def test__eval_product():
from diofant.abc import i, n
# issue 4809
a = Function('a')
assert product(2 * a(i), (i, 1, n)) == 2**n * Product(a(i), (i, 1, n))
# issue 4810
assert product(2**i, (i, 1, n)) == 2**(n / 2 + n**2 / 2)
def test_rational_products():
assert simplify(product(1 + 1/n, (n, a, b))) == (1 + b)/a
assert simplify(product(n + 1, (n, a, b))) == gamma(2 + b)/gamma(1 + a)
assert simplify(product((n + 1)/(n - 1), (n, a, b))) == b*(1 + b)/(a*(a - 1))
assert simplify(product(n/(n + 1)/(n + 2), (n, a, b))) == \
a*gamma(a + 2)/(b + 1)/gamma(b + 3)
assert simplify(product(n*(n + 1)/(n - 1)/(n - 2), (n, a, b))) == \
b**2*(b - 1)*(1 + b)/(a - 1)**2/(a*(a - 2))
def test_rational_products():
assert simplify(product(1 + 1/n, (n, a, b))) == (1 + b)/a
assert simplify(product(n + 1, (n, a, b))) == gamma(2 + b)/gamma(1 + a)
assert simplify(product((n + 1)/(n - 1), (n, a, b))) == b*(1 + b)/(a*(a - 1))
assert simplify(product(n/(n + 1)/(n + 2), (n, a, b))) == \
a*gamma(a + 2)/(b + 1)/gamma(b + 3)
assert simplify(product(n*(n + 1)/(n - 1)/(n - 2), (n, a, b))) == \
b**2*(b - 1)*(1 + b)/(a - 1)**2/(a*(a - 2))
def test_special_products():
# Wallis product
assert product((4*k)**2 / (4*k**2 - 1), (k, 1, n)) == \
4**n*factorial(n)**2/rf(Rational(1, 2), n)/rf(Rational(3, 2), n)
# Euler's product formula for sin
assert product(1 + a/k**2, (k, 1, n)) == \
rf(1 - sqrt(-a), n)*rf(1 + sqrt(-a), n)/factorial(n)**2
def test_special_products():
# Wallis product
assert product((4*k)**2 / (4*k**2 - 1), (k, 1, n)) == \
4**n*factorial(n)**2/rf(Rational(1, 2), n)/rf(Rational(3, 2), n)
# Euler's product formula for sin
assert product(1 + a/k**2, (k, 1, n)) == \
rf(1 - sqrt(-a), n)*rf(1 + sqrt(-a), n)/factorial(n)**2
def test_wallis_product():
# Wallis product, given in two different forms to ensure that Product
# can factor simple rational expressions
A = product(4 * n**2 / (4 * n**2 - 1), (n, 1, b))
B = product((2 * n) * (2 * n) / (2 * n - 1) / (2 * n + 1), (n, 1, b))
half = Rational(1, 2)
R = pi / 2 * factorial(b)**2 / factorial(b - half) / factorial(b + half)
assert A.equals(R)
assert B.equals(R)
def test_rational_products():
assert product(1 + 1 / n, (n, a, b)).equals((1 + b) / a)
assert product(n + 1, (n, a, b)).equals(gamma(2 + b) / gamma(1 + a))
assert product((n + 1) / (n - 1),
(n, a, b)).equals(b * (1 + b) / (a * (a - 1)))
assert product(n / (n + 1) / (n + 2),
(n, a, b)).equals(a * gamma(a + 2) / (b + 1) / gamma(b + 3))
assert product(n * (n + 1) / (n - 1) / (n - 2), (n, a, b)).equals(
b**2 * (b - 1) * (1 + b) / (a - 1)**2 / (a * (a - 2)))
def test__eval_product():
# issue sympy/sympy#4809
a = Function('a')
assert product(2*a(i), (i, 1, n)) == 2**n * Product(a(i), (i, 1, n))
# issue sympy/sympy#4810
assert product(2**i, (i, 1, n)) == 2**(n/2 + n**2/2)
assert (product((i - 1)*(i**6 + i - 1), (i, n, m)) ==
rf(n - 1, m - n + 1)*product(i**6 + i - 1, (i, n, m)))
assert (product(log(i)**2*cos(i)**3, (i, n, m)) ==
Product(log(i)**2*cos(i)**3, (i, n, m)))
def test_multiple_products():
assert product(x, (n, 1, k), (k, 1, m)) == x**(m**2 / 2 + m / 2)
assert product(f(n), (n, 1, m),
(m, 1, k)) == Product(f(n), (n, 1, m), (m, 1, k)).doit()
assert Product(f(n), (m, 1, k), (n, 1, k)).doit() == \
Product(Product(f(n), (m, 1, k)), (n, 1, k)).doit() == \
product(f(n), (m, 1, k), (n, 1, k)) == \
product(product(f(n), (m, 1, k)), (n, 1, k)) == \
Product(f(n)**k, (n, 1, k))
assert Product(x, (x, 1, k),
(k, 1, n)).doit() == Product(factorial(k), (k, 1, n))
assert Product(x**k, (n, 1, k), (k, 1, m)).variables == [n, k]
def test_multiple_products():
assert product(x, (n, 1, k), (k, 1, m)) == x**(m**2/2 + m/2)
assert product(f(n), (
n, 1, m), (m, 1, k)) == Product(f(n), (n, 1, m), (m, 1, k)).doit()
assert Product(f(n), (m, 1, k), (n, 1, k)).doit() == \
Product(Product(f(n), (m, 1, k)), (n, 1, k)).doit() == \
product(f(n), (m, 1, k), (n, 1, k)) == \
product(product(f(n), (m, 1, k)), (n, 1, k)) == \
Product(f(n)**k, (n, 1, k))
assert Product(
x, (x, 1, k), (k, 1, n)).doit() == Product(factorial(k), (k, 1, n))
assert Product(x**k, (n, 1, k), (k, 1, m)).variables == [n, k]
def test__eval_product():
# issue sympy/sympy#4809
a = Function('a')
assert product(2 * a(i), (i, 1, n)) == 2**n * Product(a(i), (i, 1, n))
# issue sympy/sympy#4810
assert product(2**i, (i, 1, n)) == 2**(n / 2 + n**2 / 2)
assert (product(
(i - 1) * (i**6 + i - 1),
(i, n, m)) == rf(n - 1, m - n + 1) * product(i**6 + i - 1, (i, n, m)))
assert (product(log(i)**2 * cos(i)**3,
(i, n, m)) == Product(log(i)**2 * cos(i)**3, (i, n, m)))
def test_simple_products():
assert Product(nan, (x, 1, 3)) is nan
assert product(nan, (x, 1, 3)) is nan
assert Product(x, (n, a, a)).doit() == x
assert Product(x, (x, a, a)).doit() == a
assert Product(x, (y, 1, a)).doit() == x**a
lo, hi = 1, 2
s1 = Product(n, (n, lo, hi))
s2 = Product(n, (n, hi, lo))
assert s1 != s2
# This IS correct according to Karr product convention
assert s1.doit() == 2
assert s2.doit() == 1
lo, hi = x, x + 1
s1 = Product(n, (n, lo, hi))
s2 = Product(n, (n, hi, lo))
s3 = 1 / Product(n, (n, hi + 1, lo - 1))
assert s1 != s2
# This IS correct according to Karr product convention
assert s1.doit() == x * (x + 1)
assert s2.doit() == 1
assert s3.doit() == x * (x + 1)
assert Product(Integral(2 * x, (x, 1, y)) + 2 * x,
(x, 1, 2)).doit() == (y**2 + 1) * (y**2 + 3)
assert product(2, (n, a, b)) == 2**(b - a + 1)
assert product(n, (n, 1, b)) == factorial(b)
assert product(n**3, (n, 1, b)) == factorial(b)**3
assert product(3**(2 + n),
(n, a, b)) == (3**(2 * (1 - a + b) + b / 2 + (b**2) / 2 +
a / 2 - (a**2) / 2))
assert product(cos(n), (n, 3, 5)) == cos(3) * cos(4) * cos(5)
assert product(cos(n), (n, x, x + 2)) == cos(x) * cos(x + 1) * cos(x + 2)
assert isinstance(product(cos(n), (n, x, x + Rational(1, 2))), Product)
# If Product managed to evaluate this one, it most likely got it wrong!
assert isinstance(Product(n**n, (n, 1, b)), Product)
def test_simple_products():
assert Product(nan, (x, 1, 3)) is nan
assert product(nan, (x, 1, 3)) is nan
assert Product(x, (n, a, a)).doit() == x
assert Product(x, (x, a, a)).doit() == a
assert Product(x, (y, 1, a)).doit() == x**a
lo, hi = 1, 2
s1 = Product(n, (n, lo, hi))
s2 = Product(n, (n, hi, lo))
assert s1 != s2
# This IS correct according to Karr product convention
assert s1.doit() == 2
assert s2.doit() == 1
lo, hi = x, x + 1
s1 = Product(n, (n, lo, hi))
s2 = Product(n, (n, hi, lo))
s3 = 1 / Product(n, (n, hi + 1, lo - 1))
assert s1 != s2
# This IS correct according to Karr product convention
assert s1.doit() == x*(x + 1)
assert s2.doit() == 1
assert s3.doit() == x*(x + 1)
assert Product(Integral(2*x, (x, 1, y)) + 2*x, (x, 1, 2)).doit() == \
(y**2 + 1)*(y**2 + 3)
assert product(2, (n, a, b)) == 2**(b - a + 1)
assert product(n, (n, 1, b)) == factorial(b)
assert product(n**3, (n, 1, b)) == factorial(b)**3
assert product(3**(2 + n), (n, a, b)) \
== 3**(2*(1 - a + b) + b/2 + (b**2)/2 + a/2 - (a**2)/2)
assert product(cos(n), (n, 3, 5)) == cos(3)*cos(4)*cos(5)
assert product(cos(n), (n, x, x + 2)) == cos(x)*cos(x + 1)*cos(x + 2)
assert isinstance(product(cos(n), (n, x, x + Rational(1, 2))), Product)
# If Product managed to evaluate this one, it most likely got it wrong!
assert isinstance(Product(n**n, (n, 1, b)), Product)
def maxima_product(self, var, low, high):
return product(self, (var, low, high))
def test_evalf_mul():
# diofant should not try to expand this; it should be handled term-wise
# in evalf through mpmath
assert NS(product(1 + sqrt(n)*I, (n, 1, 500)), 1) == '5.e+567 + 2.e+568*I'
def test_evalf_mul():
# diofant should not try to expand this; it should be handled term-wise
# in evalf through mpmath
assert NS(product(1 + sqrt(n)*I, (n, 1, 500)), 1) == '5.e+567 + 2.e+568*I'
def test_product_pow():
# issue 4817
assert product(2**f(k), (k, 1, n)) == 2**Sum(f(k), (k, 1, n))
assert product(2**(2 * f(k)), (k, 1, n)) == 2**Sum(2 * f(k), (k, 1, n))
def test_Product_doit():
assert Product(n * Integral(a**2), (n, 1, 3)).doit() == 2 * a**9 / 9
assert Product(n*Integral(a**2), (n, 1, 3)).doit(deep=False) == \
6*Integral(a**2)**3
assert product(n * Integral(a**2), (n, 1, 3)) == 6 * Integral(a**2)**3
def test_rational_products():
assert product(1 + 1/k, (k, 1, n)) == rf(2, n)/factorial(n)
def test_product_pow():
# issue sympy/sympy#4817
assert product(2**f(k), (k, 1, n)) == 2**Sum(f(k), (k, 1, n))
assert product(2**(2*f(k)), (k, 1, n)) == 2**Sum(2*f(k), (k, 1, n))
def test_rational_products():
assert product(1 + 1 / k, (k, 1, n)) == rf(2, n) / factorial(n)
def compute_moment(self, k):
if self.distribution == 'finite':
return sum([p * (b**k) for b, p in self.parameters])
if self.distribution == 'uniform':
l, u = self.parameters
return (u**(k + 1) - l**(k + 1)) / ((k + 1) * (u - l))
if self.distribution == 'gauss' or self.distribution == 'normal':
mu, sigma_squared = self.parameters
# For low moments avoid scipy.stats.moments as it does not support
# parametric parameters. In the future get all moments directly,
# using the following properties:
# https://math.stackexchange.com/questions/1945448/methods-for-finding-raw-moments-of-the-normal-distribution
if k == 0:
return 1
elif k == 1:
return mu
elif k == 2:
return mu**2 + sigma_squared
elif k == 3:
return mu * (mu**2 + 3 * sigma_squared)
elif k == 4:
return mu**4 + 6 * mu**2 * sigma_squared + 3 * sigma_squared**2
moment = norm(loc=mu, scale=sqrt(sigma_squared)).moment(k)
return Rational(moment)
if self.distribution == 'bernoulli':
return sympify(self.parameters[0])
if self.distribution == 'geometric':
p = sympify(self.parameters[0])
return p * polylog(-k, 1 - p)
if self.distribution == 'exponential':
lambd = sympify(self.parameters[0])
return factorial(k) / (lambd**k)
if self.distribution == 'beta':
alpha, beta = self.parameters
alpha = sympify(alpha)
beta = sympify(beta)
r = symbols('r')
return product((alpha + r) / (alpha + beta + r), (r, 0, k - 1))
if self.distribution == 'chi-squared':
n = sympify(self.parameters[0])
i = symbols('i')
return product(n + 2 * i, (i, 0, k - 1))
if self.distribution == 'rayleigh':
s = sympify(self.parameters[0])
return (2**(k / 2)) * (s**k) * gamma(1 + k / 2)
if self.distribution == 'unknown':
return sympify(f"{self.var_name}(0)^{k}")
if self.distribution == 'laplace':
mu, b = self.parameters
mu = sympify(mu)
b = sympify(b)
x = Laplace("x", mu, b)
return E(x**k)
if self.distribution == 'binomial':
n, p = self.parameters
n = sympify(n)
p = sympify(p)
x = Binomial("x", n, p)
return E(x**k)
if self.distribution == 'hypergeometric':
N, K, n = self.parameters
N = sympify(N)
K = sympify(K)
n = sympify(n)
x = Hypergeometric("x", N, K, n)
return E(x**k)
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 8301
assert product(2, (n, 1, oo)) == oo
assert product(-1, (n, 1, oo)).func is Product
def test_Product_doit():
assert Product(n*Integral(a**2), (n, 1, 3)).doit() == 2 * a**9 / 9
assert Product(n*Integral(a**2), (n, 1, 3)).doit(deep=False) == \
6*Integral(a**2)**3
assert product(n*Integral(a**2), (n, 1, 3)) == 6*Integral(a**2)**3
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)
