https://github.com/justinshenk/normaliz-notebook
Testing repository for Normaliz Tutorial
https://github.com/justinshenk/normaliz-notebook
basis cone hilbert linear-algebra matrix vector
Last synced: about 1 month ago
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Testing repository for Normaliz Tutorial
- Host: GitHub
- URL: https://github.com/justinshenk/normaliz-notebook
- Owner: JustinShenk
- Created: 2017-01-10T16:19:25.000Z (over 9 years ago)
- Default Branch: master
- Last Pushed: 2017-05-02T13:29:51.000Z (about 9 years ago)
- Last Synced: 2025-02-23T21:31:24.477Z (over 1 year ago)
- Topics: basis, cone, hilbert, linear-algebra, matrix, vector
- Language: Jupyter Notebook
- Homepage: https://www.normaliz.uni-osnabrueck.de/
- Size: 478 KB
- Stars: 0
- Watchers: 3
- Forks: 1
- Open Issues: 1
-
Metadata Files:
- Readme: README.md
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README
# PyNormaliz Web
Run `api.py`. It will start a web server listening at `localhost:8090`. Requests are expected to `localhost:8091/normaliz/:v1/:v2` where `v1` and `v2` are each comma seperated tuples of the vector's respective x and y values. E.g. `localhost:8091/normaliz/1.4,1.0/2,3`.
For local development you will also want to run `web.py` which serves the web app at `localhost:8091`.
## PyNormaliz
Normaliz is a tool for computing the Hilbert bases and enumerative data of rational cones and, more generally, sets of lattice points in rational polyhedra.
### Example 1: A cone in dimension 2
We want to investigate the cone :
This cone is defined in the input file 2cone.in:
amb_space 2
cone 2
1 3
2 1
The input tells Normaliz that the ambient space is R2, and then a cone with 2 generators is defined, namely the cone C from above.
The figure indicates the Hilbert basis, and this is our first computation goal.
If you prefer to consider the columns of a matrix as input vectors (or have got a matrix in this format from another system) you can use the input
Calculate the Hilbert basis from finite vectors.
```python
%pylab inline
from __future__ import print_function
import PyNormaliz
from ipywidgets import interact, interactive, fixed
from IPython.display import display
import ipywidgets as widgets
import numpy as np
import matplotlib.pyplot as plt
from fractions import Fraction
vectors = [[2,1],[1,4]]
cone = PyNormaliz.NmzCone("cone",vectors)
HB = PyNormaliz.NmzResult(cone,"HilbertBasis")
# Draw background lattice.
xLattice = np.arange(-1,8)
yLattice = np.arange(-1,8)
def pltHilbertBasis(theta=np.pi/10):
fig,ax = plt.subplots()
plt.ylim([-0.5,8])
plt.xlim([-0.5,8])
ax.axis('off')
vectors[0][0],vectors[0][1] = np.cos(theta)*2, np.sin(theta)*2
ratio = vectors[0][0] / vectors[0][1]
integers = Fraction.from_float(ratio).limit_denominator(100)
# TODO: Solve for >2 dimensions.
# for ind,d in enumerate(vectors[0]):
# integers = Fraction.from_float(d).limit_denominator(100)
vectors[0][0] = integers.numerator
vectors[0][1] = integers.denominator
for x in xLattice:
for y in yLattice:
plt.plot(x,y,'ko')
# Plot bounding vectors.
for v in vectors:
ax.plot([0,v[0]*2],[0,v[1]*2],'k-')
cone = PyNormaliz.NmzCone("cone",vectors)
HB = PyNormaliz.NmzResult(cone,"HilbertBasis")
xList = [x for x,y in HB]
yList = [y for x,y in HB]
# TODO: Complete fill_between function.
# x = np.arange(0,0.1,2)
# y1 = x*2
# y2 = x*3
# ax.fill_between(x,y1,y2,where=y2>=y1,facecolor='green', interpolate=True)
# Plot Hilbert basis.
ax.plot(xList,yList,'ro')
ax.annotate(str(0), (-.5,-.5))
plt.show()
return "Hilbert Bases:", HB
w = interactive(pltHilbertBasis, theta=(0.01,np.pi/2,np.pi/40))
display(w)
```
('Hilbert Bases:',
[[1L, 1L],
[1L, 2L],
[1L, 3L],
[1L, 4L],
[2L, 1L],
[3L, 1L],
[40L, 13L],
[277L, 90L]])
```python
```