def test_Identity():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
i, j = symbols('i j')
In = Identity(n)
Im = Identity(m)
assert A * Im == A
assert In * A == A
assert In.transpose() == In
assert In.inverse() == In
assert In.conjugate() == In
assert In[i, j] != 0
assert Sum(In[i, j], (i, 0, n - 1), (j, 0, n - 1)).subs(n, 3).doit() == 3
assert Sum(Sum(In[i, j], (i, 0, n - 1)), (j, 0, n - 1)).subs(n,
3).doit() == 3
# If range exceeds the limit `(0, n-1)`, do not remove `Piecewise`:
expr = Sum(In[i, j], (i, 0, n - 1))
assert expr.doit() == 1
expr = Sum(In[i, j], (i, 0, n - 2))
assert expr.doit().dummy_eq(
Piecewise((1, (j >= 0) & (j <= n - 2)), (0, True)))
expr = Sum(In[i, j], (i, 1, n - 1))
assert expr.doit().dummy_eq(
Piecewise((1, (j >= 1) & (j <= n - 1)), (0, True)))
assert Identity(3).as_explicit() == ImmutableDenseMatrix.eye(3)
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 _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 _marginalise(self, expr, rv, evaluate):
if isinstance(rv.pspace.distribution, SingleFiniteDistribution):
rv_dens = rv.pspace.distribution.pmf(rv)
else:
rv_dens = rv.pspace.distribution.pdf(rv)
rv_dom = rv.pspace.domain.set
if rv.pspace.is_Discrete or rv.pspace.is_Finite:
expr = Sum(expr * rv_dens, (rv, rv_dom._inf, rv_dom._sup))
else:
expr = Integral(expr * rv_dens, (rv, rv_dom._inf, rv_dom._sup))
if evaluate:
return expr.doit()
return expr
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 _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:]
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 test_BernoulliProcess():
B = BernoulliProcess("B", p=0.6, success=1, failure=0)
assert B.state_space == FiniteSet(0, 1)
assert B.index_set == S.Naturals0
assert B.success == 1
assert B.failure == 0
X = BernoulliProcess("X", p=Rational(1, 3), success='H', failure='T')
assert X.state_space == FiniteSet('H', 'T')
H, T = symbols("H,T")
assert E(X[1] + X[2] * X[3]
) == H**2 / 9 + 4 * H * T / 9 + H / 3 + 4 * T**2 / 9 + 2 * T / 3
t, x = symbols('t, x', positive=True, integer=True)
assert isinstance(B[t], RandomIndexedSymbol)
raises(ValueError,
lambda: BernoulliProcess("X", p=1.1, success=1, failure=0))
raises(NotImplementedError, lambda: B(t))
raises(IndexError, lambda: B[-3])
assert B.joint_distribution(B[3], B[9]) == JointDistributionHandmade(
Lambda(
(B[3], B[9]),
Piecewise((0.6, Eq(B[3], 1)), (0.4, Eq(B[3], 0)),
(0, True)) * Piecewise((0.6, Eq(B[9], 1)),
(0.4, Eq(B[9], 0)), (0, True))))
assert B.joint_distribution(2, B[4]) == JointDistributionHandmade(
Lambda(
(B[2], B[4]),
Piecewise((0.6, Eq(B[2], 1)), (0.4, Eq(B[2], 0)),
(0, True)) * Piecewise((0.6, Eq(B[4], 1)),
(0.4, Eq(B[4], 0)), (0, True))))
# Test for the sum distribution of Bernoulli Process RVs
Y = B[1] + B[2] + B[3]
assert P(Eq(Y, 0)).round(2) == Float(0.06, 1)
assert P(Eq(Y, 2)).round(2) == Float(0.43, 2)
assert P(Eq(Y, 4)).round(2) == 0
assert P(Gt(Y, 1)).round(2) == Float(0.65, 2)
# Test for independency of each Random Indexed variable
assert P(Eq(B[1], 0) & Eq(B[2], 1) & Eq(B[3], 0)
& Eq(B[4], 1)).round(2) == Float(0.06, 1)
assert E(2 * B[1] + B[2]).round(2) == Float(1.80, 3)
assert E(2 * B[1] + B[2] + 5).round(2) == Float(6.80, 3)
assert E(B[2] * B[4] + B[10]).round(2) == Float(0.96, 2)
assert E(B[2] > 0, Eq(B[1], 1) & Eq(B[2], 1)).round(2) == Float(0.60, 2)
assert E(B[1]) == 0.6
assert P(B[1] > 0).round(2) == Float(0.60, 2)
assert P(B[1] < 1).round(2) == Float(0.40, 2)
assert P(B[1] > 0, B[2] <= 1).round(2) == Float(0.60, 2)
assert P(B[12] * B[5] > 0).round(2) == Float(0.36, 2)
assert P(B[12] * B[5] > 0, B[4] < 1).round(2) == Float(0.36, 2)
assert P(Eq(B[2], 1), B[2] > 0) == 1
assert P(Eq(B[5], 3)) == 0
assert P(Eq(B[1], 1), B[1] < 0) == 0
assert P(B[2] > 0, Eq(B[2], 1)) == 1
assert P(B[2] < 0, Eq(B[2], 1)) == 0
assert P(B[2] > 0, B[2] == 7) == 0
assert P(B[5] > 0, B[5]) == BernoulliDistribution(0.6, 0, 1)
raises(ValueError, lambda: P(3))
raises(ValueError, lambda: P(B[3] > 0, 3))
# test issue 19456
expr = Sum(B[t], (t, 0, 4))
expr2 = Sum(B[t], (t, 1, 3))
expr3 = Sum(B[t]**2, (t, 1, 3))
assert expr.doit() == B[0] + B[1] + B[2] + B[3] + B[4]
assert expr2.doit() == Y
assert expr3.doit() == B[1]**2 + B[2]**2 + B[3]**2
assert B[2 * t].free_symbols == {B[2 * t], t}
assert B[4].free_symbols == {B[4]}
assert B[x * t].free_symbols == {B[x * t], x, t}
#test issue 20078
assert (2 * B[t] + 3 * B[t]).simplify() == 5 * B[t]
assert (2 * B[t] - 3 * B[t]).simplify() == -B[t]
assert (2 * (0.25 * B[t])).simplify() == 0.5 * B[t]
assert (2 * B[t] * 0.25 * B[t]).simplify() == 0.5 * B[t]**2
assert (B[t]**2 + B[t]**3).simplify() == (B[t] + 1) * B[t]**2
def _eval_rewrite_as_Sum(self, expr, **kwargs):
from sympy.concrete.summations import Sum
i = uniquely_named_symbol('i', expr)
s = Sum(self.arg[i, i], (i, 0, self.arg.rows - 1))
return s.doit()
