407 lines
13 KiB
Python
407 lines
13 KiB
Python
'''Functions returning normal forms of matrices'''
|
|
|
|
from collections import defaultdict
|
|
|
|
from .domainmatrix import DomainMatrix
|
|
from .exceptions import DMDomainError, DMShapeError
|
|
from sympy.ntheory.modular import symmetric_residue
|
|
from sympy.polys.domains import QQ, ZZ
|
|
|
|
|
|
# TODO (future work):
|
|
# There are faster algorithms for Smith and Hermite normal forms, which
|
|
# we should implement. See e.g. the Kannan-Bachem algorithm:
|
|
# <https://www.researchgate.net/publication/220617516_Polynomial_Algorithms_for_Computing_the_Smith_and_Hermite_Normal_Forms_of_an_Integer_Matrix>
|
|
|
|
|
|
def smith_normal_form(m):
|
|
'''
|
|
Return the Smith Normal Form of a matrix `m` over the ring `domain`.
|
|
This will only work if the ring is a principal ideal domain.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import ZZ
|
|
>>> from sympy.polys.matrices import DomainMatrix
|
|
>>> from sympy.polys.matrices.normalforms import smith_normal_form
|
|
>>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)],
|
|
... [ZZ(3), ZZ(9), ZZ(6)],
|
|
... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ)
|
|
>>> print(smith_normal_form(m).to_Matrix())
|
|
Matrix([[1, 0, 0], [0, 10, 0], [0, 0, -30]])
|
|
|
|
'''
|
|
invs = invariant_factors(m)
|
|
smf = DomainMatrix.diag(invs, m.domain, m.shape)
|
|
return smf
|
|
|
|
|
|
def add_columns(m, i, j, a, b, c, d):
|
|
# replace m[:, i] by a*m[:, i] + b*m[:, j]
|
|
# and m[:, j] by c*m[:, i] + d*m[:, j]
|
|
for k in range(len(m)):
|
|
e = m[k][i]
|
|
m[k][i] = a*e + b*m[k][j]
|
|
m[k][j] = c*e + d*m[k][j]
|
|
|
|
|
|
def invariant_factors(m):
|
|
'''
|
|
Return the tuple of abelian invariants for a matrix `m`
|
|
(as in the Smith-Normal form)
|
|
|
|
References
|
|
==========
|
|
|
|
[1] https://en.wikipedia.org/wiki/Smith_normal_form#Algorithm
|
|
[2] https://web.archive.org/web/20200331143852/https://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf
|
|
|
|
'''
|
|
domain = m.domain
|
|
if not domain.is_PID:
|
|
msg = "The matrix entries must be over a principal ideal domain"
|
|
raise ValueError(msg)
|
|
|
|
if 0 in m.shape:
|
|
return ()
|
|
|
|
rows, cols = shape = m.shape
|
|
m = list(m.to_dense().rep)
|
|
|
|
def add_rows(m, i, j, a, b, c, d):
|
|
# replace m[i, :] by a*m[i, :] + b*m[j, :]
|
|
# and m[j, :] by c*m[i, :] + d*m[j, :]
|
|
for k in range(cols):
|
|
e = m[i][k]
|
|
m[i][k] = a*e + b*m[j][k]
|
|
m[j][k] = c*e + d*m[j][k]
|
|
|
|
def clear_column(m):
|
|
# make m[1:, 0] zero by row and column operations
|
|
if m[0][0] == 0:
|
|
return m # pragma: nocover
|
|
pivot = m[0][0]
|
|
for j in range(1, rows):
|
|
if m[j][0] == 0:
|
|
continue
|
|
d, r = domain.div(m[j][0], pivot)
|
|
if r == 0:
|
|
add_rows(m, 0, j, 1, 0, -d, 1)
|
|
else:
|
|
a, b, g = domain.gcdex(pivot, m[j][0])
|
|
d_0 = domain.div(m[j][0], g)[0]
|
|
d_j = domain.div(pivot, g)[0]
|
|
add_rows(m, 0, j, a, b, d_0, -d_j)
|
|
pivot = g
|
|
return m
|
|
|
|
def clear_row(m):
|
|
# make m[0, 1:] zero by row and column operations
|
|
if m[0][0] == 0:
|
|
return m # pragma: nocover
|
|
pivot = m[0][0]
|
|
for j in range(1, cols):
|
|
if m[0][j] == 0:
|
|
continue
|
|
d, r = domain.div(m[0][j], pivot)
|
|
if r == 0:
|
|
add_columns(m, 0, j, 1, 0, -d, 1)
|
|
else:
|
|
a, b, g = domain.gcdex(pivot, m[0][j])
|
|
d_0 = domain.div(m[0][j], g)[0]
|
|
d_j = domain.div(pivot, g)[0]
|
|
add_columns(m, 0, j, a, b, d_0, -d_j)
|
|
pivot = g
|
|
return m
|
|
|
|
# permute the rows and columns until m[0,0] is non-zero if possible
|
|
ind = [i for i in range(rows) if m[i][0] != 0]
|
|
if ind and ind[0] != 0:
|
|
m[0], m[ind[0]] = m[ind[0]], m[0]
|
|
else:
|
|
ind = [j for j in range(cols) if m[0][j] != 0]
|
|
if ind and ind[0] != 0:
|
|
for row in m:
|
|
row[0], row[ind[0]] = row[ind[0]], row[0]
|
|
|
|
# make the first row and column except m[0,0] zero
|
|
while (any(m[0][i] != 0 for i in range(1,cols)) or
|
|
any(m[i][0] != 0 for i in range(1,rows))):
|
|
m = clear_column(m)
|
|
m = clear_row(m)
|
|
|
|
if 1 in shape:
|
|
invs = ()
|
|
else:
|
|
lower_right = DomainMatrix([r[1:] for r in m[1:]], (rows-1, cols-1), domain)
|
|
invs = invariant_factors(lower_right)
|
|
|
|
if m[0][0]:
|
|
result = [m[0][0]]
|
|
result.extend(invs)
|
|
# in case m[0] doesn't divide the invariants of the rest of the matrix
|
|
for i in range(len(result)-1):
|
|
if result[i] and domain.div(result[i+1], result[i])[1] != 0:
|
|
g = domain.gcd(result[i+1], result[i])
|
|
result[i+1] = domain.div(result[i], g)[0]*result[i+1]
|
|
result[i] = g
|
|
else:
|
|
break
|
|
else:
|
|
result = invs + (m[0][0],)
|
|
return tuple(result)
|
|
|
|
|
|
def _gcdex(a, b):
|
|
r"""
|
|
This supports the functions that compute Hermite Normal Form.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
Let x, y be the coefficients returned by the extended Euclidean
|
|
Algorithm, so that x*a + y*b = g. In the algorithms for computing HNF,
|
|
it is critical that x, y not only satisfy the condition of being small
|
|
in magnitude -- namely that |x| <= |b|/g, |y| <- |a|/g -- but also that
|
|
y == 0 when a | b.
|
|
|
|
"""
|
|
x, y, g = ZZ.gcdex(a, b)
|
|
if a != 0 and b % a == 0:
|
|
y = 0
|
|
x = -1 if a < 0 else 1
|
|
return x, y, g
|
|
|
|
|
|
def _hermite_normal_form(A):
|
|
r"""
|
|
Compute the Hermite Normal Form of DomainMatrix *A* over :ref:`ZZ`.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
A : :py:class:`~.DomainMatrix` over domain :ref:`ZZ`.
|
|
|
|
Returns
|
|
=======
|
|
|
|
:py:class:`~.DomainMatrix`
|
|
The HNF of matrix *A*.
|
|
|
|
Raises
|
|
======
|
|
|
|
DMDomainError
|
|
If the domain of the matrix is not :ref:`ZZ`.
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
|
|
(See Algorithm 2.4.5.)
|
|
|
|
"""
|
|
if not A.domain.is_ZZ:
|
|
raise DMDomainError('Matrix must be over domain ZZ.')
|
|
# We work one row at a time, starting from the bottom row, and working our
|
|
# way up.
|
|
m, n = A.shape
|
|
A = A.to_dense().rep.copy()
|
|
# Our goal is to put pivot entries in the rightmost columns.
|
|
# Invariant: Before processing each row, k should be the index of the
|
|
# leftmost column in which we have so far put a pivot.
|
|
k = n
|
|
for i in range(m - 1, -1, -1):
|
|
if k == 0:
|
|
# This case can arise when n < m and we've already found n pivots.
|
|
# We don't need to consider any more rows, because this is already
|
|
# the maximum possible number of pivots.
|
|
break
|
|
k -= 1
|
|
# k now points to the column in which we want to put a pivot.
|
|
# We want zeros in all entries to the left of the pivot column.
|
|
for j in range(k - 1, -1, -1):
|
|
if A[i][j] != 0:
|
|
# Replace cols j, k by lin combs of these cols such that, in row i,
|
|
# col j has 0, while col k has the gcd of their row i entries. Note
|
|
# that this ensures a nonzero entry in col k.
|
|
u, v, d = _gcdex(A[i][k], A[i][j])
|
|
r, s = A[i][k] // d, A[i][j] // d
|
|
add_columns(A, k, j, u, v, -s, r)
|
|
b = A[i][k]
|
|
# Do not want the pivot entry to be negative.
|
|
if b < 0:
|
|
add_columns(A, k, k, -1, 0, -1, 0)
|
|
b = -b
|
|
# The pivot entry will be 0 iff the row was 0 from the pivot col all the
|
|
# way to the left. In this case, we are still working on the same pivot
|
|
# col for the next row. Therefore:
|
|
if b == 0:
|
|
k += 1
|
|
# If the pivot entry is nonzero, then we want to reduce all entries to its
|
|
# right in the sense of the division algorithm, i.e. make them all remainders
|
|
# w.r.t. the pivot as divisor.
|
|
else:
|
|
for j in range(k + 1, n):
|
|
q = A[i][j] // b
|
|
add_columns(A, j, k, 1, -q, 0, 1)
|
|
# Finally, the HNF consists of those columns of A in which we succeeded in making
|
|
# a nonzero pivot.
|
|
return DomainMatrix.from_rep(A)[:, k:]
|
|
|
|
|
|
def _hermite_normal_form_modulo_D(A, D):
|
|
r"""
|
|
Perform the mod *D* Hermite Normal Form reduction algorithm on
|
|
:py:class:`~.DomainMatrix` *A*.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If *A* is an $m \times n$ matrix of rank $m$, having Hermite Normal Form
|
|
$W$, and if *D* is any positive integer known in advance to be a multiple
|
|
of $\det(W)$, then the HNF of *A* can be computed by an algorithm that
|
|
works mod *D* in order to prevent coefficient explosion.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
A : :py:class:`~.DomainMatrix` over :ref:`ZZ`
|
|
$m \times n$ matrix, having rank $m$.
|
|
D : :ref:`ZZ`
|
|
Positive integer, known to be a multiple of the determinant of the
|
|
HNF of *A*.
|
|
|
|
Returns
|
|
=======
|
|
|
|
:py:class:`~.DomainMatrix`
|
|
The HNF of matrix *A*.
|
|
|
|
Raises
|
|
======
|
|
|
|
DMDomainError
|
|
If the domain of the matrix is not :ref:`ZZ`, or
|
|
if *D* is given but is not in :ref:`ZZ`.
|
|
|
|
DMShapeError
|
|
If the matrix has more rows than columns.
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
|
|
(See Algorithm 2.4.8.)
|
|
|
|
"""
|
|
if not A.domain.is_ZZ:
|
|
raise DMDomainError('Matrix must be over domain ZZ.')
|
|
if not ZZ.of_type(D) or D < 1:
|
|
raise DMDomainError('Modulus D must be positive element of domain ZZ.')
|
|
|
|
def add_columns_mod_R(m, R, i, j, a, b, c, d):
|
|
# replace m[:, i] by (a*m[:, i] + b*m[:, j]) % R
|
|
# and m[:, j] by (c*m[:, i] + d*m[:, j]) % R
|
|
for k in range(len(m)):
|
|
e = m[k][i]
|
|
m[k][i] = symmetric_residue((a * e + b * m[k][j]) % R, R)
|
|
m[k][j] = symmetric_residue((c * e + d * m[k][j]) % R, R)
|
|
|
|
W = defaultdict(dict)
|
|
|
|
m, n = A.shape
|
|
if n < m:
|
|
raise DMShapeError('Matrix must have at least as many columns as rows.')
|
|
A = A.to_dense().rep.copy()
|
|
k = n
|
|
R = D
|
|
for i in range(m - 1, -1, -1):
|
|
k -= 1
|
|
for j in range(k - 1, -1, -1):
|
|
if A[i][j] != 0:
|
|
u, v, d = _gcdex(A[i][k], A[i][j])
|
|
r, s = A[i][k] // d, A[i][j] // d
|
|
add_columns_mod_R(A, R, k, j, u, v, -s, r)
|
|
b = A[i][k]
|
|
if b == 0:
|
|
A[i][k] = b = R
|
|
u, v, d = _gcdex(b, R)
|
|
for ii in range(m):
|
|
W[ii][i] = u*A[ii][k] % R
|
|
if W[i][i] == 0:
|
|
W[i][i] = R
|
|
for j in range(i + 1, m):
|
|
q = W[i][j] // W[i][i]
|
|
add_columns(W, j, i, 1, -q, 0, 1)
|
|
R //= d
|
|
return DomainMatrix(W, (m, m), ZZ).to_dense()
|
|
|
|
|
|
def hermite_normal_form(A, *, D=None, check_rank=False):
|
|
r"""
|
|
Compute the Hermite Normal Form of :py:class:`~.DomainMatrix` *A* over
|
|
:ref:`ZZ`.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import ZZ
|
|
>>> from sympy.polys.matrices import DomainMatrix
|
|
>>> from sympy.polys.matrices.normalforms import hermite_normal_form
|
|
>>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)],
|
|
... [ZZ(3), ZZ(9), ZZ(6)],
|
|
... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ)
|
|
>>> print(hermite_normal_form(m).to_Matrix())
|
|
Matrix([[10, 0, 2], [0, 15, 3], [0, 0, 2]])
|
|
|
|
Parameters
|
|
==========
|
|
|
|
A : $m \times n$ ``DomainMatrix`` over :ref:`ZZ`.
|
|
|
|
D : :ref:`ZZ`, optional
|
|
Let $W$ be the HNF of *A*. If known in advance, a positive integer *D*
|
|
being any multiple of $\det(W)$ may be provided. In this case, if *A*
|
|
also has rank $m$, then we may use an alternative algorithm that works
|
|
mod *D* in order to prevent coefficient explosion.
|
|
|
|
check_rank : boolean, optional (default=False)
|
|
The basic assumption is that, if you pass a value for *D*, then
|
|
you already believe that *A* has rank $m$, so we do not waste time
|
|
checking it for you. If you do want this to be checked (and the
|
|
ordinary, non-modulo *D* algorithm to be used if the check fails), then
|
|
set *check_rank* to ``True``.
|
|
|
|
Returns
|
|
=======
|
|
|
|
:py:class:`~.DomainMatrix`
|
|
The HNF of matrix *A*.
|
|
|
|
Raises
|
|
======
|
|
|
|
DMDomainError
|
|
If the domain of the matrix is not :ref:`ZZ`, or
|
|
if *D* is given but is not in :ref:`ZZ`.
|
|
|
|
DMShapeError
|
|
If the mod *D* algorithm is used but the matrix has more rows than
|
|
columns.
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
|
|
(See Algorithms 2.4.5 and 2.4.8.)
|
|
|
|
"""
|
|
if not A.domain.is_ZZ:
|
|
raise DMDomainError('Matrix must be over domain ZZ.')
|
|
if D is not None and (not check_rank or A.convert_to(QQ).rank() == A.shape[0]):
|
|
return _hermite_normal_form_modulo_D(A, D)
|
|
else:
|
|
return _hermite_normal_form(A)
|