def _entry(self, i, j, expand=True):
coeff, matrices = self.as_coeff_matrices()
if len(matrices) == 1: # situation like 2*X, matmul is just X
return coeff * matrices[0][i, j]
head, tail = matrices[0], matrices[1:]
if len(tail) == 0:
raise ValueError("lenth of tail cannot be 0")
X = head
Y = MatMul(*tail)
from sympy.core.symbol import Dummy
from sympy.concrete.summations import Sum
from sympy.matrices import ImmutableMatrix
k = Dummy('k', integer=True)
if X.has(ImmutableMatrix) or Y.has(ImmutableMatrix):
return coeff*Add(*[X[i, k]*Y[k, j] for k in range(X.cols)])
result = Sum(coeff*X[i, k]*Y[k, j], (k, 0, X.cols - 1))
try:
if not X.cols.is_number:
# Don't waste time in result.doit() if the sum bounds are symbolic
expand = False
except AttributeError:
pass
return result.doit() if expand else result
def test_evalf_sum():
assert Sum(n, (n, 1, 2)).evalf() == 3.
assert Sum(n, (n, 1, 2)).doit().evalf() == 3.
# the next test should return instantly
assert Sum(1 / n, (n, 1, 2)).evalf() == 1.5
# issue 8219
assert Sum(E / factorial(n), (n, 0, oo)).evalf() == (E * E).evalf()
# issue 8254
assert Sum(2**n * n / factorial(n),
(n, 0, oo)).evalf() == (2 * E * E).evalf()
# issue 8411
s = Sum(1 / x**2, (x, 100, oo))
assert s.n() == s.doit().n()
def fdiff(self, argindex=2):
from sympy.concrete.summations import Sum
if argindex == 2:
x, p = self.args
k = Dummy("k")
return self.func(x, p) * Sum(polygamma(0, x + (1 - k) / 2),
(k, 1, p))
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, alpha, x, **kwargs):
from sympy.concrete.summations import Sum
# Make sure n \in N_0
if n.is_negative or n.is_integer is False:
raise ValueError("Error: n should be a non-negative integer.")
k = Dummy("k")
kern = RisingFactorial(
-n, k) / (gamma(k + alpha + 1) * factorial(k)) * x**k
return gamma(n + alpha + 1) / factorial(n) * Sum(kern, (k, 0, n))
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 _eval_rewrite_as_polynomial(self, n, a, b, x, **kwargs):
from sympy.concrete.summations import Sum
# Make sure n \in N
if n.is_negative or n.is_integer is False:
raise ValueError("Error: n should be a non-negative integer.")
k = Dummy("k")
kern = (RisingFactorial(-n, k) * RisingFactorial(a + b + n + 1, k) * RisingFactorial(a + k + 1, n - k) /
factorial(k) * ((1 - x)/2)**k)
return 1 / factorial(n) * Sum(kern, (k, 0, n))
def test_Indexed():
# Issue #10934
if not numpy:
skip("numpy not installed")
a = IndexedBase('a')
i, j = symbols('i j')
b = numpy.array([[1, 2], [3, 4]])
assert lambdify(a, Sum(a[x, y], (x, 0, 1), (y, 0, 1)))(b) == 10
def _eval_rewrite_as_Sum(self, ap, bq, z, **kwargs):
from sympy.concrete.summations import Sum
n = Dummy("n", integer=True)
rfap = Tuple(*[RisingFactorial(a, n) for a in ap])
rfbq = Tuple(*[RisingFactorial(b, n) for b in bq])
coeff = Mul(*rfap) / Mul(*rfbq)
return Piecewise((Sum(coeff * z**n / factorial(n),
(n, 0, oo)), self.convergence_statement),
(self, True))
def test_matrix_expression_to_indices():
i, j = symbols("i, j")
i1, i2, i3 = symbols("i_1:4")
def replace_dummies(expr):
repl = {i: Symbol(i.name) for i in expr.atoms(Dummy)}
return expr.xreplace(repl)
expr = W*X*Z
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = Z.T*X.T*W.T
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[j, i2]*X[i2, i1]*Z[i1, i], (i1, 0, m-1), (i2, 0, l-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j), i) == expr
expr = W*X*Z + W*Y*Z
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = 2*W*X*Z + 3*W*Y*Z
assert replace_dummies(expr._entry(i, j)) == \
2*Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
3*Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = W*(X + Y)*Z
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[i, i1]*(X[i1, i2] + Y[i1, i2])*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = A*B**2*A
#assert replace_dummies(expr._entry(i, j)) == \
# Sum(A[i, i1]*B[i1, i2]*B[i2, i3]*A[i3, j], (i1, 0, 1), (i2, 0, 1), (i3, 0, 1))
# Check that different dummies are used in sub-multiplications:
expr = (X1*X2 + X2*X1)*X3
assert replace_dummies(expr._entry(i, j)) == \
Sum((Sum(X1[i, i2] * X2[i2, i1], (i2, 0, m - 1)) + Sum(X1[i3, i1] * X2[i, i3], (i3, 0, m - 1))) * X3[
i1, j], (i1, 0, m - 1))
def test_assoc_laguerre():
n = Symbol("n")
m = Symbol("m")
alpha = Symbol("alpha")
# generalized Laguerre polynomials:
assert assoc_laguerre(0, alpha, x) == 1
assert assoc_laguerre(1, alpha, x) == -x + alpha + 1
assert assoc_laguerre(2, alpha, x).expand() == \
(x**2/2 - (alpha + 2)*x + (alpha + 2)*(alpha + 1)/2).expand()
assert assoc_laguerre(3, alpha, x).expand() == \
(-x**3/6 + (alpha + 3)*x**2/2 - (alpha + 2)*(alpha + 3)*x/2 +
(alpha + 1)*(alpha + 2)*(alpha + 3)/6).expand()
# Test the lowest 10 polynomials with laguerre_poly, to make sure it works:
for i in range(10):
assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
X = assoc_laguerre(n, m, x)
assert isinstance(X, assoc_laguerre)
assert assoc_laguerre(n, 0, x) == laguerre(n, x)
assert assoc_laguerre(n, alpha, 0) == binomial(alpha + n, alpha)
p = Symbol("p", positive=True)
assert assoc_laguerre(p, alpha, oo) == (-1)**p * oo
assert assoc_laguerre(p, alpha, -oo) is oo
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
_k = Dummy('k')
assert diff(assoc_laguerre(n, alpha, x), alpha).dummy_eq(
Sum(assoc_laguerre(_k, alpha, x) / (-alpha + n), (_k, 0, n - 1)))
assert conjugate(assoc_laguerre(n, alpha, x)) == \
assoc_laguerre(n, conjugate(alpha), conjugate(x))
assert assoc_laguerre(n, alpha, x).rewrite('polynomial').dummy_eq(
gamma(alpha + n + 1) * Sum(
x**_k * RisingFactorial(-n, _k) /
(factorial(_k) * gamma(_k + alpha + 1)),
(_k, 0, n)) / factorial(n))
raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, x))
raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(1))
raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(4))
def _eval_expand_log(self, deep=True, **hints):
from sympy.concrete import Sum, Product
force = hints.get('force', False)
factor = hints.get('factor', False)
if (len(self.args) == 2):
return expand_log(self.func(*self.args), deep=deep, force=force)
arg = self.args[0]
if arg.is_Integer:
# remove perfect powers
p = perfect_power(arg)
logarg = None
coeff = 1
if p is not False:
arg, coeff = p
logarg = self.func(arg)
# expand as product of its prime factors if factor=True
if factor:
p = factorint(arg)
if arg not in p.keys():
logarg = sum(n*log(val) for val, n in p.items())
if logarg is not None:
return coeff*logarg
elif arg.is_Rational:
return log(arg.p) - log(arg.q)
elif arg.is_Mul:
expr = []
nonpos = []
for x in arg.args:
if force or x.is_positive or x.is_polar:
a = self.func(x)
if isinstance(a, log):
expr.append(self.func(x)._eval_expand_log(**hints))
else:
expr.append(a)
elif x.is_negative:
a = self.func(-x)
expr.append(a)
nonpos.append(S.NegativeOne)
else:
nonpos.append(x)
return Add(*expr) + log(Mul(*nonpos))
elif arg.is_Pow or isinstance(arg, exp):
if force or (arg.exp.is_extended_real and (arg.base.is_positive or ((arg.exp+1)
.is_positive and (arg.exp-1).is_nonpositive))) or arg.base.is_polar:
b = arg.base
e = arg.exp
a = self.func(b)
if isinstance(a, log):
return unpolarify(e) * a._eval_expand_log(**hints)
else:
return unpolarify(e) * a
elif isinstance(arg, Product):
if force or arg.function.is_positive:
return Sum(log(arg.function), *arg.limits)
return self.func(arg)
def test_as_dummy():
u, v, x, y, z, _0, _1 = symbols('u v x y z _0 _1')
assert Lambda(x, x + 1).as_dummy() == Lambda(_0, _0 + 1)
assert Lambda(x, x + _0).as_dummy() == Lambda(_1, _0 + _1)
eq = (1 + Sum(x, (x, 1, x)))
ans = 1 + Sum(_0, (_0, 1, x))
once = eq.as_dummy()
assert once == ans
twice = once.as_dummy()
assert twice == ans
assert Integral(x + _0, (x, x + 1),
(_0, 1, 2)).as_dummy() == Integral(_0 + _1, (_0, x + 1),
(_1, 1, 2))
for T in (Symbol, Dummy):
d = T('x', real=True)
D = d.as_dummy()
assert D != d and D.func == Dummy and D.is_real is None
assert Dummy().as_dummy().is_commutative
assert Dummy(commutative=False).as_dummy().is_commutative is False
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy.concrete.summations import Sum
# Make sure n \in N_0
if n.is_negative:
return exp(x) * self._eval_rewrite_as_polynomial(-n - 1, -x, **kwargs)
if n.is_integer is False:
raise ValueError("Error: n should be an integer.")
k = Dummy("k")
kern = RisingFactorial(-n, k) / factorial(k)**2 * x**k
return Sum(kern, (k, 0, n))
def test_mix_expression():
Y, E = Poisson('Y', 1), Exponential('E', 1)
k = Dummy('k')
expr1 = Integral(
Sum(
exp(-1) * Integral(exp(-k) * DiracDelta(k - 2),
(k, 0, oo)) / factorial(k), (k, 0, oo)),
(k, -oo, 0))
expr2 = Integral(
Sum(
exp(-1) * Integral(exp(-k) * DiracDelta(k - 2),
(k, 0, oo)) / factorial(k), (k, 0, oo)),
(k, 0, oo))
assert P(Eq(Y + E, 1)) == 0
assert P(Ne(Y + E, 2)) == 1
with ignore_warnings(
UserWarning): ### TODO: Restore tests once warnings are removed
assert P(E + Y < 2, evaluate=False).rewrite(Integral).dummy_eq(expr1)
assert P(E + Y > 2, evaluate=False).rewrite(Integral).dummy_eq(expr2)
def test_exp_rewrite():
from sympy.concrete.summations import Sum
assert exp(x).rewrite(sin) == sinh(x) + cosh(x)
assert exp(x*I).rewrite(cos) == cos(x) + I*sin(x)
assert exp(1).rewrite(cos) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(x).rewrite(tanh) == (1 + tanh(x/2))/(1 - tanh(x/2))
assert exp(pi*I/4).rewrite(sqrt) == sqrt(2)/2 + sqrt(2)*I/2
assert exp(pi*I/3).rewrite(sqrt) == S(1)/2 + sqrt(3)*I/2
assert exp(x*log(y)).rewrite(Pow) == y**x
assert exp(log(x)*log(y)).rewrite(Pow) in [x**log(y), y**log(x)]
assert exp(log(log(x))*y).rewrite(Pow) == log(x)**y
n = Symbol('n', integer=True)
assert Sum((exp(pi*I/2)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == S(4)/5 + 2*I/5
assert Sum((exp(pi*I/4)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == 1/(1 - sqrt(2)*(1 + I)/4)
assert Sum((exp(pi*I/3)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == 1/(S(3)/4 - sqrt(3)*I/4)
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 test_mul_index():
assert (A*y)[0, 0] == A[0, 0]*y[0, 0] + A[0, 1]*y[1, 0]
assert (A*B).as_mutable() == (A.as_mutable() * B.as_mutable())
X = MatrixSymbol('X', n, m)
Y = MatrixSymbol('Y', m, k)
result = (X*Y)[4,2]
expected = Sum(X[4, i]*Y[i, 2], (i, 0, m - 1))
assert result.args[0].dummy_eq(expected.args[0], i)
assert result.args[1][1:] == expected.args[1][1:]
def test_hyper_rewrite_sum():
from sympy.concrete.summations import Sum
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
_k = Dummy("k")
assert replace_dummy(hyper((1, 2), (1, 3), x).rewrite(Sum), _k) == \
Sum(x**_k / factorial(_k) * RisingFactorial(2, _k) /
RisingFactorial(3, _k), (_k, 0, oo))
assert hyper((1, 2, 3), (-1, 3), z).rewrite(Sum) == \
hyper((1, 2, 3), (-1, 3), z)
def test_legendre():
assert legendre(0, x) == 1
assert legendre(1, x) == x
assert legendre(2, x) == ((3 * x**2 - 1) / 2).expand()
assert legendre(3, x) == ((5 * x**3 - 3 * x) / 2).expand()
assert legendre(4, x) == ((35 * x**4 - 30 * x**2 + 3) / 8).expand()
assert legendre(5, x) == ((63 * x**5 - 70 * x**3 + 15 * x) / 8).expand()
assert legendre(6, x) == ((231 * x**6 - 315 * x**4 + 105 * x**2 - 5) /
16).expand()
assert legendre(10, -1) == 1
assert legendre(11, -1) == -1
assert legendre(10, 1) == 1
assert legendre(11, 1) == 1
assert legendre(10, 0) != 0
assert legendre(11, 0) == 0
assert legendre(-1, x) == 1
k = Symbol('k')
assert legendre(5 - k, x).subs(k, 2) == ((5 * x**3 - 3 * x) / 2).expand()
assert roots(legendre(4, x), x) == {
sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
}
n = Symbol("n")
X = legendre(n, x)
assert isinstance(X, legendre)
assert unchanged(legendre, n, x)
assert legendre(n,
0) == sqrt(pi) / (gamma(S.Half - n / 2) * gamma(n / 2 + 1))
assert legendre(n, 1) == 1
assert legendre(n, oo) is oo
assert legendre(-n, x) == legendre(n - 1, x)
assert legendre(n, -x) == (-1)**n * legendre(n, x)
assert unchanged(legendre, -n + k, x)
assert conjugate(legendre(n, x)) == legendre(n, conjugate(x))
assert diff(legendre(n, x), x) == \
n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
assert diff(legendre(n, x), n) == Derivative(legendre(n, x), n)
_k = Dummy('k')
assert legendre(n, x).rewrite("polynomial").dummy_eq(
Sum((-1)**_k * (S.Half - x / 2)**_k * (x / 2 + S.Half)**(-_k + n) *
binomial(n, _k)**2, (_k, 0, n)))
raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(1))
raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(3))
def pdf(self, x):
x = sympify(x)
if x.is_number:
if x.is_Integer and x >= 1 and x <= self.sides:
return Rational(1, self.sides)
return S.Zero
if x.is_Symbol:
i = Dummy('i', integer=True, positive=True)
return Sum(KroneckerDelta(x, i)/self.sides, (i, 1, self.sides))
raise ValueError("'x' expected as an argument of type 'number' or 'symbol', "
"not %s" % (type(x)))
def cdf(self, other):
assert isinstance(other, dict)
rvs = other.keys()
_set = self.domain.set
expr = self.pdf(tuple(i.args[0] for i in self.symbols))
for i in range(len(other)):
if rvs[i].is_Continuous:
density = Integral(expr, (rvs[i], _set[i].inf, other[rvs[i]]))
elif rvs[i].is_Discrete:
density = Sum(expr, (rvs[i], _set[i].inf, other[rvs[i]]))
return density
def test_hyperexpand():
# Luke, Y. L. (1969), The Special Functions and Their Approximations,
# Volume 1, section 6.2
assert hyperexpand(hyper([], [], z)) == exp(z)
assert hyperexpand(hyper([1, 1], [2], -z)*z) == log(1 + z)
assert hyperexpand(hyper([], [S.Half], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [S('3/2')], -z**2/4)) == sin(z)
assert hyperexpand(hyper([S('1/2'), S('1/2')], [S('3/2')], z**2)*z) \
== asin(z)
assert isinstance(Sum(binomial(2, z)*z**2, (z, 0, a)).doit(), Expr)
def _entry(self, i, j, expand=True):
coeff, matrices = self.as_coeff_matrices()
if len(matrices) == 1: # situation like 2*X, matmul is just X
return coeff * matrices[0][i, j]
head, tail = matrices[0], matrices[1:]
assert len(tail) != 0
X = head
Y = MatMul(*tail)
from sympy.core.symbol import Dummy
from sympy.concrete.summations import Sum
from sympy.matrices import ImmutableMatrix, MatrixBase
k = Dummy('k', integer=True)
if X.has(ImmutableMatrix) or Y.has(ImmutableMatrix):
return coeff*Add(*[X[i, k]*Y[k, j] for k in range(X.cols)])
result = Sum(coeff*X[i, k]*Y[k, j], (k, 0, X.cols - 1))
return result.doit() if expand else result
def _entry(self, i, j, expand=True):
coeff, matrices = self.as_coeff_matrices()
if len(matrices) == 1: # situation like 2*X, matmul is just X
return coeff * matrices[0][i, j]
head, tail = matrices[0], matrices[1:]
if len(tail) == 0:
raise ValueError("lenth of tail cannot be 0")
X = head
Y = MatMul(*tail)
from sympy.core.symbol import Dummy
from sympy.concrete.summations import Sum
from sympy.matrices import ImmutableMatrix
k = Dummy('k', integer=True)
if X.has(ImmutableMatrix) or Y.has(ImmutableMatrix):
return coeff * Add(*[X[i, k] * Y[k, j] for k in range(X.cols)])
result = Sum(coeff * X[i, k] * Y[k, j], (k, 0, X.cols - 1))
return result.doit() if expand else result
def test_plot_and_save_5():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
s = Sum(1/x**y, (x, 1, oo))
p = plot(s, (y, 2, 10))
filename = 'test_advanced_inf_sum.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(Sum(1/x, (x, 1, y)), (y, 2, 10), show=False)
p[0].only_integers = True
p[0].steps = True
filename = 'test_advanced_fin_sum.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_product_spaces():
X1 = Geometric('X1', S.Half)
X2 = Geometric('X2', Rational(1, 3))
#assert str(P(X1 + X2 < 3, evaluate=False)) == """Sum(Piecewise((2**(X2 - n - 2)*(2/3)**(X2 - 1)/6, """\
# + """(-X2 + n + 3 >= 1) & (-X2 + n + 3 < oo)), (0, True)), (X2, 1, oo), (n, -oo, -1))"""
n = Dummy('n')
with ignore_warnings(
UserWarning): ### TODO: Restore tests once warnings are removed
assert P(X1 + X2 < 3, evaluate=False).rewrite(Sum).dummy_eq(
Sum(Piecewise((2**(-n) / 4, n + 2 >= 1), (0, True)),
(n, -oo, -1)) / 3)
#assert str(P(X1 + X2 > 3)) == """Sum(Piecewise((2**(X2 - n - 2)*(2/3)**(X2 - 1)/6, """ +\
# """(-X2 + n + 3 >= 1) & (-X2 + n + 3 < oo)), (0, True)), (X2, 1, oo), (n, 1, oo))"""
assert P(X1 + X2 > 3).dummy_eq(
Sum(
Piecewise((2**(X2 - n - 2) *
(Rational(2, 3))**(X2 - 1) / 6, -X2 + n + 3 >= 1),
(0, True)), (X2, 1, oo), (n, 1, oo)))
# assert str(P(Eq(X1 + X2, 3))) == """Sum(Piecewise((2**(X2 - 2)*(2/3)**(X2 - 1)/6, """ +\
# """X2 <= 2), (0, True)), (X2, 1, oo))"""
assert P(Eq(X1 + X2, 3)) == Rational(1, 12)
def test_multiple_sums():
if not np:
skip("NumPy not installed")
s = Sum((x + j) * i, (i, a, b), (j, c, d))
f = lambdify((a, b, c, d, x), s, 'numpy')
a_, b_ = 0, 10
c_, d_ = 11, 21
x_ = np.linspace(-1, +1, 10)
assert np.allclose(f(a_, b_, c_, d_, x_),
sum((x_ + j_) * i_ for i_ in range(a_, b_ + 1) for j_ in range(c_, d_ + 1)))
def test_log_product():
from sympy.abc import n, m
i, j = symbols('i,j', positive=True, integer=True)
x, y = symbols('x,y', positive=True)
from sympy.concrete import Product, Sum
f, g = Function('f'), Function('g')
assert simplify(log(Product(x**i, (i, 1, n)))) == Sum(i*log(x), (i, 1, n))
assert simplify(log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))) == \
log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))
expr = log(Product(-2, (n, 0, 4)))
assert simplify(expr) == expr
def test_exp_rewrite():
from sympy.concrete.summations import Sum
assert exp(x).rewrite(sin) == sinh(x) + cosh(x)
assert exp(x * I).rewrite(cos) == cos(x) + I * sin(x)
assert exp(1).rewrite(cos) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(x).rewrite(tanh) == (1 + tanh(x / 2)) / (1 - tanh(x / 2))
assert exp(pi * I / 4).rewrite(sqrt) == sqrt(2) / 2 + sqrt(2) * I / 2
assert exp(pi * I / 3).rewrite(sqrt) == 1 / 2 + sqrt(3) * I / 2
n = Symbol('n', integer=True)
assert Sum((exp(pi * I / 2) / 2)**n,
(n, 0, oo)).rewrite(sqrt).doit() == 4 / 5 + 2 * I / 5
assert Sum(
(exp(pi * I / 4) / 2)**n,
(n, 0, oo)).rewrite(sqrt).doit() == 1 / (1 - sqrt(2) * (1 + I) / 4)
assert Sum(
(exp(pi * I / 3) / 2)**n,
(n, 0, oo)).rewrite(sqrt).doit() == 1 / (3 / 4 - sqrt(3) * I / 4)
def pdf(self, *y):
d, v, l, mu = self.delta, self.v, self.lamda, self.mu
n = Symbol('n', negative=False, integer=True)
k = len(l)
sterm1 = Pow((1 - d), n)/\
((gamma(v + n)**(k - 1))*gamma(v)*gamma(n + 1))
sterm2 = Mul.fromiter(mui*li**(-v - n) for mui, li in zip(mu, l))
term1 = sterm1 * sterm2
sterm3 = (v + n) * sum([mui * yi for mui, yi in zip(mu, y)])
sterm4 = sum([exp(mui * yi)/li for (mui, yi, li) in zip(mu, y, l)])
term2 = exp(sterm3 - sterm4)
return Pow(d, v) * Sum(term1 * term2, (n, 0, S.Infinity))
def test_log_product():
from sympy.abc import n, m
i, j = symbols('i,j', positive=True, integer=True)
x, y = symbols('x,y', positive=True)
z = symbols('z', real=True)
w = symbols('w')
expr = log(Product(x**i, (i, 1, n)))
assert simplify(expr) == expr
assert expr.expand() == Sum(i * log(x), (i, 1, n))
expr = log(Product(x**i * y**j, (i, 1, n), (j, 1, m)))
assert simplify(expr) == expr
assert expr.expand() == Sum(i * log(x) + j * log(y), (i, 1, n), (j, 1, m))
expr = log(Product(-2, (n, 0, 4)))
assert simplify(expr) == expr
assert expr.expand() == expr
assert expr.expand(force=True) == Sum(log(-2), (n, 0, 4))
expr = log(Product(exp(z * i), (i, 0, n)))
assert expr.expand() == Sum(z * i, (i, 0, n))
expr = log(Product(exp(w * i), (i, 0, n)))
assert expr.expand() == expr
assert expr.expand(force=True) == Sum(w * i, (i, 0, n))
expr = log(Product(i**2 * abs(j), (i, 1, n), (j, 1, m)))
assert expr.expand() == Sum(2 * log(i) + log(j), (i, 1, n), (j, 1, m))
def cdf(self, other):
if not isinstance(other, dict):
raise ValueError("%s should be of type dict, got %s" %
(other, type(other)))
rvs = other.keys()
_set = self.domain.set.sets
expr = self.pdf(tuple(i.args[0] for i in self.symbols))
for i in range(len(other)):
if rvs[i].is_Continuous:
density = Integral(expr, (rvs[i], _set[i].inf, other[rvs[i]]))
elif rvs[i].is_Discrete:
density = Sum(expr, (rvs[i], _set[i].inf, other[rvs[i]]))
return density