Section
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Mathematics
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Title
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Double description method over the field of algebraic numbers
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Author(-s)
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Zolotykh N.Yu.a,
Kubarev V.K.b,
Lyalin S.S.b
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Affiliations
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Nizhni Novgorod State Universitya,
Intel Corporation, Nizhni Novgorodb
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Abstract
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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.
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Keywords
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system of linear inequalities, convex hull, cone, polyhedron, double description method, algebraic extensions
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UDC
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519.61, 519.852.2
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MSC
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90-08, 52B55, 92-08
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DOI
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10.20537/vm180203
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Received
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13 April 2018
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Language
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Russian
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Citation
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Zolotykh N.Yu., Kubarev V.K., Lyalin S.S. Double description method over the field of algebraic numbers, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 161-175.
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