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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, Volume 28, Issue 2, Pages 161–175
DOI: https://doi.org/10.20537/vm180203
(Mi vuu628)
 

This article is cited in 3 scientific papers (total in 3 papers)

MATHEMATICS

Double description method over the field of algebraic numbers

N. Yu. Zolotykha, V. K. Kubarevb, S. S. Lyalinb

a Department of Algebra, Geometry and Discrete Mathematics, Nizhni Novgorod State University, pr. Gagarina, 23, Nizhni Novgorod, 603950, Russia
b Intel Corp., ul. Turgeneva, 30, Nizhni Novgorod, 603024, Russia
Full-text PDF (321 kB) Citations (3)
References:
Abstract: We consider the problem of constructing the dual representation of a convex polyhedron defined as a set of solutions to a system of linear inequalities with coefficients which are algebraic numbers. The inverse problem is equivalent (dual) to the initial problem. We propose program implementations of several variations of the well-known double description method (Motzkin–Burger method) solving this problem. The following two cases are considered: 1) the elements of the system of inequalities are arbitrary algebraic numbers, and each such number is represented by its minimal polynomial and a localizing interval; 2) the elements of the system belong to a given extension ${\mathbb Q} (\alpha)$ of ${\mathbb Q}$, and the minimal polynomial and the localizing interval are given only for $\alpha$, all elements of the system, intermediate and final results are represented as polynomials of $\alpha$. As expected, the program implementation for the second case significantly outperforms the implementation for the first one in terms of speed. In the second case, for greater acceleration, we suggest using a Boolean matrix instead of the discrepancy matrix. The results of a computational experiment show that the program is quite suitable for solving medium-scale problems.
Keywords: system of linear inequalities, convex hull, cone, polyhedron, double description method, algebraic extensions.
Funding agency Grant number
Russian Science Foundation 17-11-01336
Received: 13.04.2018
Bibliographic databases:
Document Type: Article
UDC: 519.61, 519.852.2
MSC: 90-08, 52B55, 92-08
Language: Russian
Citation: N. Yu. Zolotykh, V. K. Kubarev, S. S. Lyalin, “Double description method over the field of algebraic numbers”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:2 (2018), 161–175
Citation in format AMSBIB
\Bibitem{ZolKubLya18}
\by N.~Yu.~Zolotykh, V.~K.~Kubarev, S.~S.~Lyalin
\paper Double description method over the field of algebraic numbers
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2018
\vol 28
\issue 2
\pages 161--175
\mathnet{http://mi.mathnet.ru/vuu628}
\crossref{https://doi.org/10.20537/vm180203}
\elib{https://elibrary.ru/item.asp?id=35258684}
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  • https://www.mathnet.ru/eng/vuu/v28/i2/p161
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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    Abstract page:415
    Full-text PDF :198
    References:42
     
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