def test_against_sympy_jtheta(self):
N = 64
sigma = 2
z = sigma * randn(N) + 1.j * sigma * randn(N)
z = z.reshape((N, 1))
tau = [[1.0j]]
# jtheta inputs
w = numpy.pi * z[:, 0]
q = numpy.exp(numpy.pi * 1.0j * tau[0][0])
values1 = RiemannTheta(z, tau, epsilon=1e-16)
values2 = numpy.array([jtheta(3, wi, q) for wi in w],
dtype=numpy.complex)
rel_error = abs((values1 - values2) / values1)
rel_error_avg = numpy.mean(rel_error)
self.assertLess(rel_error_avg, 1e-14)
# repeat for different tau
tau = [[1.0 + 2.5j]]
q = numpy.exp(numpy.pi * 1.0j * tau[0][0])
values1 = RiemannTheta(z, tau, epsilon=1e-16)
values2 = numpy.array([jtheta(3, wi, q) for wi in w],
dtype=numpy.complex)
rel_error = abs((values1 - values2) / values1)
rel_error_avg = numpy.mean(rel_error)
self.assertLess(rel_error_avg, 1e-14)
def test_value(self):
z = [0, 0, 0]
Omega = self.Omega3
value = RiemannTheta(z, Omega, epsilon=1e-14)
maple = 1.2362529854204190 - 0.52099320642367818e-10j
error = abs(value - maple)
self.assertLess(error, 1e-8)
w = [0.2 + 0.5j, 0.3 - 0.1j, -0.1 + 0.2j]
value = RiemannTheta(w, Omega, epsilon=1e-14)
maple = 1.2544694041047501 - 0.77493173321770725j
error = abs(value - maple)
self.assertLess(error, 1e-8)
def test_issue159(self):
Omega = [[10j]]
z = [5j]
theta_actual = 2
theta = RiemannTheta(z, Omega)
error = abs(theta - theta_actual)
self.assertLess(error, 1e-8)
def test_against_naive_implementation_genus1(self):
# tests the genus 1 Riemann theta function against the naive
# implementation written above (directly using the summation formula).
# first test the relative error using values close to the origin,
# avoiding the double-exponential growth
N = 64
sigma = 0.1
z = sigma * randn(N) + 1.j * sigma * randn(N)
z = z.reshape((N, 1))
tau = [[1.0j]]
values1 = RiemannTheta(z, tau, epsilon=1e-16)
values2 = thetag1(z, tau[0][0])[:, 0]
rel_error = abs((values1 - values2) / values1)
rel_error_max = numpy.max(rel_error)
rel_error_avg = numpy.mean(rel_error)
self.assertLess(rel_error_max, 1e-14)
self.assertLess(rel_error_avg, 1e-14)
# next, test the relative error using larger magnitude values. we don't
# test the max error due to possible numerical roundoff issues
sigma = 3
z = sigma * randn(N) + 1.j * sigma * randn(N)
z = z.reshape((N, 1))
tau = [[1.0j]]
values1 = RiemannTheta(z, tau, epsilon=1e-16)
values2 = thetag1(z, tau[0][0])[:, 0]
rel_error = abs((values1 - values2) / values1)
rel_error_avg = numpy.mean(rel_error)
self.assertLess(rel_error_avg, 1e-14)
# repeat for different tau
tau = [[1.0 + 2.5j]]
values1 = RiemannTheta(z, tau, epsilon=1e-16)
values2 = thetag1(z, tau[0][0])[:, 0]
rel_error = abs((values1 - values2) / values1)
rel_error_avg = numpy.mean(rel_error)
self.assertLess(rel_error_avg, 1e-14)
def test_issue84_value(self):
z = [0.5 - 1.10093687j, -0.11723434j]
Omega = [[0.5 + 2j, 0.5 + 1j], [0.5 + 1j, 1 + 1.5j]]
theta_actual = 0.963179246467 - 6.2286820685j
for _ in range(1000):
theta = RiemannTheta(z, Omega)
error = abs(theta - theta_actual)
self.assertLess(
error, 1e-5, '%s not less than %s'
'\ntheta: %s\nactual: %s' %
(error, 1e-5, theta, theta_actual))
def test_gradient(self):
Omega = self.Omega3
# generate random test z-values
N = 32
u = numpy.random.rand(N, 3)
v = numpy.random.rand(N, 3)
W = u + 1.0j * v
# manually compute gradients
dz0 = RiemannTheta(W, Omega, derivs=[[1, 0, 0]])
dz1 = RiemannTheta(W, Omega, derivs=[[0, 1, 0]])
dz2 = RiemannTheta(W, Omega, derivs=[[0, 0, 1]])
grad1 = numpy.zeros_like(W, dtype=numpy.complex)
grad1[:, 0] = dz0
grad1[:, 1] = dz1
grad1[:, 2] = dz2
# compute using "gradient"
grad2 = RiemannTheta.gradient(W, Omega)
self.assertLess(numpy.linalg.norm(grad1 - grad2), 1e-14)
Omega = self.Omega4
# generate random test z-values
N = 32
u = numpy.random.rand(N, 4)
v = numpy.random.rand(N, 4)
W = u + 1.0j * v
# manually compute gradients
dz0 = RiemannTheta(W, Omega, derivs=[[1, 0, 0, 0]])
dz1 = RiemannTheta(W, Omega, derivs=[[0, 1, 0, 0]])
dz2 = RiemannTheta(W, Omega, derivs=[[0, 0, 1, 0]])
dz3 = RiemannTheta(W, Omega, derivs=[[0, 0, 0, 1]])
grad1 = numpy.zeros_like(W, dtype=numpy.complex)
grad1[:, 0] = dz0
grad1[:, 1] = dz1
grad1[:, 2] = dz2
grad1[:, 3] = dz3
# compute using "gradient"
grad2 = RiemannTheta.gradient(W, Omega)
self.assertLess(numpy.linalg.norm(grad1 - grad2), 1e-14)
def test_first_derivatives(self):
w = [0.2 + 0.5j, 0.3 - 0.1j, -0.1 + 0.2j]
Omega = self.Omega3
value_z1 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[1, 0, 0]])
value_z2 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 1, 0]])
value_z3 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 0, 1]])
maple_z1 = -5.7295900733729553 - 0.89199375315523345j
maple_z2 = -0.16300987772384356 - 0.65079269102999180j
maple_z3 = 1.0115406077003542 + 0.030528533907836019j
error_z1 = abs(value_z1 - maple_z1)
error_z2 = abs(value_z2 - maple_z2)
error_z3 = abs(value_z3 - maple_z3)
self.assertLess(error_z1, 1e-8)
self.assertLess(error_z2, 1e-8)
self.assertLess(error_z3, 1e-8)
Omega = self.Omega4
w = [0, 0, 0, 0]
value_z1 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[1, 0, 0, 0]])
value_z2 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 1, 0, 0]])
value_z3 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 0, 1, 0]])
value_z4 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 0, 0, 1]])
maple_z1 = 0.0
maple_z2 = 0.0
maple_z3 = 0.0
maple_z4 = 0.0
error_z1 = abs(value_z1 - maple_z1)
error_z2 = abs(value_z2 - maple_z2)
error_z3 = abs(value_z3 - maple_z3)
error_z4 = abs(value_z4 - maple_z4)
self.assertLess(error_z1, 1e-8)
self.assertLess(error_z2, 1e-8)
self.assertLess(error_z3, 1e-8)
self.assertLess(error_z4, 1e-8)
# different value of w
w = [
-0.37704918 - 0.18456279j, 0.63934426 + 0.42591413j,
0.54918033 + 0.09937996j, -0.21721311 - 0.07808426j
]
value_z1 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[1, 0, 0, 0]])
value_z2 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 1, 0, 0]])
value_z3 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 0, 1, 0]])
value_z4 = RiemannTheta(w, Omega, epsilon=1e-14, derivs=[[0, 0, 0, 1]])
maple_z1 = 3.3644150756 + 2.5018071784j
maple_z2 = -2.9431860155 + 5.6802762853j
maple_z3 = 8.0319838396 + 3.5491434873j
maple_z4 = -6.0837267311 - 2.4867680289j
error_z1 = abs(value_z1 - maple_z1)
error_z2 = abs(value_z2 - maple_z2)
error_z3 = abs(value_z3 - maple_z3)
error_z4 = abs(value_z4 - maple_z4)
self.assertLess(error_z1, 1e-8)
self.assertLess(error_z2, 1e-8)
self.assertLess(error_z3, 1e-8)
self.assertLess(error_z4, 1e-8)
