def as_real_imag(self, deep=True, **hints):
from sympy.core.symbol import symbols
from sympy.polys.polytools import poly
from sympy.core.function import expand_multinomial
if self.exp.is_Integer:
exp = self.exp
re, im = self.base.as_real_imag(deep=deep)
if re.func == C.re or im.func == C.im:
return self, S.Zero
a, b = symbols('a b', cls=Dummy)
if exp >= 0:
if re.is_Number and im.is_Number:
# We can be more efficient in this case
expr = expand_multinomial(self.base**exp)
return expr.as_real_imag()
expr = poly((a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp
else:
mag = re**2 + im**2
re, im = re/mag, -im/mag
if re.is_Number and im.is_Number:
# We can be more efficient in this case
expr = expand_multinomial((re + im*S.ImaginaryUnit)**-exp)
return expr.as_real_imag()
expr = poly((a + b)**-exp)
# Terms with even b powers will be real
r = [i for i in expr.terms() if not i[0][1] % 2]
re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
# Terms with odd b powers will be imaginary
r = [i for i in expr.terms() if i[0][1] % 4 == 1]
im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
r = [i for i in expr.terms() if i[0][1] % 4 == 3]
im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
return (re_part.subs({a: re, b: S.ImaginaryUnit*im}),
im_part1.subs({a: re, b: im}) + im_part3.subs({a: re, b: -im}))
elif self.exp.is_Rational:
# NOTE: This is not totally correct since for x**(p/q) with
# x being imaginary there are actually q roots, but
# only a single one is returned from here.
re, im = self.base.as_real_imag(deep=deep)
if re.func == C.re or im.func == C.im:
return self, S.Zero
r = Pow(Pow(re, 2) + Pow(im, 2), S.Half)
t = C.atan2(im, re)
rp, tp = Pow(r, self.exp), t*self.exp
return (rp*C.cos(tp), rp*C.sin(tp))
else:
if deep:
hints['complex'] = False
return (C.re(self.expand(deep, **hints)),
C.im(self.expand(deep, **hints)))
else:
return (C.re(self), C.im(self))
def expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
# TODO: support discrete sets with non integer stepsizes
if evaluate:
try:
p = poly(expr, var)
t = Dummy('t', real=True)
mgf = self.moment_generating_function(t)
deg = p.degree()
taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
result = 0
for k in range(deg + 1):
result += p.coeff_monomial(var**k) * taylor.coeff_monomial(
t**k) * factorial(k)
return result
except PolynomialError:
return summation(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
else:
return Sum(expr * self.pdf(var), (var, self.set.inf, self.set.sup),
**kwargs)
def as_real_imag(self, deep=True, **hints):
from sympy.core.symbol import symbols
from sympy.polys.polytools import poly
from sympy.core.function import expand_multinomial
if self.exp.is_Integer:
exp = self.exp
re, im = self.base.as_real_imag(deep=deep)
a, b = symbols('a b', cls=Dummy)
if exp >= 0:
if re.is_Number and im.is_Number:
# We can be more efficient in this case
expr = expand_multinomial(self.base**exp)
return expr.as_real_imag()
expr = poly((a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp
else:
mag = re**2 + im**2
re, im = re/mag, -im/mag
if re.is_Number and im.is_Number:
# We can be more efficient in this case
expr = expand_multinomial((re + im*S.ImaginaryUnit)**-exp)
return expr.as_real_imag()
expr = poly((a + b)**-exp)
# Terms with even b powers will be real
r = [i for i in expr.terms() if not i[0][1] % 2]
re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
# Terms with odd b powers will be imaginary
r = [i for i in expr.terms() if i[0][1] % 4 == 1]
im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
r = [i for i in expr.terms() if i[0][1] % 4 == 3]
im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
return (re_part.subs({a: re, b: S.ImaginaryUnit*im}),
im_part1.subs({a: re, b: im}) + im_part3.subs({a: re, b: -im}))
elif self.exp.is_Rational:
# NOTE: This is not totally correct since for x**(p/q) with
# x being imaginary there are actually q roots, but
# only a single one is returned from here.
re, im = self.base.as_real_imag(deep=deep)
r = Pow(Pow(re, 2) + Pow(im, 2), S.Half)
t = C.atan2(im, re)
rp, tp = Pow(r, self.exp), t*self.exp
return (rp*C.cos(tp), rp*C.sin(tp))
else:
if deep:
hints['complex'] = False
return (C.re(self.expand(deep, complex=False)),
C.im(self. expand(deep, **hints)))
else:
return (C.re(self), C.im(self))
def expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
if evaluate:
try:
p = poly(expr, var)
if p.is_zero:
return S.Zero
t = Dummy('t', real=True)
mgf = self._moment_generating_function(t)
if mgf is None:
return integrate(expr * self.pdf(var), (var, self.set), **kwargs)
deg = p.degree()
taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
result = 0
for k in range(deg+1):
result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k)
return result
except PolynomialError:
return integrate(expr * self.pdf(var), (var, self.set), **kwargs)
else:
return Integral(expr * self.pdf(var), (var, self.set), **kwargs)
c=symbols('c0:%d'%(Lamda.rows*Lamda.cols))
C=Matrix(Lamda.rows,Lamda.cols,c)
#-----------------------------------------
RHS_matrix=Lamda*sI_A +C #right hand side of the equation (1)
'''
-----------Converting equation (1) to a system of linear -----------
-----------equations, comparing the coefficients of the -----------
-----------polynomials in both sides of the equation (1) -----------
'''
EQ=[Eq(LHS_matrix[i,j],expand(RHS_matrix[i,j])) for i,j in product(range(LHS_matrix.rows),range(LHS_matrix.cols)) ]
eq=[]
for equation in EQ:
RHS=poly((equation).rhs,s).all_coeffs() #simplify necessary ?
LHS=poly((equation).lhs,s).all_coeffs()
if len(RHS)>len(LHS): # we add zero for each missing coefficient (greater than the degree of LHS)
LHS=(len(RHS)-len(LHS))*[0] + LHS
eq=eq+[Eq(LHS[i],RHS[i]) for i in range(len(RHS))]
SOL=solve(eq,a+c) # the coefficients of Λ and C
#----------substitute the solution in the matrices----------------
C=C.subs(SOL) #substitute(SOL)
Lamda=Lamda.subs(SOL) #substitute(SOL)
Js=(Ds+Cs*Ys+Lamda*(B0));
# for output compatibility
E=eye(A0.cols)
if do_test==True:
def _eval_simplify(self, **kwargs):
from .add import Add
from .expr import Expr
r = self
r = r.func(*[i.simplify(**kwargs) for i in r.args])
if r.is_Relational:
if not isinstance(r.lhs, Expr) or not isinstance(r.rhs, Expr):
return r
dif = r.lhs - r.rhs
# replace dif with a valid Number that will
# allow a definitive comparison with 0
v = None
if dif.is_comparable:
v = dif.n(2)
elif dif.equals(0): # XXX this is expensive
v = S.Zero
if v is not None:
r = r.func._eval_relation(v, S.Zero)
r = r.canonical
# If there is only one symbol in the expression,
# try to write it on a simplified form
free = list(
filter(lambda x: x.is_real is not False, r.free_symbols))
if len(free) == 1:
try:
from sympy.solvers.solveset import linear_coeffs
x = free.pop()
dif = r.lhs - r.rhs
m, b = linear_coeffs(dif, x)
if m.is_zero is False:
if m.is_negative:
# Dividing with a negative number, so change order of arguments
# canonical will put the symbol back on the lhs later
r = r.func(-b / m, x)
else:
r = r.func(x, -b / m)
else:
r = r.func(b, S.Zero)
except ValueError:
# maybe not a linear function, try polynomial
from sympy.polys.polyerrors import PolynomialError
from sympy.polys.polytools import gcd, Poly, poly
try:
p = poly(dif, x)
c = p.all_coeffs()
constant = c[-1]
c[-1] = 0
scale = gcd(c)
c = [ctmp / scale for ctmp in c]
r = r.func(
Poly.from_list(c, x).as_expr(), -constant / scale)
except PolynomialError:
pass
elif len(free) >= 2:
try:
from sympy.solvers.solveset import linear_coeffs
from sympy.polys.polytools import gcd
free = list(ordered(free))
dif = r.lhs - r.rhs
m = linear_coeffs(dif, *free)
constant = m[-1]
del m[-1]
scale = gcd(m)
m = [mtmp / scale for mtmp in m]
nzm = list(filter(lambda f: f[0] != 0, list(zip(m, free))))
if scale.is_zero is False:
if constant != 0:
# lhs: expression, rhs: constant
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(newexpr, -constant / scale)
else:
# keep first term on lhs
lhsterm = nzm[0][0] * nzm[0][1]
del nzm[0]
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(lhsterm, -newexpr)
else:
r = r.func(constant, S.Zero)
except ValueError:
pass
# Did we get a simplified result?
r = r.canonical
measure = kwargs['measure']
if measure(r) < kwargs['ratio'] * measure(self):
return r
else:
return self