def test_latex_PolyElement():
Ruv, u,v = ring("u,v", ZZ);
Rxyz, x,y,z = ring("x,y,z", Ruv.to_domain())
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1"
assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1"
def test_latex_PolyElement():
Ruv, u,v = ring("u,v", ZZ);
Rxyz, x,y,z = ring("x,y,z", Ruv.to_domain())
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1"
assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1"
def example_20():
"""
Example 20 from KK
"""
R, x, y, z = ring('x,y,z', RR, grevlex)
I = [z**2 + 3 * y - 7 * z,
y * z - 4 * y,
x * z - 4 * y,
y**2 - 4*y,
x*y - 4 * y,
x**5 - 8 * x**4 + 14 * x**4 + 8 * x**2 - 15 * x + 15 * y,
]
G = [z**2 + 3 * y - 7 *z,
y * z - 4 * y,
x * z - 4 * y,
y**2 - 4*y,
x *y - 4 * y,
x**2 * z - 16 * y,
x **2 * y - 16 * y,
x**3 * z - 64 * y,
x**3 * y - 64 * y,
x**4 * z - 256 * y,
x**4 * y - 256 * y,
x **5 - 8 * x **4 + 14 * x **4 + 8 * x **2 - 15 * x + 15 * y,
]
return R, I, G
def test_episode_2(s, reward):
R, x, y, z, t = sp.ring('x,y,z,t', sp.FF(32003), 'grlex')
F = [x**31 - x**6 - x - y, x**8 - z, x**10 - t]
ideal_gen = FixedIdealGenerator(F)
env = BuchbergerEnv(ideal_gen, rewards='reductions')
agent = BuchbergerAgent(selection=s)
assert run_episode(agent, env) == reward
def generate_parabolas(N=2, sigma=0.):
R, x, y = ring('x, y', RR, order=grevlex)
thetas = [2 * pi / N * t for t in xrange(N)]
I = [(x * sin(t) + y * cos(t) - 1)**2 - (x + cos(t))**2 - (y + sin(t))**2 +
sigma * randn() for t in thetas]
return R, I
def example_20():
"""
Example 20 from KK
"""
R, x, y, z = ring('x,y,z', RR, grevlex)
I = [
z**2 + 3 * y - 7 * z,
y * z - 4 * y,
x * z - 4 * y,
y**2 - 4 * y,
x * y - 4 * y,
x**5 - 8 * x**4 + 14 * x**4 + 8 * x**2 - 15 * x + 15 * y,
]
G = [
z**2 + 3 * y - 7 * z,
y * z - 4 * y,
x * z - 4 * y,
y**2 - 4 * y,
x * y - 4 * y,
x**2 * z - 16 * y,
x**2 * y - 16 * y,
x**3 * z - 64 * y,
x**3 * y - 64 * y,
x**4 * z - 256 * y,
x**4 * y - 256 * y,
x**5 - 8 * x**4 + 14 * x**4 + 8 * x**2 - 15 * x + 15 * y,
]
return R, I, G
def get_measurement_polynomials(self, noise = 0, seed = 0):
np.random.seed(seed)
k, d = self.k, self.d
params = self.get_parameters()
R = ring([x for x, _ in params], RR)[0]
names = {str(x) : R(x) for x in R.symbols}
xs = array([[names[self.x(i,j)] for j in xrange(k)] for i in xrange(d)])
params = [(names[x], v) for x, v in params]
# Second order moments (TODO: 3rd order moments)
P = zeros((d,d), dtype=np.object)
p = zeros((d,), dtype=np.object)
for i in xrange(d):
p[i] = sum(xs[i,k_] for k_ in xrange(k))# / k
for j in xrange(i, d):
P[i,j] = sum(xs[i,k_] * xs[j,k_] for k_ in xrange(k))# / k
# Project and profit
m = zeros((d,))
M = zeros((d,d))
for i in xrange(d):
m[i] = p[i].evaluate(params)
for j in xrange(i, d):
M[i,j] = P[i,j].evaluate(params)
M = M + noise * np.random.randn(d,d)
m = m + noise * np.random.randn(d)
# TODO: Something is wrong here
#m = M.sum(1)
# Finally return values.
return R, [f - f_
for f, f_ in zip(p.flatten(), m.flatten())] + [f - f_
for f, f_ in zip(P[triu_indices_from(P)], M[triu_indices_from(M)])]
def example_order_preservation():
"""
Example 3.4 from paper. 5 variables, 7 equations. Method fails
unless the lex basis is re-added to the original basis
"""
R, x0, x1, x2, x3, x4 = ring('x0,x1,x2,x3,x4', QQ, lex)
I = [x0 - x2, x0 - x3, x1 - x3, x1 - x4, x2 - x3, x3 - x4, x2**2]
return R, I
def example_simple():
"""
Example 3.1 from paper. 4 variables, 4 equations, simple, method
works
"""
R, x0, x1, x2, x3 = ring('x0,x1,x2,x3', QQ, lex)
I = [x0**4 - 1, x0**2 + x2, x1**2 + x2, x2**2 + x3]
return R, I
def test_extend():
"""
Test extension
"""
R = ring('x,y', RR)[0]
L = BorderBasedUniverse(R, [(2, 1), (1, 2)])
L.extend()
assert L._border == [(3, 1), (2, 2), (1, 3)]
def example_order_preservation():
"""
Example 3.4 from paper. 5 variables, 7 equations. Method fails
unless the lex basis is re-added to the original basis
"""
R, x0, x1, x2, x3, x4 = ring('x0,x1,x2,x3,x4', QQ, lex)
I = [x0 - x2, x0 - x3, x1 - x3, x1 - x4, x2 - x3, x3 - x4, x2**2]
return R, I
def example_simple():
"""
Example 3.1 from paper. 4 variables, 4 equations, simple, method
works
"""
R, x0, x1, x2, x3 = ring('x0,x1,x2,x3', QQ, lex)
I = [x0**4 - 1, x0**2 + x2, x1**2 + x2, x2**2 + x3]
return R, I
def test_extend():
"""
Test extension
"""
R = ring('x,y', RR)[0]
L = BorderBasedUniverse(R, [(2, 1), (1, 2)])
L.extend()
assert L._border == [(3, 1), (2, 2), (1, 3)]
def example_subset_large():
A = range(10)
S = 5
n = len(A)
syms = [Symbol('s%d'%i) for i in xrange(n+1)]
R = ring(syms, QQ, lex)[0]
I = [syms[0]] + [(syms[i] - syms[i-1]) * (syms[i] - syms[i-1] - A[i-1]) for i in xrange(1,n+1)] + [syms[n] - S]
return R, as_ring_exprs(R, I)
def _scalar_to_str(self, c):
# not an elegant way to force elements of algebraic fields be printed with sqrt
if isinstance(c, sympy.polys.polyclasses.ANP):
dummy_ring = sympy.ring([], self.domain)[0]
return f"({dummy_ring(c).as_expr()})"
if isinstance(c, sympy.polys.fields.FracElement):
return f"({c})"
return str(c)
def example_fail():
"""
Example 3.2 from paper. 2 variables, 3 equations, method fails
Failure mode: one of the ideal quotients is empty, which means that
the variety could be the empty set.
"""
R, x0, x1, x2 = ring('x0,x1,x2', QQ, lex)
I = [x0 * x1 + 1, x1 + x2, x1 * x2]
return R, I
def example_fail():
"""
Example 3.2 from paper. 2 variables, 3 equations, method fails
Failure mode: one of the ideal quotients is empty, which means that
the variety could be the empty set.
"""
R, x0, x1, x2 = ring('x0,x1,x2', QQ, lex)
I = [x0 * x1 + 1, x1 + x2, x1*x2]
return R, I
def example_subset_large():
A = range(10)
S = 5
n = len(A)
syms = [Symbol('s%d' % i) for i in xrange(n + 1)]
R = ring(syms, QQ, lex)[0]
I = [syms[0]] + [(syms[i] - syms[i - 1]) *
(syms[i] - syms[i - 1] - A[i - 1])
for i in xrange(1, n + 1)] + [syms[n] - S]
return R, as_ring_exprs(R, I)
def example_trivial():
"""
My own example.
"""
R, x, y = ring('x,y', RR, grevlex)
I = [x**2 + y**2 - 2, x + y]
return R, I
def test_episode_1(e, reward):
R, a, b, c, d = sp.ring('a,b,c,d', sp.FF(32003), 'grevlex')
F = [
a + b + c + d, a * b + b * c + c * d + d * a,
a * b * c + b * c * d + c * d * a + d * a * b, a * b * c * d - 1
]
ideal_gen = FixedIdealGenerator(F)
env = BuchbergerEnv(ideal_gen, elimination=e, rewards='reductions')
agent = BuchbergerAgent(selection=['normal', 'first'])
assert run_episode(agent, env) == reward
def generate_univariate_problem(n_common_zeros=1, max_degree=3, n_equations=5):
R, x = ring('x', RR, order=grevlex)
# set of common zeros.
V = 1 - 2 * rand(n_common_zeros)
I = [
poly_for_zeros(
R, x, np.hstack((V, 1 - 2 * rand(max_degree - n_common_zeros))))
for _ in xrange(n_equations)
]
return R, I, V
def generate_dense_polynomial(nvars, deg):
R, *variables = sp.ring([f"x{i}" for i in range(nvars)], sp.QQ)
def gen_eq_deg(d: int):
return reduce(add, [randrange(-10, 10) * reduce(mul, p) for p in itertools.product(variables, repeat=d)]) \
+ randrange(-10, 10)
def gen_eq():
return reduce(add, [gen_eq_deg(d) for d in range(1, deg + 1)])
return [gen_eq() for _ in range(nvars)]
def test_restrict_lt1():
_, x, y = ring('x,y', RR, grevlex)
I = [x + y + 1, x**2 + y + 1]
L = get_order_basis(I)
B = get_support_basis(I)
W = unitary_basis(B, I)
L_, B_, W_ = restrict_lt(L, B, W)
assert L_ == L
assert np.allclose(W_, W)
def test_restrict_lt1():
_, x, y = ring('x,y', RR, grevlex)
I = [x + y + 1, x**2 + y + 1]
L = get_order_basis(I)
B = get_support_basis(I)
W = unitary_basis(B, I)
L_, B_, W_ = restrict_lt(L, B, W)
assert L_ == L
assert np.allclose(W_, W)
def test_restrict_lt2():
_, x, y = ring('x,y', RR, grevlex)
I = [x * y**2, x**2 * y + x * y**2, x**3 * y**2]
L = get_order_basis([x**2 * y])
B = get_support_basis(I)
W = unitary_basis(B, I)
L_, B_, W_ = restrict_lt(L, B, W)
assert len(difference(L_, L)) > 0 and len(difference(L, L_)) == 0
assert len(B_) == 2
assert np.allclose(W_, W[:2, 1:])
def example():
"""
Example 4.12
"""
R, x, y, z = ring('x,y,z', RR, grevlex)
I = [
0.130 * z**2 + 0.39 * y - 0.911 * z, 0.242 * y * z - 0.97 * y,
0.243 * x * z - 0.97 * y, 0.242 * y**2 - 0.97 * y,
0.243 * x * y - 0.97 * y, 0.035 * x**5 - 0.284 * x**4 + 0.497 * x**3 +
0.284 * x**2 - 0.532 * x + 0.533 * y
]
return R, I
def example_trivial():
"""
My own example.
"""
R, x, y = ring('x,y', RR, grevlex)
I = [x**2 + y**2 - 2,
x + y
]
return R, I
def test_restrict_lt2():
_, x, y = ring('x,y', RR, grevlex)
I = [x*y**2, x**2 * y + x * y**2, x**3 * y**2]
L = get_order_basis([x**2*y])
B = get_support_basis(I)
W = unitary_basis(B, I)
L_, B_, W_ = restrict_lt(L, B, W)
assert len(difference(L_, L)) > 0 and len(difference(L, L_)) == 0
assert len(B_) == 2
assert np.allclose(W_, W[:2,1:])
def example():
"""
Example 4.12
"""
R, x, y, z = ring('x,y,z', RR, grevlex)
I = [0.130 * z**2 + 0.39 * y - 0.911 * z,
0.242 * y * z - 0.97 * y,
0.243 * x * z - 0.97 * y,
0.242 * y**2 - 0.97 * y,
0.243 * x * y - 0.97 * y,
0.035 * x**5 - 0.284 * x**4 + 0.497 * x**3 + 0.284 * x**2 - 0.532 * x + 0.533 * y
]
return R, I
def test_episode_0(s):
R, a, b, c, d, e = sp.ring('a,b,c,d,e', sp.FF(32003), 'grevlex')
F = [
a + 2 * b + 2 * c + 2 * d + 2 * e - 1,
a**2 + 2 * b**2 + 2 * c**2 + 2 * d**2 + 2 * e**2 - a,
2 * a * b + 2 * b * c + 2 * c * d + 2 * d * e - b,
b**2 + 2 * a * c + 2 * b * d + 2 * c * e - c,
2 * b * c + 2 * a * d + 2 * b * e - d
]
ideal_gen = FixedIdealGenerator(F)
env = BuchbergerEnv(ideal_gen, rewards='reductions')
agent = BuchbergerAgent(selection=s)
assert run_episode(agent, env) == -28
def generate_lifeware_conjecture(n):
"""
Generate the system from Conjecture 3.1 from https://hal.inria.fr/hal-02900798v2/document
Also is called Long Monomial in benchmarks.
"""
variables = sp.ring([f"x{i}" for i in range(n)], sp.QQ)[1:]
prod_all = 1
for i in range(n):
prod_all *= variables[i]**2
system = []
for i in range(n):
system.append(variables[(i + 1) % n]**2 + prod_all)
return system
def generate_exponential():
coef_field = sp.FractionField(
sp.QQ, ["Apsi", "Atheta", "bpsi", "btheta", "gamma", "B", "D"])
Apsi, Atheta, bpsi, btheta, gamma, B, D = coef_field.gens
R, psi, theta, exp, thinv2, u, du = sp.ring(
["psi", "theta", "exp", "thinv2", "u", "u'"], coef_field)
eqs = [
Apsi * psi + bpsi * u - D * psi * exp,
Atheta * theta + btheta * u + B * D * psi * exp
]
eqs.append(gamma * eqs[0] * thinv2 * exp)
eqs.append(eqs[0] * (-2) * thinv2 * psi)
eqs.append(du)
eqs.append(du**2)
return eqs
def test_LeadMonomialsEnv_1():
R, x, y, z = sp.ring('x,y,z', sp.FF(101), 'grevlex')
F = [y - x**2, z - x**3]
ideal_gen = FixedIdealGenerator(F)
env = LeadMonomialsEnv(ideal_gen)
state = env.reset()
assert np.array_equal(state, np.array([[2, 0, 0, 3, 0, 0]]))
state, _, done, _ = env.step(0)
assert np.array_equal(state, np.array([[2, 0, 0, 1, 1, 0]]))
assert not done
state, _, done, _ = env.step(0)
assert np.array_equal(state, np.array([[1, 1, 0, 0, 2, 0]]))
assert not done
state, _, done, _ = env.step(0)
assert done
def example_simple():
"""
Example 19 from KK
"""
R, x, y = ring('x,y', RR, grevlex)
I = [
x**3 - x, y**3 - y, x**2 * y - 0.5 * y - 0.5 * y**2,
x * y - x - 0.5 * y + x**2 - 0.5 * y**2,
x * y**2 - x - 0.5 * y + x**2 - 0.5 * y**2
]
G = [
x**2 + x * y - 0.5 * y**2 - x - 0.5 * y, y**3 - y, x * y**2 - x * y,
x**2 * y - 0.5 * y**2 - 0.5 * y
]
return R, I, G
def example_mog(sigma=0):
R, x1, x2, y1, y2 = ring('x1,x2,y1,y2', RR, lex)
m1x = x1 + x2
m1y = y1 + y2
m2xx = x1**2 + x2**2
m2xy = x1 * y1 + x2 * y2
m2yy = y1**2 + y2**2
ms = [ m1x, m1y, m2xx, m2xy,] #m2yy, ]
if True:
x1v, x2v = 1, -1
y1v, y2v = -1, 1
I = [ (m - m(x1v, x2v, y1v, y2v) + sigma * np.random.randn()) for m in ms]
return R, I
def example_simple():
"""
Example 19 from KK
"""
R, x, y = ring('x,y', RR, grevlex)
I = [x**3 - x,
y**3 - y,
x**2*y - 0.5 * y - 0.5 * y**2,
x*y - x - 0.5 * y + x**2 - 0.5 * y**2,
x * y**2 - x - 0.5 * y + x**2 - 0.5 * y**2
]
G = [x**2 + x * y - 0.5 * y**2 - x - 0.5 * y,
y**3 - y,
x*y**2 - x*y,
x**2 * y - 0.5 * y**2 - 0.5 * y
]
return R, I, G
def test_LeadMonomialsEnv_0():
R, x, y, z = sp.ring('x,y,z', sp.FF(101), 'grevlex')
F = [y - x**2, z - x**3]
ideal_gen = FixedIdealGenerator(F)
env = LeadMonomialsEnv(ideal_gen, elimination='none')
state = env.reset()
assert np.array_equal(state, np.array([[2, 0, 0, 3, 0, 0]]))
state, _, done, _ = env.step(0)
assert (np.array_equal(state,
np.array([[2, 0, 0, 1, 1, 0], [3, 0, 0, 1, 1, 0]]))
or np.array_equal(
state, np.array([[3, 0, 0, 1, 1, 0], [2, 0, 0, 1, 1, 0]])))
assert not done
action = 0 if np.array_equal(state[0], np.array([3, 0, 0, 1, 1, 0])) else 1
state, _, done, _ = env.step(action)
assert np.array_equal(state, np.array([[2, 0, 0, 1, 1, 0]]))
assert not done
for _ in range(4):
state, _, done, _ = env.step(0)
assert done
def example_mog(sigma=0):
R, x1, x2, y1, y2 = ring('x1,x2,y1,y2', RR, lex)
m1x = x1 + x2
m1y = y1 + y2
m2xx = x1**2 + x2**2
m2xy = x1 * y1 + x2 * y2
m2yy = y1**2 + y2**2
ms = [
m1x,
m1y,
m2xx,
m2xy,
] #m2yy, ]
if True:
x1v, x2v = 1, -1
y1v, y2v = -1, 1
I = [(m - m(x1v, x2v, y1v, y2v) + sigma * np.random.randn()) for m in ms]
return R, I
def get_measurement_polynomials(self, noise=0, seed=0):
np.random.seed(seed)
k, d = self.k, self.d
params = self.get_parameters()
R = ring([x for x, _ in params], RR)[0]
names = {str(x): R(x) for x in R.symbols}
xs = array([[names[self.x(i, j)] for j in xrange(k)]
for i in xrange(d)])
params = [(names[x], v) for x, v in params]
# Second order moments (TODO: 3rd order moments)
P = zeros((d, d), dtype=np.object)
p = zeros((d, ), dtype=np.object)
for i in xrange(d):
p[i] = sum(xs[i, k_] for k_ in xrange(k)) # / k
for j in xrange(i, d):
P[i, j] = sum(xs[i, k_] * xs[j, k_]
for k_ in xrange(k)) # / k
# Project and profit
m = zeros((d, ))
M = zeros((d, d))
for i in xrange(d):
m[i] = p[i].evaluate(params)
for j in xrange(i, d):
M[i, j] = P[i, j].evaluate(params)
M = M + noise * np.random.randn(d, d)
m = m + noise * np.random.randn(d)
# TODO: Something is wrong here
#m = M.sum(1)
# Finally return values.
return R, [f - f_ for f, f_ in zip(p.flatten(), m.flatten())] + [
f - f_ for f, f_ in zip(P[triu_indices_from(P)], M[
triu_indices_from(M)])
]
def test_get_order_basis():
_, x, y = ring('x,y', RR, grevlex)
f = x**2 + x * y + y
g = x**2 + x * y + x
b = get_order_basis([f, g])
assert b == [(2, 0), (1, 1), (1, 0), (0, 1), (0, 0)]
def generate_parabolas(N=2, sigma = 0.):
R, x, y = ring('x, y', RR, order=grevlex)
thetas = [2 * pi / N * t for t in xrange(N)]
I = [(x * sin(t) + y * cos(t) - 1)**2 - (x+cos(t))**2 - (y+sin(t))**2 + sigma * randn() for t in thetas]
return R, I
def test_get_order_basis():
_, x, y = ring('x,y', RR, grevlex)
f = x**2 + x*y + y
g = x**2 + x*y + x
b = get_order_basis([f,g])
assert b == [(2,0), (1,1), (1,0), (0,1), (0,0)]
# Percy's playground.
from __future__ import print_function
import sympy as sp
import numpy as np
import BorderBasis as BB
np.set_printoptions(precision=3)
from IPython.display import display, Markdown, Math
sp.init_printing()
R, x, y = sp.ring('x,y', sp.RR, order=sp.grevlex)
I = [ x**2 + y**2 - 1.0, x + y ]
R, x, y, z = sp.ring('x,y,z', sp.RR, order=sp.grevlex)
I = [ x**2 - 1, y**2 - 4, z**2 - 9]
# n = 4 takes a long time
n = 4
Rvs = sp.ring(' '.join('v'+str(i) for i in range(1, n + 1)), sp.RR, order=sp.grevlex)
R, vs = Rvs[0], Rvs[1:]
I = []
I.extend([v**2 - 1 for v in vs])
#I.extend([(v-1)**2 for v in vs])
#I.extend([v-1 for v in vs])
#I.extend([vs[i] - vs[i-1] for i in range(1, len(vs))]) # Makes it fast
print('Generating')
B = BB.BorderBasisFactory(1e-5).generate(R,I)
print('Done')
print("=== Generator Basis:")
def test_border():
R, x, y = ring('x,y', RR, grevlex)
O = get_order_basis([x, y])
dO = border(R, O)
assert dO == [(2, 0), (1, 1), (0, 2)]
def test_border_basis_divide():
R, x, y = ring('x,y', RR, grevlex)
G = [x**2 + x + 1, x *y + y, y**2 + x + 1]
f = x**3 * y**2 - x * y**2 + x**2 + 2
r = border_basis_divide(R, G, f)
assert r == -3.0 * x - 1.0
def generate_univariate_problem(n_common_zeros=1, max_degree=3, n_equations=5):
R, x = ring("x", RR, order=grevlex)
# set of common zeros.
V = 1 - 2 * rand(n_common_zeros)
I = [poly_for_zeros(R, x, np.hstack((V, 1 - 2 * rand(max_degree - n_common_zeros)))) for _ in xrange(n_equations)]
return R, I, V
from timeit import default_timer as clock
from sympy import ring, ZZ
R, x, y, z, w = ring("x y z w", ZZ)
e = (x+y+z+w)**15
t1 = clock()
f = e*(e+w)
t2 = clock()
#print f
print "Total time:", t2-t1, "s"
print "number of terms:", len(f)
def generate_lines(N=2, sigma = 0.):
R, x, y = ring('x, y', RR, order=grevlex)
thetas = [2 * pi / N * t for t in xrange(N)]
I = [x * sin(t) + y * cos(t) + sigma * randn() for t in thetas]
return R, I
def test_get_coefficient_term():
R, x, y, z = ring('x,y,z', QQ, lex)
f = R(3 * x**2 * y * z + 2 * x**2 * y**2 + z + x * y * z)
coeff = get_coefficient_term(R, f)
assert coeff == R(2 * y**2 + 3 * y * z)
def test_border():
R, x, y = ring('x,y', RR, grevlex)
O = get_order_basis([x, y])
dO = border(R, O)
assert dO == [(2,0), (1,1), (0,2)]
def test_border_basis_divide():
R, x, y = ring('x,y', RR, grevlex)
G = [x**2 + x + 1, x * y + y, y**2 + x + 1]
f = x**3 * y**2 - x * y**2 + x**2 + 2
r = border_basis_divide(R, G, f)
assert r == -3.0 * x - 1.0
def generate_lines(N=2, sigma=0.):
R, x, y = ring('x, y', RR, order=grevlex)
thetas = [2 * pi / N * t for t in xrange(N)]
I = [x * sin(t) + y * cos(t) + sigma * randn() for t in thetas]
return R, I
