def test_cg_simp_sum():
x, a, b, c, cp, alpha, beta, gamma, gammap = symbols('x a b c cp alpha beta gamma gammap')
# Varshalovich 8.7.1 Eq 1
assert cg_simp(x * Sum(CG(a,alpha,b,0,a,alpha), (alpha,-a,a))) == x*(2*a+1)*KroneckerDelta(b,0)
assert cg_simp(x * Sum(CG(a,alpha,b,0,a,alpha), (alpha,-a,a))+CG(1,0,1,0,1,0)) == x*(2*a+1)*KroneckerDelta(b,0)+CG(1,0,1,0,1,0)
assert cg_simp(2 * Sum(CG(1,alpha,0,0,1,alpha), (alpha,-1,1))) == 6
# Varshalovich 8.7.1 Eq 2
assert cg_simp(x*Sum((-1)**(a-alpha) * CG(a,alpha,a,-alpha,c,0), (alpha,-a,a))) == x*sqrt(2*a+1)*KroneckerDelta(c,0)
assert cg_simp(3*Sum((-1)**(2-alpha) * CG(2,alpha,2,-alpha,0,0), (alpha,-2,2))) == 3*sqrt(5)
# Varshalovich 8.7.2 Eq 4
assert cg_simp(Sum(CG(a,alpha,b,beta,c,gamma)*CG(a,alpha,b,beta,cp,gammap),(alpha,-a,a),(beta,-b,b))) == KroneckerDelta(c,cp)*KroneckerDelta(gamma,gammap)
assert cg_simp(Sum(CG(a,alpha,b,beta,c,gamma)*CG(a,alpha,b,beta,c,gammap),(alpha,-a,a),(beta,-b,b))) == KroneckerDelta(gamma,gammap)
assert cg_simp(Sum(CG(a,alpha,b,beta,c,gamma)*CG(a,alpha,b,beta,cp,gamma),(alpha,-a,a),(beta,-b,b))) == KroneckerDelta(c,cp)
assert cg_simp(Sum(CG(a,alpha,b,beta,c,gamma)**2,(alpha,-a,a),(beta,-b,b))) == 1
assert cg_simp(Sum(CG(2,alpha,1,beta,2,gamma)*CG(2,alpha,1,beta,2,gammap), (alpha,-2,2), (beta,-1,1))) == KroneckerDelta(gamma,gammap)
def test_cg_simp_add():
j, m1, m1p, m2, m2p = symbols('j m1 m1p m2 m2p')
# Test Varshalovich 8.7.1 Eq 1
a = CG(S(1) / 2, S(1) / 2, 0, 0, S(1) / 2, S(1) / 2)
b = CG(S(1) / 2, -S(1) / 2, 0, 0, S(1) / 2, -S(1) / 2)
c = CG(1, 1, 0, 0, 1, 1)
d = CG(1, 0, 0, 0, 1, 0)
e = CG(1, -1, 0, 0, 1, -1)
assert cg_simp(a + b) == 2
assert cg_simp(c + d + e) == 3
assert cg_simp(a + b + c + d + e) == 5
assert cg_simp(a + b + c) == 2 + c
assert cg_simp(2 * a + b) == 2 + a
assert cg_simp(2 * c + d + e) == 3 + c
assert cg_simp(5 * a + 5 * b) == 10
assert cg_simp(5 * c + 5 * d + 5 * e) == 15
assert cg_simp(-a - b) == -2
assert cg_simp(-c - d - e) == -3
assert cg_simp(-6 * a - 6 * b) == -12
assert cg_simp(-4 * c - 4 * d - 4 * e) == -12
a = CG(S(1) / 2, S(1) / 2, j, 0, S(1) / 2, S(1) / 2)
b = CG(S(1) / 2, -S(1) / 2, j, 0, S(1) / 2, -S(1) / 2)
c = CG(1, 1, j, 0, 1, 1)
d = CG(1, 0, j, 0, 1, 0)
e = CG(1, -1, j, 0, 1, -1)
assert cg_simp(a + b) == 2 * KroneckerDelta(j, 0)
assert cg_simp(c + d + e) == 3 * KroneckerDelta(j, 0)
assert cg_simp(a + b + c + d + e) == 5 * KroneckerDelta(j, 0)
assert cg_simp(a + b + c) == 2 * KroneckerDelta(j, 0) + c
assert cg_simp(2 * a + b) == 2 * KroneckerDelta(j, 0) + a
assert cg_simp(2 * c + d + e) == 3 * KroneckerDelta(j, 0) + c
assert cg_simp(5 * a + 5 * b) == 10 * KroneckerDelta(j, 0)
assert cg_simp(5 * c + 5 * d + 5 * e) == 15 * KroneckerDelta(j, 0)
assert cg_simp(-a - b) == -2 * KroneckerDelta(j, 0)
assert cg_simp(-c - d - e) == -3 * KroneckerDelta(j, 0)
assert cg_simp(-6 * a - 6 * b) == -12 * KroneckerDelta(j, 0)
assert cg_simp(-4 * c - 4 * d - 4 * e) == -12 * KroneckerDelta(j, 0)
# Test Varshalovich 8.7.1 Eq 2
a = CG(S(1) / 2, S(1) / 2, S(1) / 2, -S(1) / 2, 0, 0)
b = CG(S(1) / 2, -S(1) / 2, S(1) / 2, S(1) / 2, 0, 0)
c = CG(1, 1, 1, -1, 0, 0)
d = CG(1, 0, 1, 0, 0, 0)
e = CG(1, -1, 1, 1, 0, 0)
assert cg_simp(a - b) == sqrt(2)
assert cg_simp(c - d + e) == sqrt(3)
assert cg_simp(a - b + c - d + e) == sqrt(2) + sqrt(3)
assert cg_simp(a - b + c) == sqrt(2) + c
assert cg_simp(2 * a - b) == sqrt(2) + a
assert cg_simp(2 * c - d + e) == sqrt(3) + c
assert cg_simp(5 * a - 5 * b) == 5 * sqrt(2)
assert cg_simp(5 * c - 5 * d + 5 * e) == 5 * sqrt(3)
assert cg_simp(-a + b) == -sqrt(2)
assert cg_simp(-c + d - e) == -sqrt(3)
assert cg_simp(-6 * a + 6 * b) == -6 * sqrt(2)
assert cg_simp(-4 * c + 4 * d - 4 * e) == -4 * sqrt(3)
a = CG(S(1) / 2, S(1) / 2, S(1) / 2, -S(1) / 2, j, 0)
b = CG(S(1) / 2, -S(1) / 2, S(1) / 2, S(1) / 2, j, 0)
c = CG(1, 1, 1, -1, j, 0)
d = CG(1, 0, 1, 0, j, 0)
e = CG(1, -1, 1, 1, j, 0)
assert cg_simp(a - b) == sqrt(2) * KroneckerDelta(j, 0)
assert cg_simp(c - d + e) == sqrt(3) * KroneckerDelta(j, 0)
assert cg_simp(
a - b + c - d +
e) == sqrt(2) * KroneckerDelta(j, 0) + sqrt(3) * KroneckerDelta(j, 0)
assert cg_simp(a - b + c) == sqrt(2) * KroneckerDelta(j, 0) + c
assert cg_simp(2 * a - b) == sqrt(2) * KroneckerDelta(j, 0) + a
assert cg_simp(2 * c - d + e) == sqrt(3) * KroneckerDelta(j, 0) + c
assert cg_simp(5 * a - 5 * b) == 5 * sqrt(2) * KroneckerDelta(j, 0)
assert cg_simp(5 * c - 5 * d + 5 * e) == 5 * sqrt(3) * KroneckerDelta(j, 0)
assert cg_simp(-a + b) == -sqrt(2) * KroneckerDelta(j, 0)
assert cg_simp(-c + d - e) == -sqrt(3) * KroneckerDelta(j, 0)
assert cg_simp(-6 * a + 6 * b) == -6 * sqrt(2) * KroneckerDelta(j, 0)
assert cg_simp(-4 * c + 4 * d -
4 * e) == -4 * sqrt(3) * KroneckerDelta(j, 0)
# Test Varshalovich 8.7.2 Eq 9
# alpha=alphap,beta=betap case
# numerical
a = CG(S(1) / 2, S(1) / 2, S(1) / 2, -S(1) / 2, 1, 0)**2
b = CG(S(1) / 2, S(1) / 2, S(1) / 2, -S(1) / 2, 0, 0)**2
c = CG(1, 0, 1, 1, 1, 1)**2
d = CG(1, 0, 1, 1, 2, 1)**2
assert cg_simp(a + b) == 1
assert cg_simp(c + d) == 1
assert cg_simp(a + b + c + d) == 2
assert cg_simp(4 * a + 4 * b) == 4
assert cg_simp(4 * c + 4 * d) == 4
assert cg_simp(5 * a + 3 * b) == 3 + 2 * a
assert cg_simp(5 * c + 3 * d) == 3 + 2 * c
assert cg_simp(-a - b) == -1
assert cg_simp(-c - d) == -1
# symbolic
a = CG(S(1) / 2, m1, S(1) / 2, m2, 1, 1)**2
b = CG(S(1) / 2, m1, S(1) / 2, m2, 1, 0)**2
c = CG(S(1) / 2, m1, S(1) / 2, m2, 1, -1)**2
d = CG(S(1) / 2, m1, S(1) / 2, m2, 0, 0)**2
assert cg_simp(a + b + c + d) == 1
assert cg_simp(4 * a + 4 * b + 4 * c + 4 * d) == 4
assert cg_simp(3 * a + 5 * b + 3 * c + 4 * d) == 3 + 2 * b + d
assert cg_simp(-a - b - c - d) == -1
a = CG(1, m1, 1, m2, 2, 2)**2
b = CG(1, m1, 1, m2, 2, 1)**2
c = CG(1, m1, 1, m2, 2, 0)**2
d = CG(1, m1, 1, m2, 2, -1)**2
e = CG(1, m1, 1, m2, 2, -2)**2
f = CG(1, m1, 1, m2, 1, 1)**2
g = CG(1, m1, 1, m2, 1, 0)**2
h = CG(1, m1, 1, m2, 1, -1)**2
i = CG(1, m1, 1, m2, 0, 0)**2
assert cg_simp(a + b + c + d + e + f + g + h + i) == 1
assert cg_simp(4 * (a + b + c + d + e + f + g + h + i)) == 4
assert cg_simp(a + b + 2 * c + d + 4 * e + f + g + h + i) == 1 + c + 3 * e
assert cg_simp(-a - b - c - d - e - f - g - h - i) == -1
# alpha!=alphap or beta!=betap case
# numerical
a = CG(S(1) / 2,
S(1) / 2,
S(1) / 2, -S(1) / 2, 1, 0) * CG(
S(1) / 2, -S(1) / 2,
S(1) / 2,
S(1) / 2, 1, 0)
b = CG(S(1) / 2,
S(1) / 2,
S(1) / 2, -S(1) / 2, 0, 0) * CG(
S(1) / 2, -S(1) / 2,
S(1) / 2,
S(1) / 2, 0, 0)
c = CG(1, 1, 1, 0, 2, 1) * CG(1, 0, 1, 1, 2, 1)
d = CG(1, 1, 1, 0, 1, 1) * CG(1, 0, 1, 1, 1, 1)
assert cg_simp(a + b) == 0
assert cg_simp(c + d) == 0
# symbolic
a = CG(S(1) / 2, m1,
S(1) / 2, m2, 1, 1) * CG(S(1) / 2, m1p,
S(1) / 2, m2p, 1, 1)
b = CG(S(1) / 2, m1,
S(1) / 2, m2, 1, 0) * CG(S(1) / 2, m1p,
S(1) / 2, m2p, 1, 0)
c = CG(S(1) / 2, m1,
S(1) / 2, m2, 1, -1) * CG(S(1) / 2, m1p,
S(1) / 2, m2p, 1, -1)
d = CG(S(1) / 2, m1,
S(1) / 2, m2, 0, 0) * CG(S(1) / 2, m1p,
S(1) / 2, m2p, 0, 0)
assert cg_simp(a + b + c +
d) == KroneckerDelta(m1, m1p) * KroneckerDelta(m2, m2p)
a = CG(1, m1, 1, m2, 2, 2) * CG(1, m1p, 1, m2p, 2, 2)
b = CG(1, m1, 1, m2, 2, 1) * CG(1, m1p, 1, m2p, 2, 1)
c = CG(1, m1, 1, m2, 2, 0) * CG(1, m1p, 1, m2p, 2, 0)
d = CG(1, m1, 1, m2, 2, -1) * CG(1, m1p, 1, m2p, 2, -1)
e = CG(1, m1, 1, m2, 2, -2) * CG(1, m1p, 1, m2p, 2, -2)
f = CG(1, m1, 1, m2, 1, 1) * CG(1, m1p, 1, m2p, 1, 1)
g = CG(1, m1, 1, m2, 1, 0) * CG(1, m1p, 1, m2p, 1, 0)
h = CG(1, m1, 1, m2, 1, -1) * CG(1, m1p, 1, m2p, 1, -1)
i = CG(1, m1, 1, m2, 0, 0) * CG(1, m1p, 1, m2p, 0, 0)
assert cg_simp(a + b + c + d + e + f + g + h +
i) == KroneckerDelta(m1, m1p) * KroneckerDelta(m2, m2p)
def test_cg_simp_add():
j, m1, m1p, m2, m2p = symbols('j m1 m1p m2 m2p')
# Test Varshalovich 8.7.1 Eq 1
a = CG(S(1)/2,S(1)/2,0,0,S(1)/2,S(1)/2)
b = CG(S(1)/2,-S(1)/2,0,0,S(1)/2,-S(1)/2)
c = CG(1,1,0,0,1,1)
d = CG(1,0,0,0,1,0)
e = CG(1,-1,0,0,1,-1)
assert cg_simp(a+b) == 2
assert cg_simp(c+d+e) == 3
assert cg_simp(a+b+c+d+e) == 5
assert cg_simp(a+b+c) == 2+c
assert cg_simp(2*a+b) == 2+a
assert cg_simp(2*c+d+e) == 3+c
assert cg_simp(5*a+5*b) == 10
assert cg_simp(5*c+5*d+5*e) == 15
assert cg_simp(-a-b) == -2
assert cg_simp(-c-d-e) == -3
assert cg_simp(-6*a-6*b) == -12
assert cg_simp(-4*c-4*d-4*e) == -12
a = CG(S(1)/2,S(1)/2,j,0,S(1)/2,S(1)/2)
b = CG(S(1)/2,-S(1)/2,j,0,S(1)/2,-S(1)/2)
c = CG(1,1,j,0,1,1)
d = CG(1,0,j,0,1,0)
e = CG(1,-1,j,0,1,-1)
assert cg_simp(a+b) == 2*KroneckerDelta(j,0)
assert cg_simp(c+d+e) == 3*KroneckerDelta(j,0)
assert cg_simp(a+b+c+d+e) == 5*KroneckerDelta(j,0)
assert cg_simp(a+b+c) == 2*KroneckerDelta(j,0)+c
assert cg_simp(2*a+b) == 2*KroneckerDelta(j,0)+a
assert cg_simp(2*c+d+e) == 3*KroneckerDelta(j,0)+c
assert cg_simp(5*a+5*b) == 10*KroneckerDelta(j,0)
assert cg_simp(5*c+5*d+5*e) == 15*KroneckerDelta(j,0)
assert cg_simp(-a-b) == -2*KroneckerDelta(j,0)
assert cg_simp(-c-d-e) == -3*KroneckerDelta(j,0)
assert cg_simp(-6*a-6*b) == -12*KroneckerDelta(j,0)
assert cg_simp(-4*c-4*d-4*e) == -12*KroneckerDelta(j,0)
# Test Varshalovich 8.7.1 Eq 2
a = CG(S(1)/2,S(1)/2,S(1)/2,-S(1)/2,0,0)
b = CG(S(1)/2,-S(1)/2,S(1)/2,S(1)/2,0,0)
c = CG(1,1,1,-1,0,0)
d = CG(1,0,1,0,0,0)
e = CG(1,-1,1,1,0,0)
assert cg_simp(a-b) == sqrt(2)
assert cg_simp(c-d+e) == sqrt(3)
assert cg_simp(a-b+c-d+e) == sqrt(2)+sqrt(3)
assert cg_simp(a-b+c) == sqrt(2)+c
assert cg_simp(2*a-b) == sqrt(2)+a
assert cg_simp(2*c-d+e) == sqrt(3)+c
assert cg_simp(5*a-5*b) == 5*sqrt(2)
assert cg_simp(5*c-5*d+5*e) == 5*sqrt(3)
assert cg_simp(-a+b) == -sqrt(2)
assert cg_simp(-c+d-e) == -sqrt(3)
assert cg_simp(-6*a+6*b) == -6*sqrt(2)
assert cg_simp(-4*c+4*d-4*e) == -4*sqrt(3)
a = CG(S(1)/2,S(1)/2,S(1)/2,-S(1)/2,j,0)
b = CG(S(1)/2,-S(1)/2,S(1)/2,S(1)/2,j,0)
c = CG(1,1,1,-1,j,0)
d = CG(1,0,1,0,j,0)
e = CG(1,-1,1,1,j,0)
assert cg_simp(a-b) == sqrt(2)*KroneckerDelta(j,0)
assert cg_simp(c-d+e) == sqrt(3)*KroneckerDelta(j,0)
assert cg_simp(a-b+c-d+e) == sqrt(2)*KroneckerDelta(j,0)+sqrt(3)*KroneckerDelta(j,0)
assert cg_simp(a-b+c) == sqrt(2)*KroneckerDelta(j,0)+c
assert cg_simp(2*a-b) == sqrt(2)*KroneckerDelta(j,0)+a
assert cg_simp(2*c-d+e) == sqrt(3)*KroneckerDelta(j,0)+c
assert cg_simp(5*a-5*b) == 5*sqrt(2)*KroneckerDelta(j,0)
assert cg_simp(5*c-5*d+5*e) == 5*sqrt(3)*KroneckerDelta(j,0)
assert cg_simp(-a+b) == -sqrt(2)*KroneckerDelta(j,0)
assert cg_simp(-c+d-e) == -sqrt(3)*KroneckerDelta(j,0)
assert cg_simp(-6*a+6*b) == -6*sqrt(2)*KroneckerDelta(j,0)
assert cg_simp(-4*c+4*d-4*e) == -4*sqrt(3)*KroneckerDelta(j,0)
# Test Varshalovich 8.7.2 Eq 9
# alpha=alphap,beta=betap case
# numerical
a = CG(S(1)/2,S(1)/2,S(1)/2,-S(1)/2,1,0)**2
b = CG(S(1)/2,S(1)/2,S(1)/2,-S(1)/2,0,0)**2
c = CG(1,0,1,1,1,1)**2
d = CG(1,0,1,1,2,1)**2
assert cg_simp(a+b) == 1
assert cg_simp(c+d) == 1
assert cg_simp(a+b+c+d) == 2
assert cg_simp(4*a+4*b) == 4
assert cg_simp(4*c+4*d) == 4
assert cg_simp(5*a+3*b) == 3+2*a
assert cg_simp(5*c+3*d) == 3+2*c
assert cg_simp(-a-b) == -1
assert cg_simp(-c-d) == -1
# symbolic
a = CG(S(1)/2,m1,S(1)/2,m2,1,1)**2
b = CG(S(1)/2,m1,S(1)/2,m2,1,0)**2
c = CG(S(1)/2,m1,S(1)/2,m2,1,-1)**2
d = CG(S(1)/2,m1,S(1)/2,m2,0,0)**2
assert cg_simp(a+b+c+d) == 1
assert cg_simp(4*a+4*b+4*c+4*d) == 4
assert cg_simp(3*a+5*b+3*c+4*d) == 3+2*b+d
assert cg_simp(-a-b-c-d) == -1
a = CG(1,m1,1,m2,2,2)**2
b = CG(1,m1,1,m2,2,1)**2
c = CG(1,m1,1,m2,2,0)**2
d = CG(1,m1,1,m2,2,-1)**2
e = CG(1,m1,1,m2,2,-2)**2
f = CG(1,m1,1,m2,1,1)**2
g = CG(1,m1,1,m2,1,0)**2
h = CG(1,m1,1,m2,1,-1)**2
i = CG(1,m1,1,m2,0,0)**2
assert cg_simp(a+b+c+d+e+f+g+h+i) == 1
assert cg_simp(4*(a+b+c+d+e+f+g+h+i)) == 4
assert cg_simp(a+b+2*c+d+4*e+f+g+h+i) == 1+c+3*e
assert cg_simp(-a-b-c-d-e-f-g-h-i) == -1
# alpha!=alphap or beta!=betap case
# numerical
a = CG(S(1)/2,S(1)/2,S(1)/2,-S(1)/2,1,0)*CG(S(1)/2,-S(1)/2,S(1)/2,S(1)/2,1,0)
b = CG(S(1)/2,S(1)/2,S(1)/2,-S(1)/2,0,0)*CG(S(1)/2,-S(1)/2,S(1)/2,S(1)/2,0,0)
c = CG(1,1,1,0,2,1)*CG(1,0,1,1,2,1)
d = CG(1,1,1,0,1,1)*CG(1,0,1,1,1,1)
assert cg_simp(a+b) == 0
assert cg_simp(c+d) == 0
# symbolic
a = CG(S(1)/2,m1,S(1)/2,m2,1,1)*CG(S(1)/2,m1p,S(1)/2,m2p,1,1)
b = CG(S(1)/2,m1,S(1)/2,m2,1,0)*CG(S(1)/2,m1p,S(1)/2,m2p,1,0)
c = CG(S(1)/2,m1,S(1)/2,m2,1,-1)*CG(S(1)/2,m1p,S(1)/2,m2p,1,-1)
d = CG(S(1)/2,m1,S(1)/2,m2,0,0)*CG(S(1)/2,m1p,S(1)/2,m2p,0,0)
assert cg_simp(a+b+c+d) == KroneckerDelta(m1,m1p)*KroneckerDelta(m2,m2p)
a = CG(1,m1,1,m2,2,2)*CG(1,m1p,1,m2p,2,2)
b = CG(1,m1,1,m2,2,1)*CG(1,m1p,1,m2p,2,1)
c = CG(1,m1,1,m2,2,0)*CG(1,m1p,1,m2p,2,0)
d = CG(1,m1,1,m2,2,-1)*CG(1,m1p,1,m2p,2,-1)
e = CG(1,m1,1,m2,2,-2)*CG(1,m1p,1,m2p,2,-2)
f = CG(1,m1,1,m2,1,1)*CG(1,m1p,1,m2p,1,1)
g = CG(1,m1,1,m2,1,0)*CG(1,m1p,1,m2p,1,0)
h = CG(1,m1,1,m2,1,-1)*CG(1,m1p,1,m2p,1,-1)
i = CG(1,m1,1,m2,0,0)*CG(1,m1p,1,m2p,0,0)
assert cg_simp(a+b+c+d+e+f+g+h+i) == KroneckerDelta(m1,m1p)*KroneckerDelta(m2,m2p)