def test_number_of_real_roots():
assert number_of_real_roots(0, x) == 0
assert number_of_real_roots(7, x) == 0
f = Poly(x - 1, x)
assert number_of_real_roots(f) == 1
assert number_of_real_roots(f, sup=0) == 0
assert number_of_real_roots(f, inf=1) == 0
assert number_of_real_roots(f, sup=0, inf=1) == 1
assert number_of_real_roots(f, sup=1, inf=0) == 1
f = x**2 - 4
assert number_of_real_roots(f, x) == 2
assert number_of_real_roots(f, x, sup=0) == 1
assert number_of_real_roots(f, x, inf=-1, sup=1) == 0
raises(ValueError, "number_of_real_roots(f, x, inf=t)")
raises(ValueError, "number_of_real_roots(f, x, sup=t)")
def test_number_of_real_roots():
assert number_of_real_roots(0, x) == 0
assert number_of_real_roots(7, x) == 0
f = Poly(x - 1, x)
assert number_of_real_roots(f) == 1
assert number_of_real_roots(f, sup=0) == 0
assert number_of_real_roots(f, inf=1) == 0
assert number_of_real_roots(f, sup=0, inf=1) == 1
assert number_of_real_roots(f, sup=1, inf=0) == 1
f = x**2 - 4
assert number_of_real_roots(f, x) == 2
assert number_of_real_roots(f, x, sup=0) == 1
assert number_of_real_roots(f, x, inf=-1, sup=1) == 0
raises(ValueError, "number_of_real_roots(f, x, inf=t)")
raises(ValueError, "number_of_real_roots(f, x, sup=t)")
def log_to_real(h, q, x, t):
"""Convert complex logarithms to real functions.
Given real field K and polynomials h in K[t,x] and q in K[t],
returns real function f such that:
___
df d \ `
-- = -- ) a log(h(a, x))
dx dx /__,
a | q(a) = 0
"""
u, v = symbols('u v')
H = h.as_basic().subs({t:u+I*v}).expand()
Q = q.as_basic().subs({t:u+I*v}).expand()
H_map = collect(H, I, evaluate=False)
Q_map = collect(Q, I, evaluate=False)
a, b = H_map.get(S(1), S(0)), H_map.get(I, S(0))
c, d = Q_map.get(S(1), S(0)), Q_map.get(I, S(0))
R = Poly(resultant(c, d, v), u)
R_u = roots(R, filter='R')
if len(R_u) != number_of_real_roots(R):
return None
result = S(0)
for r_u in R_u.iterkeys():
C = Poly(c.subs({u:r_u}), v)
R_v = roots(C, filter='R')
if len(R_v) != number_of_real_roots(C):
return None
for r_v in R_v:
if not r_v.is_positive:
continue
D = d.subs({u:r_u, v:r_v})
if D.evalf(chop=True) != 0:
continue
A = Poly(a.subs({u:r_u, v:r_v}), x)
B = Poly(b.subs({u:r_u, v:r_v}), x)
AB = (A**2 + B**2).as_basic()
result += r_u*log(AB) + r_v*log_to_atan(A, B)
R_q = roots(q, filter='R')
if len(R_q) != number_of_real_roots(q):
return None
for r in R_q.iterkeys():
result += r*log(h.as_basic().subs(t, r))
return result
def log_to_real(h, q, x, t):
"""Convert complex logarithms to real functions.
Given real field K and polynomials h in K[t,x] and q in K[t],
returns real function f such that:
___
df d \ `
-- = -- ) a log(h(a, x))
dx dx /__,
a | q(a) = 0
"""
u, v = symbols('u v')
H = h.as_basic().subs({t: u + I * v}).expand()
Q = q.as_basic().subs({t: u + I * v}).expand()
H_map = collect(H, I, evaluate=False)
Q_map = collect(Q, I, evaluate=False)
a, b = H_map.get(S(1), S(0)), H_map.get(I, S(0))
c, d = Q_map.get(S(1), S(0)), Q_map.get(I, S(0))
R = Poly(resultant(c, d, v), u)
R_u = roots(R, filter='R')
if len(R_u) != number_of_real_roots(R):
return None
result = S(0)
for r_u in R_u.iterkeys():
C = Poly(c.subs({u: r_u}), v)
R_v = roots(C, filter='R')
if len(R_v) != number_of_real_roots(C):
return None
for r_v in R_v:
if not r_v.is_positive:
continue
D = d.subs({u: r_u, v: r_v})
if D.evalf(chop=True) != 0:
continue
A = Poly(a.subs({u: r_u, v: r_v}), x)
B = Poly(b.subs({u: r_u, v: r_v}), x)
AB = (A**2 + B**2).as_basic()
result += r_u * log(AB) + r_v * log_to_atan(A, B)
R_q = roots(q, filter='R')
if len(R_q) != number_of_real_roots(q):
return None
for r in R_q.iterkeys():
result += r * log(h.as_basic().subs(t, r))
return result
