def symmetrize_periods(Pa, Pb, tol=1e-4):
r"""Returns symmetric a- and b-periods `Pa_symm` and `Pb_symm`, as well as the
corresponding symplectic operator `Gamma` such that `Gamma [Pa \\ Pb] =
[Pa_symm \\ Pb_symm]`.
Parameters
----------
Pa : complex matrix
Pb : complex matrix
The a- and b-periods, respectively, of a genus `g` Riemann surface.
tol : double
(Default: 1e-4) Tolerance used to verify integrality of intermediate
matrices. Dependent on precision of period matrices.
Returns
-------
Gamma : integer matrix
The symplectic transformation operator.
Pa : complex matrix
Pb : complex matrix
Symmetric a- and b-periods, respectively, of a genus `g` Riemann surface.
Notes
-----
The algorithm described in Kalla, Klein actually operates on the transposes
of the a- and b-period matrices.
"""
# coerce from numpy, if necessary
if isinstance(Pa, numpy.ndarray):
Pa = Matrix(CDF, numpy.ascontiguousarray(Pa))
if isinstance(Pb, numpy.ndarray):
Pb = Matrix(CDF, numpy.ascontiguousarray(Pb))
# use the transposes of the period matrices and coerce to Sage matrices
Pa = Pa.T
Pb = Pb.T
# use above functions to obtain topological type matrix
g, g = Pa.dimensions()
R = involution_matrix(Pa, Pb, tol=tol)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S, tol=tol)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=tol)
# compute the corresponding symmetric periods
stacked_periods = zero_matrix(CDF, 2 * g, g)
stacked_periods[:g, :] = Pa
stacked_periods[g:, :] = Pb
stacked_symmetric_periods = Gamma * stacked_periods
Pa_symm = stacked_symmetric_periods[:g, :]
Pb_symm = stacked_symmetric_periods[g:, :]
# transpose results back
Pa_symm = Pa_symm.T
Pb_symm = Pb_symm.T
return Pa_symm, Pb_symm
def symmetrize_periods(Pa, Pb, tol=1e-4):
r"""Returns symmetric a- and b-periods `Pa_symm` and `Pb_symm`, as well as the
corresponding symplectic operator `Gamma` such that `Gamma [Pa \\ Pb] =
[Pa_symm \\ Pb_symm]`.
Parameters
----------
Pa : complex matrix
Pb : complex matrix
The a- and b-periods, respectively, of a genus `g` Riemann surface.
tol : double
(Default: 1e-4) Tolerance used to verify integrality of intermediate
matrices. Dependent on precision of period matrices.
Returns
-------
Gamma : integer matrix
The symplectic transformation operator.
Pa : complex matrix
Pb : complex matrix
Symmetric a- and b-periods, respectively, of a genus `g` Riemann surface.
Notes
-----
The algorithm described in Kalla, Klein actually operates on the transposes
of the a- and b-period matrices.
"""
# coerce from numpy, if necessary
if isinstance(Pa, numpy.ndarray):
Pa = Matrix(CDF, numpy.ascontiguousarray(Pa))
if isinstance(Pb, numpy.ndarray):
Pb = Matrix(CDF, numpy.ascontiguousarray(Pb))
# use the transposes of the period matrices and coerce to Sage matrices
Pa = Pa.T
Pb = Pb.T
# use above functions to obtain topological type matrix
g,g = Pa.dimensions()
R = involution_matrix(Pa, Pb, tol=tol)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S, tol=tol)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=tol)
# compute the corresponding symmetric periods
stacked_periods = zero_matrix(CDF, 2*g, g)
stacked_periods[:g,:] = Pa
stacked_periods[g:,:] = Pb
stacked_symmetric_periods = Gamma*stacked_periods
Pa_symm = stacked_symmetric_periods[:g,:]
Pb_symm = stacked_symmetric_periods[g:,:]
# transpose results back
Pa_symm = Pa_symm.T
Pb_symm = Pb_symm.T
return Pa_symm, Pb_symm
def coeff_reduce(C, B, SB, Detail=0, debug=False):
"""B is a list of lists of rationals holding an invertible dxd matrix
C is a list of lists of rationals holding an nxd matrix
C*B remains invariant
SB = Sturm bound: the first SB rows of C should span the whole row space over Z
Computes invertible U and returns (C', B') = (C*U, U^(-1)*B), so
that the entries of C' are integers and 'small'
"""
t0 = time.time()
if Detail > 1:
print("Before LLL, coefficients:\n{}".format(C))
# convert to actual matrices:
d = len(B)
nan = len(C)
B = Matrix(B)
C = Matrix(C)
if debug:
print("B has size {}".format(B.dimensions()))
#print("B={}".format(B))
print("C has size {}".format(C.dimensions()))
#print("C={}".format(C))
CB = C * B
# Make integral:
C1, den = C._clear_denom()
B1 = B / den
t1 = time.time()
if Detail:
print("Cleared denominator = {} in {:0.3f}".format(den, t1 - t0))
# Make primitive:
if debug:
print("C1 is in {}".format(C1.parent()))
#print("C1 = {}".format(C1))
C1 = pari(C1)
# if debug:
# print("B1={} in {}".format(B1, B1.parent()))
B1 = pari(B1)
V1V2S = C1.matsnf(1)
t2 = time.time()
V2 = V1V2S[1]
S = pari_row_slice(nan - d + 1, nan)(V1V2S[2])
if Detail:
print("Computed Smith form in {:0.3f}".format(t2 - t1))
if debug:
print("About to invert matrix of size {}".format(V2.matsize()))
C1 *= V2
B1 = V2**(-1) * B1
if debug:
assert CB == C1 * B1
scales = [S[i, i] for i in range(d)]
if not all(s == 1 for s in scales):
if Detail:
print("scale factors {}".format(scales))
C1 *= S**(-1)
B1 = S * B1
if debug:
assert CB == C1 * B1
# LLL-reduce
U = C1.qflll()
t3 = time.time()
if Detail:
print("LLL done in {:0.3f}".format(t3 - t2))
if debug:
assert U.matdet() in [1, -1]
C1 *= U
B1 = U**(-1) * B1
if debug:
assert CB == C1 * B1
print("found Bred, Cred")
# Convert back to lists of lists
Cred = [ci.list() for ci in C1.mattranspose()]
Bred = [bi.list() for bi in B1.mattranspose()]
if Detail > 1:
print("After LLL, coefficients:\n{}\nbasis={}".format(Cred, Bred))
return Cred, Bred
