42 research outputs found
Polymake and Lattice Polytopes
The polymake software system deals with convex polytopes and related objects
from geometric combinatorics. This note reports on a new implementation of a
subclass for lattice polytopes. The features displayed are enabled by recent
changes to the polymake core, which will be discussed briefly.Comment: 12 pages, 1 figur
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
The subdivision of large simplicial cones in Normaliz
Normaliz is an open-source software for the computation of lattice points in
rational polyhedra, or, in a different language, the solutions of linear
diophantine systems. The two main computational goals are (i) finding a system
of generators of the set of lattice points and (ii) counting elements
degree-wise in a generating function, the Hilbert Series. In the homogeneous
case, in which the polyhedron is a cone, the set of generators is the Hilbert
basis of the intersection of the cone and the lattice, an affine monoid.
We will present some improvements to the Normaliz algorithm by subdividing
simplicial cones with huge volumes. In the first approach the subdivision
points are found by integer programming techniques. For this purpose we
interface to the integer programming solver SCIP to our software. In the second
approach we try to find good subdivision points in an approximating overcone
that is faster to compute.Comment: To appear in the proceedings of the ICMS 2016, published by Springer
as Volume 9725 of Lecture Notes in Computer Science (LNCS
Toric Geometry in OSCAR
We report on the computer implementation for toric geometry in the computer
algebra system .
The main architectural feature of is that its four
fundamental tools (Hecke, Nemo), ,
and are ,
rather than external software. Toric geometry benefits greatly from this
architecture.
is a high-performance programming language designed for
numerical and scientific computing. The growing ecosystem of
packages ensures its continued viability for scientific computing and data
analysis. Indeed, is written in . This implies
that the performance of should be comparable or even better
than many other implementations.Comment: 6 pages, prepared for the ComputerAlgebraRundbrief (March 2023
polymake and Lattice Polytopes
The software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes to the core, which will be discussed briefly