def get_full_christoffel(psi, p_expo):
"""
This computes and returns the "second" Christoffel symbols
(i.e. Gamma^i_jk) built from the original, "full," nonconformal
metric (called gamma_ij most of the time). It will compute the
second Christoffels (C2) built from the conformal metric it they
have not already been computed. The notation should be understood
such that for C3[i,j,k] the first index (i) is the "up" index
and the last two (j,k) are the "down" indices. As this uses the
metric, it assumes the metric has been set, e.g.,
dendro_ccz4.set_metric(gt);
"""
global metric, inv_metric, undef, C1, C2, C3, d
#global metric, inv_metric, undef, C2, C3, d
if C3 == undef:
C3 = MutableDenseNDimArray(range(27), (3, 3, 3))
if C2 == undef:
get_second_christoffel()
for k in e_i:
for j in e_i:
for i in e_i:
# C3[i, j, k] = C2[i, j, k] - 1/(2*chi)*(KroneckerDelta(i, j) * d(k, chi) +
#C3[i, j, k] = C2[i, j, k] + 0.5*p_expo*(psi**(p_expo))*(KroneckerDelta(i, j) * d(k, psi) +
C3[i, j, k] = C2[i, j, k] + 0.5*p_expo/psi*(KroneckerDelta(i, j) * d(k, psi) +
KroneckerDelta(i, k) * d(j, psi) -
metric[j, k]*sum([inv_metric[i, m]*d(m, psi) for m in e_i])
)
return C3
def test_complicated_derivative_with_Indexed():
x, y = symbols("x,y", cls=IndexedBase)
sigma = symbols("sigma")
i, j, k = symbols("i,j,k")
m0, m1, m2, m3, m4, m5 = symbols("m0:6")
f = Function("f")
expr = f((x[i] - y[i])**2 / sigma)
_xi_1 = symbols("xi_1", cls=Dummy)
assert expr.diff(x[m0]).dummy_eq(
(x[i] - y[i])*KroneckerDelta(i, m0)*\
2*Subs(
Derivative(f(_xi_1), _xi_1),
(_xi_1,),
((x[i] - y[i])**2/sigma,)
)/sigma
)
assert expr.diff(x[m0]).diff(x[m1]).dummy_eq(
2*KroneckerDelta(i, m0)*\
KroneckerDelta(i, m1)*Subs(
Derivative(f(_xi_1), _xi_1),
(_xi_1,),
((x[i] - y[i])**2/sigma,)
)/sigma + \
4*(x[i] - y[i])**2*KroneckerDelta(i, m0)*KroneckerDelta(i, m1)*\
Subs(
Derivative(f(_xi_1), _xi_1, _xi_1),
(_xi_1,),
((x[i] - y[i])**2/sigma,)
)/sigma**2
)
def apply(given, i=None, j=None):
assert given.is_Equality
x_set_comprehension, interval = given.args
n = interval.max() + 1
assert interval.min() == 0
assert len(x_set_comprehension.limits) == 1
k, a, b = x_set_comprehension.limits[0]
assert b - a == n - 1
x = LAMBDA(x_set_comprehension.function.arg,
*x_set_comprehension.limits).simplify()
if j is None:
j = Symbol.j(domain=[0, n - 1], integer=True, given=True)
if i is None:
i = Symbol.i(domain=[0, n - 1], integer=True, given=True)
assert j >= 0 and j < n
assert i >= 0 and i < n
index = index_function(n)
di = index[i](x[:n])
dj = index[j](x[:n])
return Equality(KroneckerDelta(di, dj), KroneckerDelta(i, j), given=given)
def _eval_derivative(self, v):
from sympy import Sum, symbols, Dummy
if not isinstance(v, MatrixElement):
from sympy import MatrixBase
if isinstance(self.parent, MatrixBase):
return self.parent.diff(v)[self.i, self.j]
return S.Zero
M = self.args[0]
if M == v.args[0]:
return KroneckerDelta(self.args[1], v.args[1])*KroneckerDelta(self.args[2], v.args[2])
if isinstance(M, Inverse):
i, j = self.args[1:]
i1, i2 = symbols("z1, z2", cls=Dummy)
Y = M.args[0]
r1, r2 = Y.shape
return -Sum(M[i, i1]*Y[i1, i2].diff(v)*M[i2, j], (i1, 0, r1-1), (i2, 0, r2-1))
if self.has(v.args[0]):
return None
return S.Zero
def _eval_innerproduct_JzBra(self, bra, **hints):
result = KroneckerDelta(self.j, bra.j)
if bra.dual_class() is not self.__class__:
result *= self._represent_JzOp(None)[bra.j-bra.m]
else:
result *= KroneckerDelta(self.j, bra.j) * KroneckerDelta(self.m, bra.m)
return result
def get_complete_christoffel(chi):
"""
Computes and returns the second Christoffel Symbols. Assumes the metric has been set. Will compute the first/second
Christoffel if not already computed. e.g.,
dendro.set_metric(gt);
C2_spatial = dendro.get_complete_christoffel();
"""
global metric, inv_metric, undef, C1, C2, C3, d
if C3 == undef:
C3 = MutableDenseNDimArray(range(27), (3, 3, 3))
if C2 == undef:
get_second_christoffel()
for k in e_i:
for j in e_i:
for i in e_i:
# C3[i, j, k] = C2[i, j, k] - 1/(2*chi)*(KroneckerDelta(i, j) * d(k, chi) +
C3[i, j, k] = C2[i, j, k] - 0.5/(chi)*(KroneckerDelta(i, j) * d(k, chi) +
KroneckerDelta(i, k) * d(j, chi) -
metric[j, k]*sum([inv_metric[i, m]*d(m, chi) for m in e_i])
)
return C3
def test_matrixelement_diff():
dexpr = diff((D*w)[k,0], w[p,0])
assert w[k, p].diff(w[k, p]) == 1
assert w[k, p].diff(w[0, 0]) == KroneckerDelta(0, k, (0, n-1))*KroneckerDelta(0, p, (0, 0))
_i_1 = Dummy("_i_1")
assert dexpr.dummy_eq(Sum(KroneckerDelta(_i_1, p, (0, n-1))*D[k, _i_1], (_i_1, 0, n - 1)))
assert dexpr.doit() == D[k, p]
def _check_varsh_872_9(term_list):
# Sum( CG(a,alpha,b,beta,c,gamma)*CG(a,alpha',b,beta',c,gamma), (gamma, -c, c), (c, abs(a-b), a+b))
a, alpha, alphap, b, beta, betap, c, gamma, lt = map(
Wild,
('a', 'alpha', 'alphap', 'b', 'beta', 'betap', 'c', 'gamma', 'lt'))
# Case alpha==alphap, beta==betap
# For numerical alpha,beta
expr = lt * CG(a, alpha, b, beta, c, gamma)**2
simp = 1
sign = lt / abs(lt)
x = abs(a - b)
y = abs(alpha + beta)
build_expr = a + b + 1 - Piecewise((x, x > y), (0, Eq(x, y)), (y, y > x))
index_expr = a + b - c
term_list, other1 = _check_cg_simp(expr, simp, sign, lt, term_list,
(a, alpha, b, beta, c, gamma, lt),
(a, alpha, b, beta), build_expr,
index_expr)
# For symbolic alpha,beta
x = abs(a - b)
y = a + b
build_expr = (y + 1 - x) * (x + y + 1)
index_expr = (c - x) * (x + c) + c + gamma
term_list, other2 = _check_cg_simp(expr, simp, sign, lt, term_list,
(a, alpha, b, beta, c, gamma, lt),
(a, alpha, b, beta), build_expr,
index_expr)
# Case alpha!=alphap or beta!=betap
# Note: this only works with leading term of 1, pattern matching is unable to match when there is a Wild leading term
# For numerical alpha,alphap,beta,betap
expr = CG(a, alpha, b, beta, c, gamma) * CG(a, alphap, b, betap, c, gamma)
simp = KroneckerDelta(alpha, alphap) * KroneckerDelta(beta, betap)
sign = sympify(1)
x = abs(a - b)
y = abs(alpha + beta)
build_expr = a + b + 1 - Piecewise((x, x > y), (0, Eq(x, y)), (y, y > x))
index_expr = a + b - c
term_list, other3 = _check_cg_simp(
expr, simp, sign, sympify(1), term_list,
(a, alpha, alphap, b, beta, betap, c, gamma),
(a, alpha, alphap, b, beta, betap), build_expr, index_expr)
# For symbolic alpha,alphap,beta,betap
x = abs(a - b)
y = a + b
build_expr = (y + 1 - x) * (x + y + 1)
index_expr = (c - x) * (x + c) + c + gamma
term_list, other4 = _check_cg_simp(
expr, simp, sign, sympify(1), term_list,
(a, alpha, alphap, b, beta, betap, c, gamma),
(a, alpha, alphap, b, beta, betap), build_expr, index_expr)
return term_list, other1 + other2 + other4
def column_transformation(*limits):
n = limits[0][-1] + 1
(i, *_), (j, *_) = limits
# return Identity(n) + LAMBDA[j:n, i:n](Piecewise((0, i < n - 1), (KroneckerDelta(j, n - 1) - 1, True)))
# return Identity(n) + LAMBDA[j:n, i:n](Piecewise((KroneckerDelta(j, n - 1) - 1, Equality(i, n - 1)), (0, True)))
return Identity(n) + LAMBDA[j:n, i:n](KroneckerDelta(i, n - 1) *
(KroneckerDelta(j, n - 1) - 1))
return LAMBDA(
Piecewise((KroneckerDelta(i, j), i < n - 1),
(2 * KroneckerDelta(j, n - 1) - 1, True)), *limits)
def test_codegen_array_parse():
expr = M[i, j]
assert _codegen_array_parse(expr) == (M, (i, j))
expr = M[i, j] * N[k, l]
assert _codegen_array_parse(expr) == (CodegenArrayTensorProduct(M, N),
(i, j, k, l))
expr = M[i, j] * N[j, k]
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(
CodegenArrayTensorProduct(M, N), (1, 2)), (i, k, j))
expr = Sum(M[i, j] * N[j, k], (j, 0, k - 1))
assert _codegen_array_parse(expr) == (CodegenArrayContraction(
CodegenArrayTensorProduct(M, N), (1, 2)), (i, k))
expr = M[i, j] + N[i, j]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(M, N),
(i, j))
expr = M[i, j] + N[j, i]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(
M, CodegenArrayPermuteDims(N, Permutation([1, 0]))), (i, j))
expr = M[i, j] + M[j, i]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(
M, CodegenArrayPermuteDims(M, Permutation([1, 0]))), (i, j))
expr = (M * N * P)[i, j]
assert _codegen_array_parse(expr) == (CodegenArrayContraction(
CodegenArrayTensorProduct(M, N, P), (1, 2), (3, 4)), (i, j))
expr = expr.function # Disregard summation in previous expression
ret1, ret2 = _codegen_array_parse(expr)
assert ret1 == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P),
(1, 2), (3, 4))
assert str(ret2) == "(i, j, _i_1, _i_2)"
expr = KroneckerDelta(i, j) * M[i, k]
assert _codegen_array_parse(expr) == (M, ({i, j}, k))
expr = KroneckerDelta(i, j) * KroneckerDelta(j, k) * M[i, l]
assert _codegen_array_parse(expr) == (M, ({i, j, k}, l))
expr = KroneckerDelta(j, k) * (M[i, j] * N[k, l] + N[i, j] * M[k, l])
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(
CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N),
Permutation(0, 2)(1, 3))),
(1, 2)), (i, l, frozenset({j, k})))
expr = KroneckerDelta(j, m) * KroneckerDelta(
m, k) * (M[i, j] * N[k, l] + N[i, j] * M[k, l])
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(
CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N),
Permutation(0, 2)(1, 3))),
(1, 2)), (i, l, frozenset({j, m, k})))
expr = KroneckerDelta(i, j) * KroneckerDelta(j, k) * KroneckerDelta(
k, m) * M[i, 0] * KroneckerDelta(m, n)
assert _codegen_array_parse(expr) == (M, ({i, j, k, m, n}, 0))
expr = M[i, i]
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(M,
(0, 1)), (i, ))
def apply(W):
n = W.shape[0]
k = Symbol.k(integer=True)
return Equality(
Concatenate(
Concatenate(W.T, ZeroMatrix(n)).T,
LAMBDA[k:n + 1](KroneckerDelta(k, n))),
Concatenate(
Concatenate(W, ZeroMatrix(n)).T,
LAMBDA[k:n + 1](KroneckerDelta(k, n))).T)
def _eval_derivative(self, v):
if not isinstance(v, MatrixElement):
from sympy import MatrixBase
if isinstance(self.parent, MatrixBase):
return self.parent.diff(v)[self.i, self.j]
return S.Zero
if self.args[0] != v.args[0]:
return S.Zero
return KroneckerDelta(self.args[1], v.args[1])*KroneckerDelta(self.args[2], v.args[2])
def test_arrayexpr_convert_index_to_array_support_function():
expr = M[i, j]
assert _convert_indexed_to_array(expr) == (M, (i, j))
expr = M[i, j] * N[k, l]
assert _convert_indexed_to_array(expr) == (ArrayTensorProduct(M, N),
(i, j, k, l))
expr = M[i, j] * N[j, k]
assert _convert_indexed_to_array(expr) == (ArrayDiagonal(
ArrayTensorProduct(M, N), (1, 2)), (i, k, j))
expr = Sum(M[i, j] * N[j, k], (j, 0, k - 1))
assert _convert_indexed_to_array(expr) == (ArrayContraction(
ArrayTensorProduct(M, N), (1, 2)), (i, k))
expr = M[i, j] + N[i, j]
assert _convert_indexed_to_array(expr) == (ArrayAdd(M, N), (i, j))
expr = M[i, j] + N[j, i]
assert _convert_indexed_to_array(expr) == (ArrayAdd(
M, PermuteDims(N, Permutation([1, 0]))), (i, j))
expr = M[i, j] + M[j, i]
assert _convert_indexed_to_array(expr) == (ArrayAdd(
M, PermuteDims(M, Permutation([1, 0]))), (i, j))
expr = (M * N * P)[i, j]
assert _convert_indexed_to_array(expr) == (_array_contraction(
ArrayTensorProduct(M, N, P), (1, 2), (3, 4)), (i, j))
expr = expr.function # Disregard summation in previous expression
ret1, ret2 = _convert_indexed_to_array(expr)
assert ret1 == ArrayDiagonal(ArrayTensorProduct(M, N, P), (1, 2), (3, 4))
assert str(ret2) == "(i, j, _i_1, _i_2)"
expr = KroneckerDelta(i, j) * M[i, k]
assert _convert_indexed_to_array(expr) == (M, ({i, j}, k))
expr = KroneckerDelta(i, j) * KroneckerDelta(j, k) * M[i, l]
assert _convert_indexed_to_array(expr) == (M, ({i, j, k}, l))
expr = KroneckerDelta(j, k) * (M[i, j] * N[k, l] + N[i, j] * M[k, l])
assert _convert_indexed_to_array(expr) == (_array_diagonal(
_array_add(
ArrayTensorProduct(M, N),
_permute_dims(ArrayTensorProduct(M, N),
Permutation(0, 2)(1, 3))),
(1, 2)), (i, l, frozenset({j, k})))
expr = KroneckerDelta(j, m) * KroneckerDelta(
m, k) * (M[i, j] * N[k, l] + N[i, j] * M[k, l])
assert _convert_indexed_to_array(expr) == (_array_diagonal(
_array_add(
ArrayTensorProduct(M, N),
_permute_dims(ArrayTensorProduct(M, N),
Permutation(0, 2)(1, 3))),
(1, 2)), (i, l, frozenset({j, m, k})))
expr = KroneckerDelta(i, j) * KroneckerDelta(j, k) * KroneckerDelta(
k, m) * M[i, 0] * KroneckerDelta(m, n)
assert _convert_indexed_to_array(expr) == (M, ({i, j, k, m, n}, 0))
expr = M[i, i]
assert _convert_indexed_to_array(expr) == (ArrayDiagonal(M, (0, 1)), (i, ))
def get_complete_christoffel(chi):
"""Computes and returns the complete Christoffel Symbols
This function will take the metric, inverse metric,
first derivative, and then calculate the complete Christoffel
Symbols. It will store these symbols internally for
other calculations as well.
If the second Christoffel Symbols were not already computed,
it will compute them and store them internally.
If these complete Christoffel Symbols were already computed,
it will just return the already-stored Symbols.
Parameters
----------
chi : sympy.Scalar
The input scalar used in the calculation
Returns
-------
sympy.Matrix
The 3x3 matrix returned is the complete Christoffel Symbols
Example
-------
>>> C3 = dendrosym.nr.get_complete_christoffel()
"""
global metric, inv_metric, undef, C1, C2, C3, d
if C3 == undef:
C3 = sym.tensor.array.MutableDenseNDimArray(range(27), (3, 3, 3))
if C2 == undef:
get_second_christoffel()
for kk in e_i:
for jj in e_i:
for ii in e_i:
# C3[i, j, k] = C2[i, j, k] - \
# 1/(2*chi)*(KroneckerDelta(i, j) * d(k, chi) +
C3[ii, jj, kk] = C2[ii, jj, kk] - 0.5 / (chi) * (
KroneckerDelta(ii, jj) * d(kk, chi) +
KroneckerDelta(ii, kk) * d(jj, chi) - metric[jj, kk] *
sum([inv_metric[ii, mm] * d(mm, chi) for mm in e_i]))
return C3
def _eval_derivative(self, wrt):
if self == wrt:
return S.One
elif isinstance(
wrt,
IndexedFunc.IndexedFuncValue) and wrt.base == self.base:
if len(self.indices) != len(wrt.indices):
msg = "Different # of indices: d({!s})/d({!s})".format(
self, wrt)
raise IndexException(msg)
elif self.functional_form != wrt.functional_form:
msg = "Different function form d({!s})/d({!s})".format(
self.functional_form, wrt.functional_form)
raise IndexException(msg)
result = S.One
for index1, index2 in zip(self.indices, wrt.indices):
result *= KroneckerDelta(index1, index2)
return result
else:
# f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s)
i = 0
l = []
funcof = self._get_iter_func()
for a in funcof:
i += 1
da = a.diff(wrt)
if da is S.Zero:
continue
df = self._get_df(a, wrt)
l.append(df * da)
return Add(*l)
def _entry(self, i, j):
eq = Eq(i, j)
if eq is S.false:
return S.Zero
elif eq is S.true:
return self.arg[i, i]
return self.arg[i, j]*KroneckerDelta(i, j)
def test_SHOKet():
assert SHOKet('k').dual_class() == SHOBra
assert SHOBra('b').dual_class() == SHOKet
assert InnerProduct(b, k).doit() == KroneckerDelta(k.n, b.n)
assert k.hilbert_space == ComplexSpace(S.Infinity)
assert k3_rep[k3.n, 0] == Integer(1)
assert b3_rep[0, b3.n] == Integer(1)
def test_states():
assert PIABKet(n).dual_class() == PIABBra
assert PIABKet(n).hilbert_space ==\
L2(Interval(S.NegativeInfinity,S.Infinity))
assert represent(PIABKet(n)) == sqrt(2 / L) * sin(n * pi * x / L)
assert (PIABBra(i) * PIABKet(j)).doit() == KroneckerDelta(i, j)
assert PIABBra(n).dual_class() == PIABKet
def _entry(self, i, j, **kwargs):
eq = Eq(i, j)
if eq is S.true:
return S.One
elif eq is S.false:
return S.Zero
return KroneckerDelta(i, j)
def _check_varsh_sum_872_4(e):
a = Wild('a')
alpha = Wild('alpha')
b = Wild('b')
beta = Wild('beta')
c = Wild('c')
cp = Wild('cp')
gamma = Wild('gamma')
gammap = Wild('gammap')
match1 = e.match(Sum(CG(a,alpha,b,beta,c,gamma)*CG(a,alpha,b,beta,cp,gammap),(alpha,-a,a),(beta,-b,b)))
if match1 is not None and len(match1) == 8:
return (KroneckerDelta(c,cp)*KroneckerDelta(gamma,gammap)).subs(match1)
match2 = e.match(Sum(CG(a,alpha,b,beta,c,gamma)**2,(alpha,-a,a),(beta,-b,b)))
if match2 is not None and len(match2) == 6:
return 1
return e
def _check_varsh_sum_871_2(e):
a = Wild('a')
alpha = symbols('alpha')
c = Wild('c')
match = e.match(Sum((-1)**(a-alpha)*CG(a,alpha,a,-alpha,c,0),(alpha,-a,a)))
if match is not None and len(match) == 2:
return (sqrt(2*a+1)*KroneckerDelta(c,0)).subs(match)
return e
def _check_varsh_sum_871_1(e):
a = Wild('a')
alpha = symbols('alpha')
b = Wild('b')
match = e.match(Sum(CG(a, alpha, b, 0, a, alpha), (alpha, -a, a)))
if match is not None and len(match) == 2:
return ((2 * a + 1) * KroneckerDelta(b, 0)).subs(match)
return e
def _entry(self, i, j, **kwargs):
if self._iscolumn:
result = self._vector._entry(i, 0, **kwargs)
else:
result = self._vector._entry(0, j, **kwargs)
if i != j:
result *= KroneckerDelta(i, j)
return result
def _check_varsh_871_2(term_list):
# Sum((-1)**(a-alpha)*CG(a,alpha,a,-alpha,c,0),(alpha,-a,a))
a, alpha, c, lt = map(Wild, ('a', 'alpha', 'c', 'lt'))
expr = lt*CG(a, alpha, a, -alpha, c, 0)
simp = sqrt(2*a + 1)*KroneckerDelta(c, 0)
sign = (-1)**(a - alpha)*lt/abs(lt)
build_expr = 2*a + 1
index_expr = a + alpha
return _check_cg_simp(expr, simp, sign, lt, term_list, (a, alpha, c, lt), (a, c), build_expr, index_expr)
def _check_varsh_871_1(term_list):
# Sum( CG(a,alpha,b,0,a,alpha), (alpha, -a, a)) == KroneckerDelta(b,0)
a, alpha, b, lt = map(Wild, ('a', 'alpha', 'b', 'lt'))
expr = lt*CG(a, alpha, b, 0, a, alpha)
simp = (2*a + 1)*KroneckerDelta(b, 0)
sign = lt/abs(lt)
build_expr = 2*a + 1
index_expr = a + alpha
return _check_cg_simp(expr, simp, sign, lt, term_list, (a, alpha, b, lt), (a, b), build_expr, index_expr)
def _check_varsh_sum_872_4(e):
alpha = symbols('alpha')
beta = symbols('beta')
a = Wild('a')
b = Wild('b')
c = Wild('c')
cp = Wild('cp')
gamma = Wild('gamma')
gammap = Wild('gammap')
cg1 = CG(a, alpha, b, beta, c, gamma)
cg2 = CG(a, alpha, b, beta, cp, gammap)
match1 = e.match(Sum(cg1 * cg2, (alpha, -a, a), (beta, -b, b)))
if match1 is not None and len(match1) == 6:
return (KroneckerDelta(c, cp) *
KroneckerDelta(gamma, gammap)).subs(match1)
match2 = e.match(Sum(cg1**2, (alpha, -a, a), (beta, -b, b)))
if match2 is not None and len(match2) == 4:
return S.One
return e
def Lij(l, a, b, axis):
if axis == "z":
return KroneckerDelta(a, b) * b
elif axis == "x":
return (1. / 2) * (Lij(l, a, b, "Lp") + Lij(l, a, b, "Lm"))
elif axis == "y":
return (-1j / 2) * (Lij(l, a, b, "Lp") - Lij(l, a, b, "Lm"))
elif axis == "Lp":
if b == l:
return 0
else:
return KroneckerDelta(a, b + 1) * np.sqrt((l - b) * (l + b + 1))
elif axis == "Lm":
if b == -l:
return 0
else:
return KroneckerDelta(a, b - 1) * np.sqrt((l + b) * (l - b + 1))
else:
return None
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 _eval_derivative(self, wrt):
if isinstance(wrt, Indexed) and wrt.base == self.base:
if len(self.indices) != len(wrt.indices):
msg = "Different # of indices: d({!s})/d({!s})".format(
self, wrt)
raise IndexException(msg)
result = S.One
for index1, index2 in zip(self.indices, wrt.indices):
result *= KroneckerDelta(index1, index2)
return result
else:
return S.Zero
def _entry(self, i, j):
if self.diagonal_length is not None:
if Ge(i, self.diagonal_length) is S.true:
return S.Zero
elif Ge(j, self.diagonal_length) is S.true:
return S.Zero
eq = Eq(i, j)
if eq is S.true:
return self.arg[i, i]
elif eq is S.false:
return S.Zero
return self.arg[i, j]*KroneckerDelta(i, j)