def test_simple_products():
assert product(2, (k, a, n)) == 2**(n - a + 1)
assert product(k, (k, 1, n)) == factorial(n)
assert product(k**3, (k, 1, n)) == factorial(n)**3
assert product(k + 1, (k, 0, n - 1)) == factorial(n)
assert product(k + 1, (k, a, n - 1)) == rf(1 + a, n - a)
assert product(cos(k),
(k, 0, 5)) == cos(1) * cos(2) * cos(3) * cos(4) * cos(5)
assert product(cos(k), (k, 3, 5)) == cos(3) * cos(4) * cos(5)
assert product(cos(k), (k, 1, Rational(5, 2))) != cos(1) * cos(2)
assert isinstance(product(k**k, (k, 1, n)), Product)
assert Product(x**k, (k, 1, n)).variables == [k]
pytest.raises(ValueError, lambda: Product(n))
pytest.raises(ValueError, lambda: Product(n * k))
pytest.raises(ValueError, lambda: Product(n, k))
pytest.raises(ValueError, lambda: Product(n, k, 1))
pytest.raises(ValueError, lambda: Product(n, k, 1, 10))
pytest.raises(ValueError, lambda: Product(n, (k, 1)))
assert product(1, (n, 1, oo)) == 1 # issue sympy/sympy#8301
assert product(2, (n, 1, oo)) == oo
assert isinstance(product(-1, (n, 1, oo)), Product)
assert product(Kd(n, m), (m, 1, 3)) == 0
assert product(Kd(n, m), (m, 1, 1)) == Kd(n, 1)
def test_deltasummation_trivial():
assert ds(x, (j, 1, 0)) == 0
assert ds(x, (j, 1, 3)) == 3 * x
assert ds(x + y, (j, 1, 3)) == 3 * (x + y)
assert ds(x * y, (j, 1, 3)) == 3 * x * y
assert ds(Kd(i, j), (k, 1, 3)) == 3 * Kd(i, j)
assert ds(x * Kd(i, j), (k, 1, 3)) == 3 * x * Kd(i, j)
assert ds(x * y * Kd(i, j), (k, 1, 3)) == 3 * x * y * Kd(i, j)
def test_deltaproduct_trivial():
assert dp(x, (j, 1, 0)) == 1
assert dp(x, (j, 1, 3)) == x**3
assert dp(x + y, (j, 1, 3)) == (x + y)**3
assert dp(x * y, (j, 1, 3)) == (x * y)**3
assert dp(Kd(i, j), (k, 1, 3)) == Kd(i, j)
assert dp(x * Kd(i, j), (k, 1, 3)) == x**3 * Kd(i, j)
assert dp(x * y * Kd(i, j), (k, 1, 3)) == (x * y)**3 * Kd(i, j)
def test_deltasummation_mul_x_kd():
assert ds(x*Kd(i, j), (j, 1, 3)) == \
Piecewise((x, And(Integer(1) <= i, i <= 3)), (0, True))
assert ds(x * Kd(i, j), (j, 1, 1)) == Piecewise((x, Eq(i, 1)), (0, True))
assert ds(x * Kd(i, j), (j, 2, 2)) == Piecewise((x, Eq(i, 2)), (0, True))
assert ds(x * Kd(i, j), (j, 3, 3)) == Piecewise((x, Eq(i, 3)), (0, True))
assert ds(x*Kd(i, j), (j, 1, k)) == \
Piecewise((x, And(Integer(1) <= i, i <= k)), (0, True))
assert ds(x*Kd(i, j), (j, k, 3)) == \
Piecewise((x, And(k <= i, i <= 3)), (0, True))
assert ds(x*Kd(i, j), (j, k, l)) == \
Piecewise((x, And(k <= i, i <= l)), (0, True))
def test_deltasummation_mul_x_add_y_kd():
assert ds(x*(y + Kd(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(Integer(1) <= i, i <= 3)), (3*x*y, True))
assert ds(x*(y + Kd(i, j)), (j, 1, 1)) == \
Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
assert ds(x*(y + Kd(i, j)), (j, 2, 2)) == \
Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
assert ds(x*(y + Kd(i, j)), (j, 3, 3)) == \
Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
assert ds(x*(y + Kd(i, j)), (j, 1, k)) == \
Piecewise((k*x*y + x, And(Integer(1) <= i, i <= k)), (k*x*y, True))
assert ds(x*(y + Kd(i, j)), (j, k, 3)) == \
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x * (y + Kd(i, j)), (j, k, l)) == Piecewise(
((l - k + 1) * x * y + x, And(k <= i, i <= l)),
((l - k + 1) * x * y, True))
def test_deltaproduct_mul_add_x_y_add_kd_kd():
assert dp((x + y) * (Kd(i, k) + Kd(j, k)), (k, 1, 3)) == 0
assert dp((x + y)*(Kd(i, k) + Kd(j, k)), (k, 1, 1)) == \
(x + y)*(Kd(i, 1) + Kd(j, 1))
assert dp((x + y)*(Kd(i, k) + Kd(j, k)), (k, 2, 2)) == \
(x + y)*(Kd(i, 2) + Kd(j, 2))
assert dp((x + y)*(Kd(i, k) + Kd(j, k)), (k, 3, 3)) == \
(x + y)*(Kd(i, 3) + Kd(j, 3))
assert dp((x + y)*(Kd(i, k) + Kd(j, k)), (k, 1, l)) == Kd(l, 0) + \
(x + y)*Kd(i, 1)*Kd(l, 1) + (x + y)*Kd(j, 1)*Kd(l, 1) + \
(x + y)**2*Kd(i, 1)*Kd(j, 2)*Kd(l, 2) + \
(x + y)**2*Kd(j, 1)*Kd(i, 2)*Kd(l, 2)
assert dp((x + y)*(Kd(i, k) + Kd(j, k)), (k, l, 3)) == Kd(l, 4) + \
(x + y)*Kd(i, 3)*Kd(l, 3) + (x + y)*Kd(j, 3)*Kd(l, 3) + \
(x + y)**2*Kd(i, 2)*Kd(j, 3)*Kd(l, 2) + \
(x + y)**2*Kd(i, 3)*Kd(j, 2)*Kd(l, 2)
assert dp((x + y)*(Kd(i, k) + Kd(j, k)), (k, l, m)) == Kd(l, m + 1) + \
(x + y)*Kd(i, m)*Kd(l, m) + (x + y)*Kd(j, m)*Kd(l, m) + \
(x + y)**2*Kd(i, m - 1)*Kd(j, m)*Kd(l, m - 1) + \
(x + y)**2*Kd(i, m)*Kd(j, m - 1)*Kd(l, m - 1)
def test_deltaproduct_add_kd_kd():
assert dp(Kd(i, k) + Kd(j, k), (k, 1, 3)) == 0
assert dp(Kd(i, k) + Kd(j, k), (k, 1, 1)) == Kd(i, 1) + Kd(j, 1)
assert dp(Kd(i, k) + Kd(j, k), (k, 2, 2)) == Kd(i, 2) + Kd(j, 2)
assert dp(Kd(i, k) + Kd(j, k), (k, 3, 3)) == Kd(i, 3) + Kd(j, 3)
assert dp(Kd(i, k) + Kd(j, k), (k, 1, l)) == Kd(l, 0) + \
Kd(i, 1)*Kd(l, 1) + Kd(j, 1)*Kd(l, 1) + \
Kd(i, 1)*Kd(j, 2)*Kd(l, 2) + Kd(j, 1)*Kd(i, 2)*Kd(l, 2)
assert dp(Kd(i, k) + Kd(j, k), (k, l, 3)) == Kd(l, 4) + \
Kd(i, 3)*Kd(l, 3) + Kd(j, 3)*Kd(l, 3) + \
Kd(i, 2)*Kd(j, 3)*Kd(l, 2) + Kd(i, 3)*Kd(j, 2)*Kd(l, 2)
assert dp(Kd(i, k) + Kd(j, k), (k, l, m)) == Kd(l, m + 1) + \
Kd(i, m)*Kd(l, m) + Kd(j, m)*Kd(l, m) + \
Kd(i, m)*Kd(j, m - 1)*Kd(l, m - 1) + Kd(i, m - 1)*Kd(j, m)*Kd(l, m - 1)
def test_deltasummation_mul_add_x_kd_add_y_kd():
assert ds((x + Kd(i, k)) * (y + Kd(i, j)), (j, 1, 3)) == piecewise_fold(
Piecewise((Kd(i, k) + x, And(Integer(1) <= i, i <= 3)), (0, True)) +
3 * (Kd(i, k) + x) * y)
assert ds((x + Kd(i, k)) * (y + Kd(i, j)), (j, 1, 1)) == piecewise_fold(
Piecewise((Kd(i, k) + x, Eq(i, 1)), (0, True)) + (Kd(i, k) + x) * y)
assert ds((x + Kd(i, k)) * (y + Kd(i, j)), (j, 2, 2)) == piecewise_fold(
Piecewise((Kd(i, k) + x, Eq(i, 2)), (0, True)) + (Kd(i, k) + x) * y)
assert ds((x + Kd(i, k)) * (y + Kd(i, j)), (j, 3, 3)) == piecewise_fold(
Piecewise((Kd(i, k) + x, Eq(i, 3)), (0, True)) + (Kd(i, k) + x) * y)
assert ds((x + Kd(i, k)) * (y + Kd(i, j)), (j, 1, k)) == piecewise_fold(
Piecewise((Kd(i, k) + x, And(Integer(1) <= i, i <= k)), (0, True)) +
k * (Kd(i, k) + x) * y)
assert ds((x + Kd(i, k)) * (y + Kd(i, j)), (j, k, 3)) == piecewise_fold(
Piecewise((Kd(i, k) + x, And(k <= i, i <= 3)), (0, True)) + (4 - k) *
(Kd(i, k) + x) * y)
assert ds((x + Kd(i, k)) * (y + Kd(i, j)), (j, k, l)) == piecewise_fold(
Piecewise((Kd(i, k) + x, And(k <= i, i <= l)), (0, True)) +
(l - k + 1) * (Kd(i, k) + x) * y)
def test_deltaproduct_mul_add_x_y_kd():
assert dp((x + y) * Kd(i, j), (j, 1, 3)) == 0
assert dp((x + y) * Kd(i, j), (j, 1, 1)) == (x + y) * Kd(i, 1)
assert dp((x + y) * Kd(i, j), (j, 2, 2)) == (x + y) * Kd(i, 2)
assert dp((x + y) * Kd(i, j), (j, 3, 3)) == (x + y) * Kd(i, 3)
assert dp((x + y)*Kd(i, j), (j, 1, k)) == \
(x + y)*Kd(i, 1)*Kd(k, 1) + Kd(k, 0)
assert dp((x + y)*Kd(i, j), (j, k, 3)) == \
(x + y)*Kd(i, 3)*Kd(k, 3) + Kd(k, 4)
assert dp((x + y)*Kd(i, j), (j, k, l)) == \
(x + y)*Kd(i, l)*Kd(k, l) + Kd(k, l + 1)
def test_deltasummation_mul_add_x_y_add_kd_kd():
assert ds((x + y) * (Kd(i, k) + Kd(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x + y, And(Integer(1) <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(Integer(1) <= j, j <= 3)), (0, True)))
assert ds((x + y) * (Kd(i, k) + Kd(j, k)), (k, 1, 1)) == piecewise_fold(
Piecewise((x + y, Eq(i, 1)), (0, True)) +
Piecewise((x + y, Eq(j, 1)), (0, True)))
assert ds((x + y) * (Kd(i, k) + Kd(j, k)), (k, 2, 2)) == piecewise_fold(
Piecewise((x + y, Eq(i, 2)), (0, True)) +
Piecewise((x + y, Eq(j, 2)), (0, True)))
assert ds((x + y) * (Kd(i, k) + Kd(j, k)), (k, 3, 3)) == piecewise_fold(
Piecewise((x + y, Eq(i, 3)), (0, True)) +
Piecewise((x + y, Eq(j, 3)), (0, True)))
assert ds((x + y) * (Kd(i, k) + Kd(j, k)), (k, 1, l)) == piecewise_fold(
Piecewise((x + y, And(Integer(1) <= i, i <= l)), (0, True)) +
Piecewise((x + y, And(Integer(1) <= j, j <= l)), (0, True)))
assert ds((x + y) * (Kd(i, k) + Kd(j, k)), (k, l, 3)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= 3)), (0, True)))
assert ds((x + y) * (Kd(i, k) + Kd(j, k)), (k, l, m)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= m)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_basic_symbolic():
assert ds(Kd(exp(i), 0), (i, 1, 3)) == 0
assert ds(Kd(exp(i), 0), (i, -1, 3)) == 0
assert ds(Kd(exp(i), 1), (i, 0, 3)) == 1
assert ds(Kd(exp(i), 1), (i, 1, 3)) == 0
assert ds(Kd(exp(i), 1), (i, -10, 3)) == 1
assert ds(Kd(i, j), (j, 1, 3)) == \
Piecewise((1, And(Integer(1) <= i, i <= 3)), (0, True))
assert ds(Kd(i, j), (j, 1, 1)) == Piecewise((1, Eq(i, 1)), (0, True))
assert ds(Kd(i, j), (j, 2, 2)) == Piecewise((1, Eq(i, 2)), (0, True))
assert ds(Kd(i, j), (j, 3, 3)) == Piecewise((1, Eq(i, 3)), (0, True))
assert ds(Kd(i, j), (j, 1, k)) == \
Piecewise((1, And(Integer(1) <= i, i <= k)), (0, True))
assert ds(Kd(i, j), (j, k, 3)) == \
Piecewise((1, And(k <= i, i <= 3)), (0, True))
assert ds(Kd(i, j), (j, k, l)) == \
Piecewise((1, And(k <= i, i <= l)), (0, True))
def test_deltasummation_basic_numerical():
n = symbols('n', integer=True, nonzero=True)
assert ds(Kd(n, 0), (n, 1, 3)) == 0
# return unevaluated, until it gets implemented
assert ds(Kd(i**2, j**2), (j, -oo, oo)) == \
Sum(Kd(i**2, j**2), (j, -oo, oo))
assert Piecewise((Kd(i, k), And(Integer(1) <= i, i <= 3)), (0, True)) == \
ds(Kd(i, j)*Kd(j, k), (j, 1, 3)) == \
ds(Kd(j, k)*Kd(i, j), (j, 1, 3))
assert ds(Kd(i, k), (k, -oo, oo)) == 1
assert ds(Kd(i, k), (k, 0, oo)) == Piecewise((1, Integer(0) <= i),
(0, True))
assert ds(Kd(i, k), (k, 1, 3)) == \
Piecewise((1, And(Integer(1) <= i, i <= 3)), (0, True))
assert ds(k * Kd(i, j) * Kd(j, k), (k, -oo, oo)) == j * Kd(i, j)
assert ds(j * Kd(i, j), (j, -oo, oo)) == i
assert ds(i * Kd(i, j), (i, -oo, oo)) == j
assert ds(x, (i, 1, 3)) == 3 * x
assert ds((i + j) * Kd(i, j), (j, -oo, oo)) == 2 * i
def test_deltaproduct_basic():
assert dp(Kd(i, j), (j, 1, 3)) == 0
assert dp(Kd(i, j), (j, 1, 1)) == Kd(i, 1)
assert dp(Kd(i, j), (j, 2, 2)) == Kd(i, 2)
assert dp(Kd(i, j), (j, 3, 3)) == Kd(i, 3)
assert dp(Kd(i, j), (j, 1, k)) == Kd(i, 1) * Kd(k, 1) + Kd(k, 0)
assert dp(Kd(i, j), (j, k, 3)) == Kd(i, 3) * Kd(k, 3) + Kd(k, 4)
assert dp(Kd(i, j), (j, k, l)) == Kd(i, l) * Kd(k, l) + Kd(k, l + 1)
assert dp(Kd(i, 1), (i, j**2, k**2)) == (Kd(1, j**2) * Kd(j**2, k**2) +
Kd(k**2, j**2 - 1))
def test_deltaproduct_mul_add_x_kd_add_y_kd():
assert dp((x + Kd(i, k))*(y + Kd(i, j)), (j, 1, 3)) == \
Kd(i, 1)*(Kd(i, k) + x)*((Kd(i, k) + x)*y)**2 + \
Kd(i, 2)*(Kd(i, k) + x)*y*(Kd(i, k) + x)**2*y + \
Kd(i, 3)*((Kd(i, k) + x)*y)**2*(Kd(i, k) + x) + \
((Kd(i, k) + x)*y)**3
assert dp((x + Kd(i, k))*(y + Kd(i, j)), (j, 1, 1)) == \
(x + Kd(i, k))*(y + Kd(i, 1))
assert dp((x + Kd(i, k))*(y + Kd(i, j)), (j, 2, 2)) == \
(x + Kd(i, k))*(y + Kd(i, 2))
assert dp((x + Kd(i, k))*(y + Kd(i, j)), (j, 3, 3)) == \
(x + Kd(i, k))*(y + Kd(i, 3))
assert dp((x + Kd(i, k))*(y + Kd(i, j)), (j, 1, k)) == \
((x + Kd(i, k))*y)**k + Piecewise(
(((x + Kd(i, k))*y)**(i - 1)*(x + Kd(i, k)) *
((x + Kd(i, k))*y)**(-i + k), And(Integer(1) <= i, i <= k)),
(0, True)
)
assert dp((x + Kd(i, k))*(y + Kd(i, j)), (j, k, 3)) == \
((x + Kd(i, k))*y)**(4 - k) + Piecewise(
(((x + Kd(i, k))*y)**(i - k)*(x + Kd(i, k)) *
((x + Kd(i, k))*y)**(-i + 3), And(k <= i, i <= 3)),
(0, True)
)
assert dp((x + Kd(i, k))*(y + Kd(i, j)), (j, k, l)) == \
((x + Kd(i, k))*y)**(-k + l + 1) + Piecewise(
(((x + Kd(i, k))*y)**(i - k)*(x + Kd(i, k)) *
((x + Kd(i, k))*y)**(-i + l), And(k <= i, i <= l)),
(0, True)
)
def test_deltaproduct_mul_add_x_y_add_y_kd():
assert dp((x + y)*(y + Kd(i, j)), (j, 1, 3)) == ((x + y)*y)**3 + \
(x + y)*((x + y)*y)**2*Kd(i, 1) + \
(x + y)*y*(x + y)**2*y*Kd(i, 2) + \
((x + y)*y)**2*(x + y)*Kd(i, 3)
assert dp((x + y) * (y + Kd(i, j)), (j, 1, 1)) == (x + y) * (y + Kd(i, 1))
assert dp((x + y) * (y + Kd(i, j)), (j, 2, 2)) == (x + y) * (y + Kd(i, 2))
assert dp((x + y) * (y + Kd(i, j)), (j, 3, 3)) == (x + y) * (y + Kd(i, 3))
assert dp((x + y)*(y + Kd(i, j)), (j, 1, k)) == \
((x + y)*y)**k + Piecewise(
(((x + y)*y)**(i - 1)*(x + y)*((x + y)*y)**(k - i),
And(Integer(1) <= i, i <= k)),
(0, True)
)
assert dp((x + y)*(y + Kd(i, j)), (j, k, 3)) == \
((x + y)*y)**(-k + 4) + Piecewise(
(((x + y)*y)**(i - k)*(x + y)*((x + y)*y)**(3 - i),
And(k <= i, i <= 3)),
(0, True)
)
assert dp((x + y)*(y + Kd(i, j)), (j, k, l)) == \
((x + y)*y)**(-k + l + 1) + Piecewise(
(((x + y)*y)**(i - k)*(x + y)*((x + y)*y)**(l - i),
And(k <= i, i <= l)),
(0, True)
)
def test_deltaproduct_mul_x_add_y_twokd():
assert dp(x*(y + 2*Kd(i, j)), (j, 1, 3)) == (x*y)**3 + \
2*x*(x*y)**2*Kd(i, 1) + 2*x*y*x*x*y*Kd(i, 2) + 2*(x*y)**2*x*Kd(i, 3)
assert dp(x * (y + 2 * Kd(i, j)), (j, 1, 1)) == x * (y + 2 * Kd(i, 1))
assert dp(x * (y + 2 * Kd(i, j)), (j, 2, 2)) == x * (y + 2 * Kd(i, 2))
assert dp(x * (y + 2 * Kd(i, j)), (j, 3, 3)) == x * (y + 2 * Kd(i, 3))
assert dp(x*(y + 2*Kd(i, j)), (j, 1, k)) == \
(x*y)**k + Piecewise(
(2*(x*y)**(i - 1)*x*(x*y)**(k - i), And(Integer(1) <= i, i <= k)),
(0, True)
)
assert dp(x*(y + 2*Kd(i, j)), (j, k, 3)) == \
(x*y)**(-k + 4) + Piecewise(
(2*(x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
(0, True)
)
assert dp(x*(y + 2*Kd(i, j)), (j, k, l)) == \
(x*y)**(-k + l + 1) + Piecewise(
(2*(x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
(0, True)
)