1,502 research outputs found

    Geodesic continued fractions and LLL

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    We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to dd real numbers α1,,αd\alpha_1,\ldots,\alpha_d. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter tt as t0t\downarrow0. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in tt

    A Modified KZ Reduction Algorithm

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    The Korkine-Zolotareff (KZ) reduction has been used in communications and cryptography. In this paper, we modify a very recent KZ reduction algorithm proposed by Zhang et al., resulting in a new algorithm, which can be much faster and more numerically reliable, especially when the basis matrix is ill conditioned.Comment: has been accepted by IEEE ISIT 201

    A Canonical Form for Positive Definite Matrices

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    We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension nn, but it is practical and more efficient than canonical forms based on Minkowski reduction

    Computation of Atomic Fibers of Z-Linear Maps

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    For given matrix AZd×nA\in\Z^{d\times n}, the set Pb={z:Az=b,zZ+n}P_{b}=\{z:Az=b,z\in\Z^n_+\} describes the preimage or fiber of bZdb\in\Z^d under the Z\Z-linear map fA:Z+nZdf_A:\Z^n_+\to\Z^d, xAxx\mapsto Ax. The fiber PbP_{b} is called atomic, if Pb=Pb1+Pb2P_{b}=P_{b_1}+P_{b_2} implies b=b1b=b_1 or b=b2b=b_2. In this paper we present a novel algorithm to compute such atomic fibers. An algorithmic solution to appearing subproblems, computational examples and applications are included as well.Comment: 27 page

    Primitive Zonotopes

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    We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type BdB_d. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension dd whose coordinates are integers between 00 and kk, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde

    Implicitization of curves and (hyper)surfaces using predicted support

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    We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance. In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive
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