hsnf

使用转换矩阵计算Hermite正交形式和Smith正交形式。

• 文档：https://hsnf.readthedocs.io/en/latest/
• 开发文档：https://lan496.github.io/hsnf/develop
• Github：https://github.com/lan496/hsnf
• PyPI：https://pypi.ac.cn/project/hsnf

使用方法

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

安装

hsnf与Python3.8+兼容，可以通过PyPI安装

pip install hsnf

或本地安装

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

参考资料

• http://www.dlfer.xyz/post/2016-10-27-smith-normal-form/
  • 感谢D. L. Ferrario博士的指导性博客文章及其允许引用其脚本的批准。
• CSE206A: Lattices Algorithms and Applications (Spring 2014)
• Henri Cohen, A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1993).