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quaternion_algebra_cython.pyx
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quaternion_algebra_cython.pyx
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"""
Optimized Cython code needed by quaternion algebras.
This is a collection of miscellaneous routines that are in Cython for
speed purposes and are used by the quaternion algebra code. For
example, there are functions for quickly constructing an n x 4 matrix
from a list of n rational quaternions.
AUTHORS:
- William Stein
"""
########################################################################
# Copyright (C) 2009 William Stein <wstein@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# The full text of the GPL is available at:
#
# http://www.gnu.org/licenses/
########################################################################
include "sage/ext/stdsage.pxi"
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.integer cimport Integer
from sage.matrix.matrix_space import MatrixSpace
from sage.matrix.matrix_integer_dense cimport Matrix_integer_dense
from sage.matrix.matrix_rational_dense cimport Matrix_rational_dense
from quaternion_algebra_element cimport QuaternionAlgebraElement_rational_field
from sage.libs.gmp.mpz cimport mpz_t, mpz_lcm, mpz_init, mpz_set, mpz_clear, mpz_init_set, mpz_mul, mpz_fdiv_q, mpz_cmp_si
from sage.libs.gmp.mpq cimport mpq_set_num, mpq_set_den, mpq_canonicalize
def integral_matrix_and_denom_from_rational_quaternions(v, reverse=False):
r"""
Given a list of rational quaternions, return matrix `A` over `\ZZ`
and denominator `d`, such that the rows of `(1/d)A` are the
entries of the quaternions.
INPUT:
- v -- a list of quaternions in a rational quaternion algebra
- reverse -- whether order of the coordinates as well as the
order of the list v should be reversed.
OUTPUT:
- a matrix over ZZ
- an integer (the common denominator)
EXAMPLES::
sage: A.<i,j,k>=QuaternionAlgebra(-4,-5)
sage: sage.algebras.quatalg.quaternion_algebra_cython.integral_matrix_and_denom_from_rational_quaternions([i/2,1/3+j+k])
(
[0 3 0 0]
[2 0 6 6], 6
)
sage: sage.algebras.quatalg.quaternion_algebra_cython.integral_matrix_and_denom_from_rational_quaternions([i/2,1/3+j+k], reverse=True)
(
[6 6 0 2]
[0 0 3 0], 6
)
"""
# This function is an optimized version of
# MatrixSpace(QQ,len(v),4)([x.coefficient_tuple() for x in v], coerce=False)._clear_denom
cdef Py_ssize_t i, n=len(v)
M = MatrixSpace(ZZ, n, 4)
cdef Matrix_integer_dense A = M.zero_matrix().__copy__()
if n == 0: return A
# Find least common multiple of the denominators
cdef QuaternionAlgebraElement_rational_field x
cdef Integer d = Integer()
# set denom to the denom of the first quaternion
x = v[0]; mpz_set(d.value, x.d)
for x in v[1:]:
mpz_lcm(d.value, d.value, x.d)
# Now fill in each row x of A, multiplying it by q = d/denom(x)
cdef mpz_t q
cdef mpz_t* row
cdef mpz_t tmp
mpz_init(q)
mpz_init(tmp)
for i in range(n):
x = v[i]
mpz_fdiv_q(q, d.value, x.d)
if reverse:
mpz_mul(tmp, q, x.x)
A.set_unsafe_mpz(n-i-1,3,tmp)
mpz_mul(tmp, q, x.y)
A.set_unsafe_mpz(n-i-1,2,tmp)
mpz_mul(tmp, q, x.z)
A.set_unsafe_mpz(n-i-1,1,tmp)
mpz_mul(tmp, q, x.w)
A.set_unsafe_mpz(n-i-1,0,tmp)
else:
mpz_mul(tmp, q, x.x)
A.set_unsafe_mpz(i,0,tmp)
mpz_mul(tmp, q, x.y)
A.set_unsafe_mpz(i,1,tmp)
mpz_mul(tmp, q, x.z)
A.set_unsafe_mpz(i,2,tmp)
mpz_mul(tmp, q, x.w)
A.set_unsafe_mpz(i,3,tmp)
mpz_clear(q)
mpz_clear(tmp)
return A, d
def rational_matrix_from_rational_quaternions(v, reverse=False):
"""
Return matrix over the rationals whose rows have entries the
coefficients of the rational quaternions in v.
INPUT:
- v -- a list of quaternions in a rational quaternion algebra
- reverse -- whether order of the coordinates as well as the
order of the list v should be reversed.
OUTPUT:
- a matrix over QQ
EXAMPLES::
sage: A.<i,j,k>=QuaternionAlgebra(-4,-5)
sage: sage.algebras.quatalg.quaternion_algebra_cython.rational_matrix_from_rational_quaternions([i/2,1/3+j+k])
[ 0 1/2 0 0]
[1/3 0 1 1]
sage: sage.algebras.quatalg.quaternion_algebra_cython.rational_matrix_from_rational_quaternions([i/2,1/3+j+k], reverse=True)
[ 1 1 0 1/3]
[ 0 0 1/2 0]
"""
cdef Py_ssize_t i, j, n=len(v)
M = MatrixSpace(QQ, n, 4)
cdef Matrix_rational_dense A = M.zero_matrix().__copy__()
if n == 0: return A
cdef QuaternionAlgebraElement_rational_field x
if reverse:
for i in range(n):
x = v[i]
mpq_set_num(A._matrix[n-i-1][3], x.x)
mpq_set_num(A._matrix[n-i-1][2], x.y)
mpq_set_num(A._matrix[n-i-1][1], x.z)
mpq_set_num(A._matrix[n-i-1][0], x.w)
if mpz_cmp_si(x.d,1):
for j in range(4):
mpq_set_den(A._matrix[n-i-1][j], x.d)
mpq_canonicalize(A._matrix[n-i-1][j])
else:
for i in range(n):
x = v[i]
mpq_set_num(A._matrix[i][0], x.x)
mpq_set_num(A._matrix[i][1], x.y)
mpq_set_num(A._matrix[i][2], x.z)
mpq_set_num(A._matrix[i][3], x.w)
if mpz_cmp_si(x.d,1):
for j in range(4):
mpq_set_den(A._matrix[i][j], x.d)
mpq_canonicalize(A._matrix[i][j])
return A
def rational_quaternions_from_integral_matrix_and_denom(A, Matrix_integer_dense H, Integer d, reverse=False):
"""
Given an integral matrix and denominator, returns a list of rational quaternions.
INPUT:
- A -- rational quaternion algebra
- H -- matrix over the integers
- d -- integer
- reverse -- whether order of the coordinates as well as the
order of the list v should be reversed.
OUTPUT:
- list of H.nrows() elements of A
EXAMPLES::
sage: A.<i,j,k>=QuaternionAlgebra(-1,-2)
sage: f = sage.algebras.quatalg.quaternion_algebra_cython.rational_quaternions_from_integral_matrix_and_denom
sage: f(A, matrix([[1,2,3,4],[-1,2,-4,3]]), 3)
[1/3 + 2/3*i + j + 4/3*k, -1/3 + 2/3*i - 4/3*j + k]
sage: f(A, matrix([[3,-4,2,-1],[4,3,2,1]]), 3, reverse=True)
[1/3 + 2/3*i + j + 4/3*k, -1/3 + 2/3*i - 4/3*j + k]
"""
#
# This is an optimized version of the following interpreted Python code.
# H2 = H.change_ring(QQ)._rmul_(1/d)
# return [A(v.list()) for v in H2.rows()]
#
cdef QuaternionAlgebraElement_rational_field x
v = []
cdef Integer a, b
a = Integer(A.invariants()[0])
b = Integer(A.invariants()[1])
cdef Py_ssize_t i, j
cdef mpz_t tmp
mpz_init(tmp)
if reverse:
rng = range(H.nrows()-1,-1,-1)
else:
rng = range(H.nrows())
for i in rng:
x = <QuaternionAlgebraElement_rational_field> PY_NEW(QuaternionAlgebraElement_rational_field)
x._parent = A
mpz_set(x.a, a.value)
mpz_set(x.b, b.value)
if reverse:
H.get_unsafe_mpz(i,3,tmp)
mpz_init_set(x.x, tmp)
H.get_unsafe_mpz(i,2,tmp)
mpz_init_set(x.y, tmp)
H.get_unsafe_mpz(i,1,tmp)
mpz_init_set(x.z, tmp)
H.get_unsafe_mpz(i,0,tmp)
mpz_init_set(x.w, tmp)
else:
H.get_unsafe_mpz(i,0,tmp)
mpz_init_set(x.x, tmp)
H.get_unsafe_mpz(i,1,tmp)
mpz_init_set(x.y, tmp)
H.get_unsafe_mpz(i,2,tmp)
mpz_init_set(x.z, tmp)
H.get_unsafe_mpz(i,3,tmp)
mpz_init_set(x.w, tmp)
mpz_init_set(x.d, d.value)
# WARNING -- we do *not* canonicalize the entries in the quaternion. This is
# I think _not_ needed for quaternion_element.pyx
v.append(x)
mpz_clear(tmp)
return v
from sage.rings.rational_field import QQ
MS_16_4 = MatrixSpace(QQ,16,4)