. sem. POMI, 481, POMI, SPb., 2019, 63–73
Murru N., Terracini L., “On P-Adic Multidimensional Continued Fractions”, Math. Comput., 88:320 (2019), 2913–2934
V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions”, J. Math. Sci. (N. Y.), 225:6 (2017), 924–949
A. D. Bryuno, “Universalnoe obobschenie algoritma tsepnoi drobi”, Chebyshevskii sb., 16:2 (2015), 35–65
Murru N., “on the Periodic Writing of Cubic Irrationals and a Generalization of Redei Functions”, Int. J. Number Theory, 11:3 (2015), 779–799
Fritz Schweiger, “Variations of the Poincaré Map”, Integers, 12:2 (2012)
D. I. Bodnar, R. A. Zators'kyi, “Generalization of continued fractions. I”, J Math Sci, 183:1 (2012), 54
V. I. Parusnikov, “Chetyrekhmernoe obobschenie tsepnykh drobei”, Preprinty IPM im. M. V. Keldysha, 2011, 078, 16 pp.
A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii sb., 11:1 (2010), 68–73
Alexander D. Bruno, “New generalization of continued fraction, I”, Funct. Approx. Comment. Math., 43:1 (2010)
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
Karpenkov, ON, “Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions”, Monatshefte fur Mathematik, 152:3 (2007), 217
V. I. Parusnikov, “Klein polyhedra for three extremal cubic forms”, Math. Notes, 77:4 (2005), 523–538
Bruno, AD, “Structure of best diophantine approximations”, Doklady Mathematics, 71:3 (2005), 396
L. D. Pustyl'nikov, “Generalized continued fractions and ergodic theory”, Russian Math. Surveys, 58:1 (2003), 109–159
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
Bruno, AD, “Power expansions of solutions to a single algebraic or differential equation”, Doklady Mathematics, 64:2 (2001), 160

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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