def deltaproduct(f, limit):
"""
Handle products containing a KroneckerDelta.
See Also
========
deltasummation
sympy.functions.special.tensor_functions.KroneckerDelta
sympy.concrete.products.product
"""
from sympy.concrete.products import product
if ((limit[2] - limit[1]) < 0) == True:
return S.One
if not f.has(KroneckerDelta):
return product(f, limit)
if f.is_Add:
# Identify the term in the Add that has a simple KroneckerDelta
delta = None
terms = []
for arg in sorted(f.args, key=default_sort_key):
if delta is None and _has_simple_delta(arg, limit[0]):
delta = arg
else:
terms.append(arg)
newexpr = f.func(*terms)
k = Dummy("kprime", integer=True)
if isinstance(limit[1], int) and isinstance(limit[2], int):
result = deltaproduct(newexpr, limit) + sum([
deltaproduct(newexpr, (limit[0], limit[1], ik - 1)) *
delta.subs(limit[0], ik) *
deltaproduct(newexpr, (limit[0], ik + 1, limit[2]))
for ik in range(int(limit[1]), int(limit[2] + 1))
])
else:
result = deltaproduct(newexpr, limit) + deltasummation(
deltaproduct(newexpr, (limit[0], limit[1], k - 1)) *
delta.subs(limit[0], k) *
deltaproduct(newexpr, (limit[0], k + 1, limit[2])),
(k, limit[1], limit[2]),
no_piecewise=_has_simple_delta(newexpr, limit[0]))
return _remove_multiple_delta(result)
delta, _ = _extract_delta(f, limit[0])
if not delta:
g = _expand_delta(f, limit[0])
if f != g:
from sympy import factor
try:
return factor(deltaproduct(g, limit))
except AssertionError:
return deltaproduct(g, limit)
return product(f, limit)
return _remove_multiple_delta(f.subs(limit[0], limit[1])*KroneckerDelta(limit[2], limit[1])) + \
S.One*_simplify_delta(KroneckerDelta(limit[2], limit[1] - 1))
def test_funcmatrix_creation():
i, j, k = symbols('i j k')
assert FunctionMatrix(2, 2, Lambda((i, j), 0))
assert FunctionMatrix(0, 0, Lambda((i, j), 0))
raises(ValueError, lambda: FunctionMatrix(-1, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2.0, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2j, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, -1, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, 2.0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, 2j, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda(i, 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, lambda i, j: 0))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i,), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i, j, k), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, i+j))
assert FunctionMatrix(2, 2, "lambda i, j: 0") == \
FunctionMatrix(2, 2, Lambda((i, j), 0))
m = FunctionMatrix(2, 2, KroneckerDelta)
assert m.as_explicit() == Identity(2).as_explicit()
assert m.args[2] == Lambda((i, j), KroneckerDelta(i, j))
n = symbols('n')
assert FunctionMatrix(n, n, Lambda((i, j), 0))
n = symbols('n', integer=False)
raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
n = symbols('n', negative=True)
raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
def _remove_multiple_delta(expr):
"""
Evaluate products of KroneckerDelta's.
"""
from sympy.solvers import solve
if expr.is_Add:
return expr.func(*list(map(_remove_multiple_delta, expr.args)))
if not expr.is_Mul:
return expr
eqs = []
newargs = []
for arg in expr.args:
if isinstance(arg, KroneckerDelta):
eqs.append(arg.args[0] - arg.args[1])
else:
newargs.append(arg)
if not eqs:
return expr
solns = solve(eqs, dict=True)
if len(solns) == 0:
return S.Zero
elif len(solns) == 1:
for key in solns[0].keys():
newargs.append(KroneckerDelta(key, solns[0][key]))
expr2 = expr.func(*newargs)
if expr != expr2:
return _remove_multiple_delta(expr2)
return expr
def test_PythonCodePrinter():
prntr = PythonCodePrinter()
assert not prntr.module_imports
assert prntr.doprint(x**y) == 'x**y'
assert prntr.doprint(Mod(x, 2)) == 'x % 2'
assert prntr.doprint(And(x, y)) == 'x and y'
assert prntr.doprint(Or(x, y)) == 'x or y'
assert not prntr.module_imports
assert prntr.doprint(pi) == 'math.pi'
assert prntr.module_imports == {'math': {'pi'}}
assert prntr.doprint(x**Rational(1, 2)) == 'math.sqrt(x)'
assert prntr.doprint(sqrt(x)) == 'math.sqrt(x)'
assert prntr.module_imports == {'math': {'pi', 'sqrt'}}
assert prntr.doprint(acos(x)) == 'math.acos(x)'
assert prntr.doprint(Assignment(x, 2)) == 'x = 2'
assert prntr.doprint(Piecewise(
(1, Eq(x, 0)),
(2, x > 6))) == '((1) if (x == 0) else (2) if (x > 6) else None)'
assert prntr.doprint(Piecewise((2, Le(x, 0)),
(3, Gt(x, 0)), evaluate=False)) == '((2) if (x <= 0) else'\
' (3) if (x > 0) else None)'
assert prntr.doprint(sign(x)) == '(0.0 if x == 0 else math.copysign(1, x))'
assert prntr.doprint(p[0, 1]) == 'p[0, 1]'
assert prntr.doprint(KroneckerDelta(x, y)) == '(1 if x == y else 0)'
def test_funcmatrix_creation():
i, j, k = symbols('i j k')
assert FunctionMatrix(2, 2, Lambda((i, j), 0))
assert FunctionMatrix(0, 0, Lambda((i, j), 0))
raises(ValueError, lambda: FunctionMatrix(-1, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2.0, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2j, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, -1, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, 2.0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, 2j, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda(i, 0)))
with warns(SymPyDeprecationWarning, test_stacklevel=False):
# This raises a deprecation warning from sympify()
raises(ValueError, lambda: FunctionMatrix(2, 2, lambda i, j: 0))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i,), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i, j, k), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, i+j))
assert FunctionMatrix(2, 2, "lambda i, j: 0") == \
FunctionMatrix(2, 2, Lambda((i, j), 0))
m = FunctionMatrix(2, 2, KroneckerDelta)
assert m.as_explicit() == Identity(2).as_explicit()
assert m.args[2].dummy_eq(Lambda((i, j), KroneckerDelta(i, j)))
n = symbols('n')
assert FunctionMatrix(n, n, Lambda((i, j), 0))
n = symbols('n', integer=False)
raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
n = symbols('n', negative=True)
raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
def _simplify_delta(expr):
"""
Rewrite a KroneckerDelta's indices in its simplest form.
"""
from sympy.solvers import solve
if isinstance(expr, KroneckerDelta):
try:
slns = solve(expr.args[0] - expr.args[1], dict=True)
if slns and len(slns) == 1:
return Mul(*[KroneckerDelta(*(key, value))
for key, value in slns[0].items()])
except NotImplementedError:
pass
return expr
def test_KroneckerDelta():
from sympy.functions import KroneckerDelta
assert mcode(KroneckerDelta(x, y)) == "double(x == y)"
assert mcode(KroneckerDelta(x, y + 1)) == "double(x == (y + 1))"
assert mcode(KroneckerDelta(2**x, y)) == "double((2.^x) == y)"
def test_latex_KroneckerDelta():
assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}"
assert latex(KroneckerDelta(x, y)**2) == r"\left(\delta_{x y}\right)^{2}"
assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}"
# issue 3479
assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}"
def _entry(self, i, j, **kwargs):
perm = self.args[0]
return KroneckerDelta(perm.apply(i), j)
def test_latex_KroneckerDelta():
assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}"
assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}"
# issue 3479
assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}"
