def __cmp__(self, right):
"""
Compare self to right.
EXAMPLES::
sage: x = Gamma0(18)([19,1,18,1])
sage: x.__cmp__(3) is not 0
True
sage: x.__cmp__(x)
0
sage: x = Gamma0(5)([1,1,0,1])
sage: x == 0
False
This once caused a segfault (see trac #5443)::
sage: r,s,t,u,v = map(SL2Z, [[1, 1, 0, 1], [-1, 0, 0, -1], [1, -1, 0, 1], [1, -1, 2, -1], [-1, 1, -2, 1]])
sage: v == s*u
True
sage: s*u == v
True
"""
from sage.misc.functional import parent
if parent(self) != parent(right):
return cmp(type(self), type(right))
else:
return cmp(self.__x, right.__x)
def __cmp__(self, right):
"""
Compare self to right.
EXAMPLES::
sage: x = Gamma0(18)([19,1,18,1])
sage: x.__cmp__(3) is not 0
True
sage: x.__cmp__(x)
0
sage: x = Gamma0(5)([1,1,0,1])
sage: x == 0
False
This once caused a segfault (see trac #5443)::
sage: r,s,t,u,v = map(SL2Z, [[1, 1, 0, 1], [-1, 0, 0, -1], [1, -1, 0, 1], [1, -1, 2, -1], [-1, 1, -2, 1]])
sage: v == s*u
True
sage: s*u == v
True
"""
from sage.misc.functional import parent
if parent(self) != parent(right):
return cmp(type(self), type(right))
else:
return cmp(self.__x, right.__x)
def _negaconvolution_fft(L1, L2, n):
"""
Returns negacyclic convolution of lists L1 and L2, using FFT
algorithm. L1 and L2 must both be length `2^n`, where
`n \geq 3`. Assumes all entries of L1 and L2 belong to the
same ring.
EXAMPLES::
sage: from sage.rings.polynomial.convolution import _negaconvolution_naive
sage: from sage.rings.polynomial.convolution import _negaconvolution_fft
sage: _negaconvolution_naive(range(8), range(5, 13))
[-224, -234, -224, -192, -136, -54, 56, 196]
sage: _negaconvolution_fft(range(8), range(5, 13), 3)
[-224, -234, -224, -192, -136, -54, 56, 196]
::
sage: for n in range(3, 10):
... L1 = [ZZ.random_element(100) for _ in range(1 << n)]
... L2 = [ZZ.random_element(100) for _ in range(1 << n)]
... assert _negaconvolution_naive(L1, L2) == _negaconvolution_fft(L1, L2, n)
"""
assert n >= 3
R = parent(L1[0])
# split into 2^m pieces of 2^(k-1) coefficients each, with k as small
# as possible, subject to m <= k (so that the ring of Fourier coefficients
# has enough roots of unity)
m = (n + 1) >> 1
k = n + 1 - m
M = 1 << m
K = 1 << k
# split inputs into polynomials
L1 = _split(L1, m, k)
L2 = _split(L2, m, k)
# fft each input
_fft(L1, K, 0, m, K >> 1)
_fft(L2, K, 0, m, K >> 1)
# pointwise multiply
L3 = [_negaconvolution(L1[i], L2[i], k) for i in range(M)]
# inverse fft
_ifft(L3, K, 0, m, K >> 1)
# combine back into a single list
L3 = _nega_combine(L3, m, k)
# normalise
return [R(x / M) for x in L3]
def _negaconvolution_fft(L1, L2, n):
"""
Returns negacyclic convolution of lists L1 and L2, using FFT
algorithm. L1 and L2 must both be length `2^n`, where
`n \geq 3`. Assumes all entries of L1 and L2 belong to the
same ring.
EXAMPLES::
sage: from sage.rings.polynomial.convolution import _negaconvolution_naive
sage: from sage.rings.polynomial.convolution import _negaconvolution_fft
sage: _negaconvolution_naive(range(8), range(5, 13))
[-224, -234, -224, -192, -136, -54, 56, 196]
sage: _negaconvolution_fft(range(8), range(5, 13), 3)
[-224, -234, -224, -192, -136, -54, 56, 196]
::
sage: for n in range(3, 10):
... L1 = [ZZ.random_element(100) for _ in range(1 << n)]
... L2 = [ZZ.random_element(100) for _ in range(1 << n)]
... assert _negaconvolution_naive(L1, L2) == _negaconvolution_fft(L1, L2, n)
"""
assert n >= 3
R = parent(L1[0])
# split into 2^m pieces of 2^(k-1) coefficients each, with k as small
# as possible, subject to m <= k (so that the ring of Fourier coefficients
# has enough roots of unity)
m = (n + 1) >> 1
k = n + 1 - m
M = 1 << m
K = 1 << k
# split inputs into polynomials
L1 = _split(L1, m, k)
L2 = _split(L2, m, k)
# fft each input
_fft(L1, K, 0, m, K >> 1)
_fft(L2, K, 0, m, K >> 1)
# pointwise multiply
L3 = [_negaconvolution(L1[i], L2[i], k) for i in range(M)]
# inverse fft
_ifft(L3, K, 0, m, K >> 1)
# combine back into a single list
L3 = _nega_combine(L3, m, k)
# normalise
return [R(x / M) for x in L3]
def __contains__(self, x):
r"""
Check in the element x is in the mathematical parent self.
EXAMPLES::
sage: D5 = FiniteCoxeterGroups().example()
sage: D5.an_element() in D5
True
sage: 1 in D5
False
(also tested by :meth:`test_an_element` :meth:`test_some_elements`)
"""
from sage.misc.functional import parent
return parent(x) is self
def __contains__(self, x):
r"""
Check in the element x is in the mathematical parent self.
EXAMPLES::
sage: D5 = FiniteCoxeterGroups().example()
sage: D5.an_element() in D5
True
sage: 1 in D5
False
(also tested by :meth:`test_an_element` :meth:`test_some_elements`)
"""
from sage.misc.functional import parent
return parent(x) is self
def _coerce_impl(self, x):
"""
Coerce x into self, if possible.
EXAMPLES::
sage: J = J0(37) ; J.Hom(J)._coerce_impl(matrix(ZZ,4,[5..20]))
Abelian variety endomorphism of Abelian variety J0(37) of dimension 2
sage: K = J0(11) * J0(11) ; J.Hom(K)._coerce_impl(matrix(ZZ,4,[5..20]))
Abelian variety morphism:
From: Abelian variety J0(37) of dimension 2
To: Abelian variety J0(11) x J0(11) of dimension 2
"""
if self.matrix_space().has_coerce_map_from(parent(x)):
return self(x)
else:
return HomsetWithBase._coerce_impl(self, x)
def _coerce_impl(self, x):
"""
Coerce x into self, if possible.
EXAMPLES::
sage: J = J0(37) ; J.Hom(J)._coerce_impl(matrix(ZZ,4,[5..20]))
Abelian variety endomorphism of Abelian variety J0(37) of dimension 2
sage: K = J0(11) * J0(11) ; J.Hom(K)._coerce_impl(matrix(ZZ,4,[5..20]))
Abelian variety morphism:
From: Abelian variety J0(37) of dimension 2
To: Abelian variety J0(11) x J0(11) of dimension 2
"""
if self.matrix_space().has_coerce_map_from(parent(x)):
return self(x)
else:
return HomsetWithBase._coerce_impl(self, x)
def _split(L, m, k):
"""
Assumes L is a list of length `2^{m+k-1}`. Splits it into
`2^m` lists of length `2^{k-1}`, returned as a list
of lists. Each list is zero padded up to length `2^k`.
EXAMPLES::
sage: from sage.rings.polynomial.convolution import _split
sage: _split([1, 2, 3, 4, 5, 6, 7, 8], 2, 2)
[[1, 2, 0, 0], [3, 4, 0, 0], [5, 6, 0, 0], [7, 8, 0, 0]]
sage: _split([1, 2, 3, 4, 5, 6, 7, 8], 1, 3)
[[1, 2, 3, 4, 0, 0, 0, 0], [5, 6, 7, 8, 0, 0, 0, 0]]
sage: _split([1, 2, 3, 4, 5, 6, 7, 8], 3, 1)
[[1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0]]
"""
K = 1 << (k-1)
zero = parent(L[0])(0)
zeroes = [zero] * K
return [[L[i+j] for j in range(K)] + zeroes for i in range(0, K << m, K)]
def _split(L, m, k):
"""
Assumes L is a list of length `2^{m+k-1}`. Splits it into
`2^m` lists of length `2^{k-1}`, returned as a list
of lists. Each list is zero padded up to length `2^k`.
EXAMPLES::
sage: from sage.rings.polynomial.convolution import _split
sage: _split([1, 2, 3, 4, 5, 6, 7, 8], 2, 2)
[[1, 2, 0, 0], [3, 4, 0, 0], [5, 6, 0, 0], [7, 8, 0, 0]]
sage: _split([1, 2, 3, 4, 5, 6, 7, 8], 1, 3)
[[1, 2, 3, 4, 0, 0, 0, 0], [5, 6, 7, 8, 0, 0, 0, 0]]
sage: _split([1, 2, 3, 4, 5, 6, 7, 8], 3, 1)
[[1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0]]
"""
K = 1 << (k - 1)
zero = parent(L[0])(0)
zeroes = [zero] * K
return [[L[i + j] for j in range(K)] + zeroes for i in range(0, K << m, K)]
def _convolution_fft(L1, L2):
"""
Returns convolution of non-empty lists L1 and L2, using FFT
algorithm. L1 and L2 may have arbitrary lengths `\geq 4`.
Assumes all entries of L1 and L2 belong to the same ring.
EXAMPLES::
sage: from sage.rings.polynomial.convolution import _convolution_naive
sage: from sage.rings.polynomial.convolution import _convolution_fft
sage: _convolution_naive([1, 2, 3], [4, 5, 6])
[4, 13, 28, 27, 18]
sage: _convolution_fft([1, 2, 3], [4, 5, 6])
[4, 13, 28, 27, 18]
::
sage: for len1 in range(4, 30):
... for len2 in range(4, 30):
... L1 = [ZZ.random_element(100) for _ in range(len1)]
... L2 = [ZZ.random_element(100) for _ in range(len2)]
... assert _convolution_naive(L1, L2) == _convolution_fft(L1, L2)
"""
R = parent(L1[0])
# choose n so that output convolution length is 2^n
len1 = len(L1)
len2 = len(L2)
outlen = len1 + len2 - 1
n = int(ceil(log(outlen) / log(2.0)))
# split into 2^m pieces of 2^(k-1) coefficients each, with k as small
# as possible, subject to m <= k + 1 (so that the ring of Fourier
# coefficients has enough roots of unity)
m = (n >> 1) + 1
k = n + 1 - m
N = 1 << n
M = 1 << m
K = 1 << k
# zero pad inputs up to length N
zero = R(0)
L1 = L1 + [zero] * (N - len(L1))
L2 = L2 + [zero] * (N - len(L2))
# split inputs into polynomials
L1 = _split(L1, m, k)
L2 = _split(L2, m, k)
# fft each input
_fft(L1, K, 0, m, K)
_fft(L2, K, 0, m, K)
# pointwise multiply
L3 = [_negaconvolution(L1[i], L2[i], k) for i in range(M)]
# inverse fft
_ifft(L3, K, 0, m, K)
# combine back into a single list
L3 = _combine(L3, m, k)
# normalise, and truncate to correct length
return [R(L3[i] / M) for i in range(outlen)]
def __call__(self, M):
r"""
Create a homomorphism in this space from M. M can be any of the
following:
- a Morphism of abelian varieties
- a matrix of the appropriate size
(i.e. 2\*self.domain().dimension() x
2\*self.codomain().dimension()) whose entries are coercible
into self.base_ring()
- anything that can be coerced into self.matrix_space()
EXAMPLES::
sage: H = Hom(J0(11), J0(22))
sage: phi = H(matrix(ZZ,2,4,[5..12])) ; phi
Abelian variety morphism:
From: Abelian variety J0(11) of dimension 1
To: Abelian variety J0(22) of dimension 2
sage: phi.matrix()
[ 5 6 7 8]
[ 9 10 11 12]
sage: phi.matrix().parent()
Full MatrixSpace of 2 by 4 dense matrices over Integer Ring
::
sage: H = J0(22).Hom(J0(11)*J0(11))
sage: m1 = J0(22).degeneracy_map(11,1).matrix() ; m1
[ 0 1]
[-1 1]
[-1 0]
[ 0 -1]
sage: m2 = J0(22).degeneracy_map(11,2).matrix() ; m2
[ 1 -2]
[ 0 -2]
[ 1 -1]
[ 0 -1]
sage: m = m1.transpose().stack(m2.transpose()).transpose() ; m
[ 0 1 1 -2]
[-1 1 0 -2]
[-1 0 1 -1]
[ 0 -1 0 -1]
sage: phi = H(m) ; phi
Abelian variety morphism:
From: Abelian variety J0(22) of dimension 2
To: Abelian variety J0(11) x J0(11) of dimension 2
sage: phi.matrix()
[ 0 1 1 -2]
[-1 1 0 -2]
[-1 0 1 -1]
[ 0 -1 0 -1]
"""
if isinstance(M, morphism.Morphism):
if M.parent() is self:
return M
elif M.domain() == self.domain() and M.codomain() == self.codomain():
M = M.matrix()
else:
raise ValueError, "cannot convert %s into %s" % (M, self)
elif is_Matrix(M):
if M.base_ring() != ZZ:
M = M.change_ring(ZZ)
if M.nrows() != 2*self.domain().dimension() or M.ncols() != 2*self.codomain().dimension():
raise TypeError, "matrix has wrong dimension"
elif self.matrix_space().has_coerce_map_from(parent(M)):
M = self.matrix_space()(M)
else:
raise TypeError, "can only coerce in matrices or morphisms"
return morphism.Morphism(self, M)
def _convolution_fft(L1, L2):
"""
Returns convolution of non-empty lists L1 and L2, using FFT
algorithm. L1 and L2 may have arbitrary lengths `\geq 4`.
Assumes all entries of L1 and L2 belong to the same ring.
EXAMPLES::
sage: from sage.rings.polynomial.convolution import _convolution_naive
sage: from sage.rings.polynomial.convolution import _convolution_fft
sage: _convolution_naive([1, 2, 3], [4, 5, 6])
[4, 13, 28, 27, 18]
sage: _convolution_fft([1, 2, 3], [4, 5, 6])
[4, 13, 28, 27, 18]
::
sage: for len1 in range(4, 30):
... for len2 in range(4, 30):
... L1 = [ZZ.random_element(100) for _ in range(len1)]
... L2 = [ZZ.random_element(100) for _ in range(len2)]
... assert _convolution_naive(L1, L2) == _convolution_fft(L1, L2)
"""
R = parent(L1[0])
# choose n so that output convolution length is 2^n
len1 = len(L1)
len2 = len(L2)
outlen = len1 + len2 - 1
n = int(ceil(log(outlen) / log(2.0)))
# split into 2^m pieces of 2^(k-1) coefficients each, with k as small
# as possible, subject to m <= k + 1 (so that the ring of Fourier
# coefficients has enough roots of unity)
m = (n >> 1) + 1
k = n + 1 - m
N = 1 << n
M = 1 << m
K = 1 << k
# zero pad inputs up to length N
zero = R(0)
L1 = L1 + [zero] * (N - len(L1))
L2 = L2 + [zero] * (N - len(L2))
# split inputs into polynomials
L1 = _split(L1, m, k)
L2 = _split(L2, m, k)
# fft each input
_fft(L1, K, 0, m, K)
_fft(L2, K, 0, m, K)
# pointwise multiply
L3 = [_negaconvolution(L1[i], L2[i], k) for i in range(M)]
# inverse fft
_ifft(L3, K, 0, m, K)
# combine back into a single list
L3 = _combine(L3, m, k)
# normalise, and truncate to correct length
return [R(L3[i] / M) for i in range(outlen)]
def __call__(self, M):
r"""
Create a homomorphism in this space from M. M can be any of the
following:
- a Morphism of abelian varieties
- a matrix of the appropriate size
(i.e. 2\*self.domain().dimension() x
2\*self.codomain().dimension()) whose entries are coercible
into self.base_ring()
- anything that can be coerced into self.matrix_space()
EXAMPLES::
sage: H = Hom(J0(11), J0(22))
sage: phi = H(matrix(ZZ,2,4,[5..12])) ; phi
Abelian variety morphism:
From: Abelian variety J0(11) of dimension 1
To: Abelian variety J0(22) of dimension 2
sage: phi.matrix()
[ 5 6 7 8]
[ 9 10 11 12]
sage: phi.matrix().parent()
Full MatrixSpace of 2 by 4 dense matrices over Integer Ring
::
sage: H = J0(22).Hom(J0(11)*J0(11))
sage: m1 = J0(22).degeneracy_map(11,1).matrix() ; m1
[ 0 1]
[-1 1]
[-1 0]
[ 0 -1]
sage: m2 = J0(22).degeneracy_map(11,2).matrix() ; m2
[ 1 -2]
[ 0 -2]
[ 1 -1]
[ 0 -1]
sage: m = m1.transpose().stack(m2.transpose()).transpose() ; m
[ 0 1 1 -2]
[-1 1 0 -2]
[-1 0 1 -1]
[ 0 -1 0 -1]
sage: phi = H(m) ; phi
Abelian variety morphism:
From: Abelian variety J0(22) of dimension 2
To: Abelian variety J0(11) x J0(11) of dimension 2
sage: phi.matrix()
[ 0 1 1 -2]
[-1 1 0 -2]
[-1 0 1 -1]
[ 0 -1 0 -1]
"""
if isinstance(M, morphism.Morphism):
if M.parent() is self:
return M
elif M.domain() == self.domain() and M.codomain() == self.codomain(
):
M = M.matrix()
else:
raise ValueError, "cannot convert %s into %s" % (M, self)
elif is_Matrix(M):
if M.base_ring() != ZZ:
M = M.change_ring(ZZ)
if M.nrows() != 2 * self.domain().dimension() or M.ncols(
) != 2 * self.codomain().dimension():
raise TypeError, "matrix has wrong dimension"
elif self.matrix_space().has_coerce_map_from(parent(M)):
M = self.matrix_space()(M)
else:
raise TypeError, "can only coerce in matrices or morphisms"
return morphism.Morphism(self, M)