hsnf

Computing Hermite normal form and Smith normal form with transformation matrices.

• Document: https://hsnf.readthedocs.io/en/latest/

• Document(develop): https://lan496.github.io/hsnf/develop

• Github: https://github.com/lan496/hsnf

• PyPI: https://pypi.org/project/hsnf

Usage

import numpy as np
from hsnf import column_style_hermite_normal_form, row_style_hermite_normal_form, smith_normal_form

# Integer matrix to be decomposed
M = np.array(
    [
        [-6, 111, -36, 6],
        [5, -672, 210, 74],
        [0, -255, 81, 24],
    ]
)

# Smith normal form
D, L, R = smith_normal_form(M)
"""
D = array([
[   1    0    0    0]
[   0    3    0    0]
[   0    0 2079    0]])
"""
assert np.allclose(L @ M @ R, D)
assert np.around(np.abs(np.linalg.det(L))) == 1  # unimodular
assert np.around(np.abs(np.linalg.det(R))) == 1  # unimodular

# Row-style hermite normal form
H, L = row_style_hermite_normal_form(M)
"""
H = array([
[     1      0    420  -2522]
[     0      3   1809 -10860]
[     0      0   2079 -12474]])
"""
assert np.allclose(L @ M, H)
assert np.around(np.abs(np.linalg.det(L))) == 1  # unimodular

# Column-style hermite normal form
H, R = column_style_hermite_normal_form(M)
"""
H = array([
[   3    0    0    0]
[   0    1    0    0]
[1185  474 2079    0]])
"""
assert np.allclose(np.dot(M, R), H)
assert np.around(np.abs(np.linalg.det(R))) == 1  # unimodular

Installation

hsnf works with Python3.8+ and can be installed via PyPI:

pip install hsnf

or in local:

git clone git@github.com:lan496/hsnf.git
cd hsnf
pip install -e .[dev,docs]

References

• http://www.dlfer.xyz/post/2016-10-27-smith-normal-form/

  • I appreciate Dr. D. L. Ferrario’s instructive blog post and his approval for referring his scripts.

• CSE206A: Lattices Algorithms and Applications (Spring 2014)

• Henri Cohen, A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1993).