def basis_of_short_vectors(self, show_lengths=False, safe_flag=True):
"""
Return a basis for `ZZ^n` made of vectors with minimal lengths Q(`v`).
The safe_flag allows us to select whether we want a copy of the
output, or the original output. By default safe_flag = True, so
we return a copy of the cached information. If this is set to
False, then the routine is much faster but the return values are
vulnerable to being corrupted by the user.
OUTPUT:
a list of vectors, and optionally a list of values for each vector.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.basis_of_short_vectors()
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]
sage: Q.basis_of_short_vectors(True)
([(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)], [1, 3, 5, 7])
"""
## Try to use the cached results
try:
## Return the appropriate result
if show_lengths:
if safe_flag:
return deep_copy(self.__basis_of_short_vectors), deepcopy(self.__basis_of_short_vectors_lengths)
else:
return self.__basis_of_short_vectors, self.__basis_of_short_vectors_lengths
else:
if safe_flag:
return deepcopy(self.__basis_of_short_vectors)
else:
return deepcopy(self.__basis_of_short_vectors)
except Exception:
pass
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
## Run through all possible minimal lengths to find a spanning set of vectors
n = self.dim()
# MS = MatrixSpace(QQ, n)
M1 = Matrix([[0]])
vec_len = 0
while M1.rank() < n:
## DIAGONSTIC
# print
# print "Starting with vec_len = ", vec_len
# print "M_str = ", M_str
vec_len += 1
gp_mat = gp.qfminim(M_str, vec_len, Max_number_of_vectors)[3].mattranspose()
number_of_vecs = ZZ(gp_mat.matsize()[1])
vector_list = []
for i in range(number_of_vecs):
# print "List at", i, ":", list(gp_mat[i+1,])
new_vec = vector([ZZ(x) for x in list(gp_mat[i + 1,])])
vector_list.append(new_vec)
## DIAGNOSTIC
# print "number_of_vecs = ", number_of_vecs
# print "vector_list = ", vector_list
## Make a matrix from the short vectors
if len(vector_list) > 0:
M1 = Matrix(vector_list)
## DIAGNOSTIC
# print "matrix of vectors = \n", M1
# print "rank of the matrix = ", M1.rank()
# print " vec_len = ", vec_len
# print M1
## Organize these vectors by length (and also introduce their negatives)
max_len = vec_len / 2
vector_list_by_length = [[] for _ in range(max_len + 1)]
for v in vector_list:
l = self(v)
vector_list_by_length[l].append(v)
vector_list_by_length[l].append(vector([-x for x in v]))
## Make a matrix from the column vectors (in order of ascending length).
sorted_list = []
for i in range(len(vector_list_by_length)):
for v in vector_list_by_length[i]:
sorted_list.append(v)
sorted_matrix = Matrix(sorted_list).transpose()
## Determine a basis of vectors of minimal length
pivots = sorted_matrix.pivots()
basis = [sorted_matrix.column(i) for i in pivots]
pivot_lengths = [self(v) for v in basis]
## DIAGNOSTIC
# print "basis = ", basis
# print "pivot_lengths = ", pivot_lengths
## Cache the result
self.__basis_of_short_vectors = basis
self.__basis_of_short_vectors_lenghts = pivot_lengths
## Return the appropriate result
if show_lengths:
return basis, pivot_lengths
else:
return basis
def basis_of_short_vectors(self, show_lengths=False, safe_flag=None):
"""
Return a basis for `ZZ^n` made of vectors with minimal lengths Q(`v`).
OUTPUT: a tuple of vectors, and optionally a tuple of values for
each vector.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.basis_of_short_vectors()
((1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1))
sage: Q.basis_of_short_vectors(True)
(((1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)), (1, 3, 5, 7))
The returned vectors are immutable::
sage: v = Q.basis_of_short_vectors()[0]
sage: v
(1, 0, 0, 0)
sage: v[0] = 0
Traceback (most recent call last):
...
ValueError: vector is immutable; please change a copy instead (use copy())
"""
if safe_flag is not None:
from sage.misc.superseded import deprecation
deprecation(
18673,
"The safe_flag argument to basis_of_short_vectors() is deprecated and no longer used"
)
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix for the associated Hessian matrix
M_pari = self._pari_()
## Run through all possible minimal lengths to find a spanning set of vectors
n = self.dim()
M1 = Matrix([[0]])
vec_len = 0
while M1.rank() < n:
vec_len += 1
pari_mat = M_pari.qfminim(vec_len, Max_number_of_vectors)[2]
number_of_vecs = ZZ(pari_mat.matsize()[1])
vector_list = []
for i in range(number_of_vecs):
new_vec = vector([ZZ(x) for x in list(pari_mat[i])])
vector_list.append(new_vec)
## Make a matrix from the short vectors
if len(vector_list) > 0:
M1 = Matrix(vector_list)
## Organize these vectors by length (and also introduce their negatives)
max_len = vec_len // 2
vector_list_by_length = [[] for _ in range(max_len + 1)]
for v in vector_list:
l = self(v)
vector_list_by_length[l].append(v)
vector_list_by_length[l].append(vector([-x for x in v]))
## Make a matrix from the column vectors (in order of ascending length).
sorted_list = []
for i in range(len(vector_list_by_length)):
for v in vector_list_by_length[i]:
sorted_list.append(v)
sorted_matrix = Matrix(sorted_list).transpose()
## Determine a basis of vectors of minimal length
pivots = sorted_matrix.pivots()
basis = tuple(sorted_matrix.column(i) for i in pivots)
for v in basis:
v.set_immutable()
## Return the appropriate result
if show_lengths:
pivot_lengths = tuple(self(v) for v in basis)
return basis, pivot_lengths
else:
return basis
def basis_of_short_vectors(self, show_lengths=False, safe_flag=True):
"""
Return a basis for `ZZ^n` made of vectors with minimal lengths Q(`v`).
The safe_flag allows us to select whether we want a copy of the
output, or the original output. By default safe_flag = True, so
we return a copy of the cached information. If this is set to
False, then the routine is much faster but the return values are
vulnerable to being corrupted by the user.
OUTPUT:
a list of vectors, and optionally a list of values for each vector.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.basis_of_short_vectors()
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]
sage: Q.basis_of_short_vectors(True)
([(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)], [1, 3, 5, 7])
"""
## Try to use the cached results
try:
## Return the appropriate result
if show_lengths:
if safe_flag:
return deep_copy(self.__basis_of_short_vectors), deepcopy(self.__basis_of_short_vectors_lengths)
else:
return self.__basis_of_short_vectors, self.__basis_of_short_vectors_lengths
else:
if safe_flag:
return deepcopy(self.__basis_of_short_vectors)
else:
return deepcopy(self.__basis_of_short_vectors)
except Exception:
pass
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix string for the associated Hessian matrix
M_str = str(gp(self.matrix()))
## Run through all possible minimal lengths to find a spanning set of vectors
n = self.dim()
#MS = MatrixSpace(QQ, n)
M1 = Matrix([[0]])
vec_len = 0
while M1.rank() < n:
## DIAGONSTIC
#print
#print "Starting with vec_len = ", vec_len
#print "M_str = ", M_str
vec_len += 1
gp_mat = gp.qfminim(M_str, vec_len, Max_number_of_vectors)[3].mattranspose()
number_of_vecs = ZZ(gp_mat.matsize()[1])
vector_list = []
for i in range(number_of_vecs):
#print "List at", i, ":", list(gp_mat[i+1,])
new_vec = vector([ZZ(x) for x in list(gp_mat[i+1,])])
vector_list.append(new_vec)
## DIAGNOSTIC
#print "number_of_vecs = ", number_of_vecs
#print "vector_list = ", vector_list
## Make a matrix from the short vectors
if len(vector_list) > 0:
M1 = Matrix(vector_list)
## DIAGNOSTIC
#print "matrix of vectors = \n", M1
#print "rank of the matrix = ", M1.rank()
#print " vec_len = ", vec_len
#print M1
## Organize these vectors by length (and also introduce their negatives)
max_len = vec_len // 2
vector_list_by_length = [[] for _ in range(max_len + 1)]
for v in vector_list:
l = self(v)
vector_list_by_length[l].append(v)
vector_list_by_length[l].append(vector([-x for x in v]))
## Make a matrix from the column vectors (in order of ascending length).
sorted_list = []
for i in range(len(vector_list_by_length)):
for v in vector_list_by_length[i]:
sorted_list.append(v)
sorted_matrix = Matrix(sorted_list).transpose()
## Determine a basis of vectors of minimal length
pivots = sorted_matrix.pivots()
basis = [sorted_matrix.column(i) for i in pivots]
pivot_lengths = [self(v) for v in basis]
## DIAGNOSTIC
#print "basis = ", basis
#print "pivot_lengths = ", pivot_lengths
## Cache the result
self.__basis_of_short_vectors = basis
self.__basis_of_short_vectors_lenghts = pivot_lengths
## Return the appropriate result
if show_lengths:
return basis, pivot_lengths
else:
return basis
def basis_of_short_vectors(self, show_lengths=False, safe_flag=None):
"""
Return a basis for `ZZ^n` made of vectors with minimal lengths Q(`v`).
OUTPUT: a tuple of vectors, and optionally a tuple of values for
each vector.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.basis_of_short_vectors()
((1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1))
sage: Q.basis_of_short_vectors(True)
(((1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)), (1, 3, 5, 7))
The returned vectors are immutable::
sage: v = Q.basis_of_short_vectors()[0]
sage: v
(1, 0, 0, 0)
sage: v[0] = 0
Traceback (most recent call last):
...
ValueError: vector is immutable; please change a copy instead (use copy())
"""
if safe_flag is not None:
from sage.misc.superseded import deprecation
deprecation(18673, "The safe_flag argument to basis_of_short_vectors() is deprecated and no longer used")
## Set an upper bound for the number of vectors to consider
Max_number_of_vectors = 10000
## Generate a PARI matrix for the associated Hessian matrix
M_pari = self.__pari__()
## Run through all possible minimal lengths to find a spanning set of vectors
n = self.dim()
M1 = Matrix([[0]])
vec_len = 0
while M1.rank() < n:
vec_len += 1
pari_mat = M_pari.qfminim(vec_len, Max_number_of_vectors)[2]
number_of_vecs = ZZ(pari_mat.matsize()[1])
vector_list = []
for i in range(number_of_vecs):
new_vec = vector([ZZ(x) for x in list(pari_mat[i])])
vector_list.append(new_vec)
## Make a matrix from the short vectors
if len(vector_list) > 0:
M1 = Matrix(vector_list)
## Organize these vectors by length (and also introduce their negatives)
max_len = vec_len // 2
vector_list_by_length = [[] for _ in range(max_len + 1)]
for v in vector_list:
l = self(v)
vector_list_by_length[l].append(v)
vector_list_by_length[l].append(vector([-x for x in v]))
## Make a matrix from the column vectors (in order of ascending length).
sorted_list = []
for i in range(len(vector_list_by_length)):
for v in vector_list_by_length[i]:
sorted_list.append(v)
sorted_matrix = Matrix(sorted_list).transpose()
## Determine a basis of vectors of minimal length
pivots = sorted_matrix.pivots()
basis = tuple(sorted_matrix.column(i) for i in pivots)
for v in basis:
v.set_immutable()
## Return the appropriate result
if show_lengths:
pivot_lengths = tuple(self(v) for v in basis)
return basis, pivot_lengths
else:
return basis
