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

This article is cited in 18 scientific papers (total in 19 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

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.


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

Bibliographic databases:

UDC: 511.36+514.172.45
Received: 14.11.1995
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
\by A.~D.~Bruno, V.~I.~Parusnikov
\paper Comparison of various generalizations of continued fractions
\jour Mat. Zametki
\yr 1997
\vol 61
\issue 3
\pages 339--348
\jour Math. Notes
\yr 1997
\vol 61
\issue 3
\pages 278--286

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    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  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Bruno, AD, “Newton polyhedra and power transformations”, Mathematics and Computers in Simulation, 45:5–6 (1998), 429  crossref  mathscinet  zmath  isi
    3. V. I. Parusnikov, “Klein polyhedra for the fourth extremal cubic form”, Math. Notes, 67:1 (2000), 87–102  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  isi
    5. Bruno, AD, “Power expansions of solutions to a single algebraic or differential equation”, Doklady Mathematics, 64:2 (2001), 160  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    7. L. D. Pustyl'nikov, “Generalized continued fractions and ergodic theory”, Russian Math. Surveys, 58:1 (2003), 109–159  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. V. I. Parusnikov, “Klein polyhedra for three extremal cubic forms”, Math. Notes, 77:4 (2005), 523–538  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Bruno, AD, “Structure of best diophantine approximations”, Doklady Mathematics, 71:3 (2005), 396  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet  elib
    11. Karpenkov, ON, “Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions”, Monatshefte fur Mathematik, 152:3 (2007), 217  crossref  mathscinet  zmath  isi  scopus  scopus
    12. A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii sb., 11:1 (2010), 68–73  mathnet  mathscinet
    13. V. I. Parusnikov, “Chetyrekhmernoe obobschenie tsepnykh drobei”, Preprinty IPM im. M. V. Keldysha, 2011, 078, 16 pp.  mathnet
    14. A. D. Bryuno, “Universalnoe obobschenie algoritma tsepnoi drobi”, Chebyshevskii sb., 16:2 (2015), 35–65  mathnet  elib
    15. Murru N., “on the Periodic Writing of Cubic Irrationals and a Generalization of Redei Functions”, Int. J. Number Theory, 11:3 (2015), 779–799  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    16. 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  mathnet  crossref  mathscinet
    17. 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  mathnet
    18. Murru N., Terracini L., “On P-Adic Multidimensional Continued Fractions”, Math. Comput., 88:320 (2019), 2913–2934  crossref  isi
    19. Yu. A. Basalov, “O russkoi nauchnoi shkole diofantovykh priblizhenii”, Chebyshevskii sb., 21:1 (2020), 388–403  mathnet  crossref
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