Normaliz - a tool for discrete convex geometry
Normaliz is a open source tool for computations in affine monoids, vector configurations, rational polyhedra and rational cones. Normaliz now computes rational and algebraic polyhedra, i.e., polyhedra defined over real algebraic extensions of QQ.
Computation goals
- convex hulls and dual cones
- conversion from generators to constraints and vice versa
- projections of cones and polyhedra
- triangulations, disjoint decompositions and Stanley decompositions
- Hilbert bases of rational, not necessarily pointed cones
- normalizations of affine monoids
- lattice points of polytopes and (unbounded) polyhedra
- automorphism groups (euclidean, integral, rational/algebraic, combinatorial)
- face lattices and f-vectors
- Euclidean and lattice normalized volumes of polytopes
- Hilbert (or Ehrhart) series and (quasi) polynomials under Z-gradings (for example, for rational polytopes)
- generalized (or weighted) Ehrhart series and Lebesgue integrals of - polynomials over rational polytopes
Normaliz offers the API libnormaliz that allows the user to access Normaliz computations from any C++ program.
The frontend Normaliz reads input files and writes output files. There is a wide variety of input types to specify polyhedra and lattices by generators (vertices, extreme rays, bases) or by constraints (inequalities, equations and congruences). The user sets computation goals and chooses algorithmic variants through command line options or the input file.
Online exploration of Normaliz: https://mybinder.org/v2/gh/Normaliz/NormalizJupyter/master
Sample input and output
The file 2cone.in from the directory example contains
amb_space 2
cone 2
1 3
2 1
It defines a cone in two-dimensional real space by its extreme rays.
The command
normaliz example/2cone
runs Normaliz with its default computation goals. It produces the output file 2cone.out (here typeset in two columns):
4 Hilbert basis elements embedding dimension = 2
2 extreme rays rank = 2 (maximal)
2 support hyperplanes external index = 1
internal index = 5
original monoid is not integrally closed
size of triangulation = 1 rank of class group = 0
resulting sum of |det|s = 5 finite cyclic summands:
5: 1
No implicit grading found
***********************************************************************
4 Hilbert basis elements: 2 extreme rays:
1 1 1 3
1 2 2 1
1 3
2 1 2 support hyperplanes:
-1 2
3 -1
The main point was the computation of the Hilbert basis (encircled in red in the figure).
Platforms
Each release contains executables for Linux 64, MacOS X and MS Windows 64.
Interfaces
Normaliz can be called from several other systems:
The Python package PyNormaliz by Sebastian Gutsche provides an environment for interactive access. It is contained in the source package of Normaliz.
jNormaliz by Vicinius Almendra and Bogdan Ichim provides a GUI to Normaliz
Optional packages
For its basic functionality Normaliz needs only GMP. Parallelization is based on OpenMP. For the computation of integrals CoCoALib is used.
For algebraic polyhedra Normaliz needs Flint, arb, antic and e-antic
The computation of automorphism groups uses nauty.
Installation
All files can be found at https://github.com/Normaliz/Normaliz/releases.
For the binary package (ready made executable program) download and extract
- the executable for your system (
normaliz-x.y.zLinux64.zip,normaliz-x.y.zMacOS.zipornormaliz-x.y.zLinux64.zip).
For the source package download and extract
normaliz-x.y.z.zip(or tar.gz)
From the source one can compile Normaliz oneself on Linux or MacOS by one of the installation scripts
install_normaliz_with_opt.sh(only rational polyhedra)install_normaliz_with_eantic.sh(with algebraic polyhedra)
Docker image
available from https://hub.docker.com/r/normaliz/normaliz/
Distributions
Normaliz is available as a Debian, Gentoo and Ubuntu package, as well as from Conda (Linux, MacOS, MS Windows).