1,502 research outputs found
Geodesic continued fractions and LLL
We discuss a proposal for a continued fraction-like algorithm to determine
simultaneous rational approximations to real numbers
. It combines an algorithm of Hermite and Lagarias
with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with
parameter as . The new idea in this paper is that checking
the LLL-conditions consists of solving linear equations in
A Modified KZ Reduction Algorithm
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
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 , but it is practical and more efficient than
canonical forms based on Minkowski reduction
Computation of Atomic Fibers of Z-Linear Maps
For given matrix , the set
describes the preimage or fiber of under the -linear map
, . The fiber is called atomic, if
implies or . 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
We introduce and study a family of polytopes which can be seen as a
generalization of the permutahedron of type . We highlight connections
with the largest possible diameter of the convex hull of a set of points in
dimension whose coordinates are integers between and , 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
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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