def extract_coeff(x1, x2, x3, cos_theta12, cos_theta23, cos_theta13, d12, d23,
d13):
"""
Extract coefficients of a polynomial
Args:
x1, x2, x3: symbols representing the unknown camera-object distance
cos_theta12, cos_theta23, cos_theta13: cos values of the inter-point angles
d12, d23, d13: square of inter-point distances
Returns:
a: the coefficients of the polynomial of x1
"""
f12 = x1**2 + x2**2 - 2 * x1 * x2 * cos_theta12 - d12
f23 = x2**2 + x3**2 - 2 * x2 * x3 * cos_theta23 - d23
f13 = x1**2 + x3**2 - 2 * x1 * x3 * cos_theta13 - d13
matrix = subresultants_qq_zz.sylvester(f23, f13, x3)
f12_ = matrix.det()
f1 = subresultants_qq_zz.sylvester(f12, f12_, x2).det()
a1 = f1.func(*[term for term in f1.args if not term.free_symbols])
a2 = f1.coeff(x1**2)
a3 = f1.coeff(x1**4)
a4 = f1.coeff(x1**6)
a5 = f1.coeff(x1**8)
a = np.array([a1, a2, a3, a4, a5])
return a
def test_bezout():
x = var("x")
p = -2 * x ** 5 + 7 * x ** 3 + 9 * x ** 2 - 3 * x + 1
q = -10 * x ** 4 + 21 * x ** 2 + 18 * x - 3
assert bezout(p, q, x, "bz").det() == sylvester(p, q, x, 2).det()
assert bezout(p, q, x, "bz").det() != sylvester(p, q, x, 1).det()
assert bezout(p, q, x, "prs") == backward_eye(5) * bezout(p, q, x, "bz") * backward_eye(5)
def test_bezout():
x = var('x')
p = -2*x**5+7*x**3+9*x**2-3*x+1
q = -10*x**4+21*x**2+18*x-3
assert bezout(p, q, x, 'bz').det() == sylvester(p, q, x, 2).det()
assert bezout(p, q, x, 'bz').det() != sylvester(p, q, x, 1).det()
assert bezout(p, q, x, 'prs') == backward_eye(5) * bezout(p, q, x, 'bz') * backward_eye(5)
def test_bezout():
x = var('x')
p = -2*x**5+7*x**3+9*x**2-3*x+1
q = -10*x**4+21*x**2+18*x-3
assert bezout(p, q, x, 'bz').det() == sylvester(p, q, x, 2).det()
assert bezout(p, q, x, 'bz').det() != sylvester(p, q, x, 1).det()
assert bezout(p, q, x, 'prs') == backward_eye(5) * bezout(p, q, x, 'bz') * backward_eye(5)
def test_bezout():
x = var("x")
p = -2 * x**5 + 7 * x**3 + 9 * x**2 - 3 * x + 1
q = -10 * x**4 + 21 * x**2 + 18 * x - 3
assert bezout(p, q, x, "bz").det() == sylvester(p, q, x, 2).det()
assert bezout(p, q, x, "bz").det() != sylvester(p, q, x, 1).det()
assert bezout(
p, q, x,
"prs") == backward_eye(5) * bezout(p, q, x, "bz") * backward_eye(5)
def test_euclid_amv():
x = var('x')
p = x**3 - 7*x + 7
q = 3*x**2 - 7
assert euclid_amv(p, q, x)[-1] == sylvester(p, q, x).det()
assert euclid_amv(p, q, x) == subresultants_amv(p, q, x)
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
assert euclid_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
sam_factors = [1, 1, -1, -1, 1, 1]
assert euclid_amv(p, q, x) == [i*j for i,j in zip(sam_factors, sturm_amv(p, q, x))]
def test_euclid_pg():
x = var('x')
p = x**6+x**5-x**4-x**3+x**2-x+1
q = 6*x**5+5*x**4-4*x**3-3*x**2+2*x-1
assert euclid_pg(p, q, x)[-1] == sylvester(p, q, x).det()
assert euclid_pg(p, q, x) == subresultants_pg(p, q, x)
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
assert euclid_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
sam_factors = [1, 1, -1, -1, 1, 1]
assert euclid_pg(p, q, x) == [i*j for i,j in zip(sam_factors, sturm_pg(p, q, x))]
def test_euclid_amv():
x = var("x")
p = x ** 3 - 7 * x + 7
q = 3 * x ** 2 - 7
assert euclid_amv(p, q, x)[-1] == sylvester(p, q, x).det()
assert euclid_amv(p, q, x) == subresultants_amv(p, q, x)
p = x ** 8 + x ** 6 - 3 * x ** 4 - 3 * x ** 3 + 8 * x ** 2 + 2 * x - 5
q = 3 * x ** 6 + 5 * x ** 4 - 4 * x ** 2 - 9 * x + 21
assert euclid_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
sam_factors = [1, 1, -1, -1, 1, 1]
assert euclid_amv(p, q, x) == [i * j for i, j in zip(sam_factors, sturm_amv(p, q, x))]
def test_euclid_pg():
x = var("x")
p = x**6 + x**5 - x**4 - x**3 + x**2 - x + 1
q = 6 * x**5 + 5 * x**4 - 4 * x**3 - 3 * x**2 + 2 * x - 1
assert euclid_pg(p, q, x)[-1] == sylvester(p, q, x).det()
assert euclid_pg(p, q, x) == subresultants_pg(p, q, x)
p = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5
q = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21
assert euclid_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
sam_factors = [1, 1, -1, -1, 1, 1]
assert euclid_pg(
p, q, x) == [i * j for i, j in zip(sam_factors, sturm_pg(p, q, x))]
def test_sturm_amv():
x = var('x')
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
sam_factors = [1, 1, -1, -1, 1, 1]
assert sturm_amv(p, q, x) == [i*j for i,j in zip(sam_factors, euclid_amv(p, q, x))]
p = -9*x**5 - 5*x**3 - 9
q = -45*x**4 - 15*x**2
assert sturm_amv(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det()
assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
assert sturm_amv(-p, q, x)[-1] == sylvester(-p, q, x, 2).det()
assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x)
def test_sturm_amv():
x = var('x')
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
sam_factors = [1, 1, -1, -1, 1, 1]
assert sturm_amv(p, q, x) == [i*j for i,j in zip(sam_factors, euclid_amv(p, q, x))]
p = -9*x**5 - 5*x**3 - 9
q = -45*x**4 - 15*x**2
assert sturm_amv(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det()
assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
assert sturm_amv(-p, q, x)[-1] == sylvester(-p, q, x, 2).det()
assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x)
def test_subresultants_vv_2():
x = var('x')
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
assert subresultants_vv_2(p, q, x) == subresultants(p, q, x)
assert subresultants_vv_2(p, q, x)[-1] == sylvester(p, q, x).det()
assert subresultants_vv_2(p, q, x) != euclid_amv(p, q, x)
amv_factors = [1, 1, -1, 1, -1, 1]
assert subresultants_vv_2(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
p = x**3 - 7*x + 7
q = 3*x**2 - 7
assert subresultants_vv_2(p, q, x) == euclid_amv(p, q, x)
def test_modified_subresultants_amv():
x = var('x')
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
amv_factors = [1, 1, -1, 1, -1, 1]
assert modified_subresultants_amv(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))]
assert modified_subresultants_amv(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
assert modified_subresultants_amv(p, q, x) != sturm_amv(p, q, x)
p = x**3 - 7*x + 7
q = 3*x**2 - 7
assert modified_subresultants_amv(p, q, x) == sturm_amv(p, q, x)
assert modified_subresultants_amv(-p, q, x) != sturm_amv(-p, q, x)
def test_subresultants_bezout():
x = var('x')
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
assert subresultants_bezout(p, q, x) == subresultants(p, q, x)
assert subresultants_bezout(p, q, x)[-1] == sylvester(p, q, x).det()
assert subresultants_bezout(p, q, x) != euclid_amv(p, q, x)
amv_factors = [1, 1, -1, 1, -1, 1]
assert subresultants_bezout(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
p = x**3 - 7*x + 7
q = 3*x**2 - 7
assert subresultants_bezout(p, q, x) == euclid_amv(p, q, x)
def test_modified_subresultants_amv():
x = var('x')
p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
amv_factors = [1, 1, -1, 1, -1, 1]
assert modified_subresultants_amv(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))]
assert modified_subresultants_amv(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
assert modified_subresultants_amv(p, q, x) != sturm_amv(p, q, x)
p = x**3 - 7*x + 7
q = 3*x**2 - 7
assert modified_subresultants_amv(p, q, x) == sturm_amv(p, q, x)
assert modified_subresultants_amv(-p, q, x) != sturm_amv(-p, q, x)
def _roots_using_eliminator_method(system, variable, other_variable):
"""
For solving a 2 polynomial system of 2 unknowns. Calculates the real roots
of the first unknown using Sylvester's resultant or the eliminator.
Note that there is a constrain that the roots must be between 0 and 1.
Attributes
----------
system: list
A list of size 2 with the polynomials
variable: symbol
The variable for which we are returning the roots
other_variable: symbol
The second variable of the system
"""
matrix = subresultants_qq_zz.sylvester(system[0], system[1],
other_variable)
matrix = sym.N(matrix, 9)
resultant = matrix.det(method="berkowitz")
num, den = sym.fraction(resultant.factor())
# candidate roots
coeffs = sym.Poly(num, variable).all_coeffs()
roots = np.roots(coeffs)
feasible_roots = set()
for root in roots:
if not np.iscomplex(root):
if root >= 0 and root <= 1:
feasible_roots.add(root)
if den != 1:
for root in feasible_roots:
if den.subs({variable: root}) == 0:
feasible_roots.remove(root)
return feasible_roots
def test_sylvester():
x = var('x')
assert sylvester(x**3 -7, 0, x) == sylvester(x**3 -7, 0, x, 1) == Matrix([[0]])
assert sylvester(0, x**3 -7, x) == sylvester(0, x**3 -7, x, 1) == Matrix([[0]])
assert sylvester(x**3 -7, 0, x, 2) == Matrix([[0]])
assert sylvester(0, x**3 -7, x, 2) == Matrix([[0]])
assert sylvester(x**3 -7, 7, x).det() == sylvester(x**3 -7, 7, x, 1).det() == 343
assert sylvester(7, x**3 -7, x).det() == sylvester(7, x**3 -7, x, 1).det() == 343
assert sylvester(x**3 -7, 7, x, 2).det() == -343
assert sylvester(7, x**3 -7, x, 2).det() == -343
assert sylvester(3, 7, x).det() == sylvester(3, 7, x, 1).det() == sylvester(3, 7, x, 2).det() == 1
assert sylvester(3, 0, x).det() == sylvester(3, 0, x, 1).det() == sylvester(3, 0, x, 2).det() == 1
assert sylvester(x - 3, x - 8, x) == sylvester(x - 3, x - 8, x, 1) == sylvester(x - 3, x - 8, x, 2) == Matrix([[1, -3], [1, -8]])
assert sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x) == sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x, 1) == Matrix([[1, 0, -7, 7, 0], [0, 1, 0, -7, 7], [3, 0, -7, 0, 0], [0, 3, 0, -7, 0], [0, 0, 3, 0, -7]])
assert sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x, 2) == Matrix([
[1, 0, -7, 7, 0, 0], [0, 3, 0, -7, 0, 0], [0, 1, 0, -7, 7, 0], [0, 0, 3, 0, -7, 0], [0, 0, 1, 0, -7, 7], [0, 0, 0, 3, 0, -7]])
def test_sylvester():
x = var('x')
assert sylvester(x**3 - 7, 0, x) == sylvester(x**3 - 7, 0, x, 1) == Matrix(
[[0]])
assert sylvester(0, x**3 - 7, x) == sylvester(0, x**3 - 7, x, 1) == Matrix(
[[0]])
assert sylvester(x**3 - 7, 0, x, 2) == Matrix([[0]])
assert sylvester(0, x**3 - 7, x, 2) == Matrix([[0]])
assert sylvester(x**3 - 7, 7, x).det() == sylvester(x**3 - 7, 7, x,
1).det() == 343
assert sylvester(7, x**3 - 7, x).det() == sylvester(7, x**3 - 7, x,
1).det() == 343
assert sylvester(x**3 - 7, 7, x, 2).det() == -343
assert sylvester(7, x**3 - 7, x, 2).det() == 343
assert sylvester(3, 7, x).det() == sylvester(
3, 7, x, 1).det() == sylvester(3, 7, x, 2).det() == 1
assert sylvester(3, 0, x).det() == sylvester(
3, 0, x, 1).det() == sylvester(3, 0, x, 2).det() == 1
assert sylvester(x - 3, x - 8,
x) == sylvester(x - 3, x - 8, x, 1) == sylvester(
x - 3, x - 8, x, 2) == Matrix([[1, -3], [1, -8]])
assert sylvester(x**3 - 7 * x + 7, 3 * x**2 - 7, x) == sylvester(
x**3 - 7 * x + 7, 3 * x**2 - 7, x, 1) == Matrix(
[[1, 0, -7, 7, 0], [0, 1, 0, -7, 7], [3, 0, -7, 0, 0],
[0, 3, 0, -7, 0], [0, 0, 3, 0, -7]])
assert sylvester(x**3 - 7 * x + 7, 3 * x**2 - 7, x,
2) == Matrix([[1, 0, -7, 7, 0, 0], [0, 3, 0, -7, 0, 0],
[0, 1, 0, -7, 7, 0], [0, 0, 3, 0, -7, 0],
[0, 0, 1, 0, -7, 7], [0, 0, 0, 3, 0, -7]])