def expands_power_base(expression):
'''
Invoke identity (1) and (2)
The function expand the power of the
base in the expression
'''
return expand_power_base(expression)
def calc_eruption_density():
_ = calc_hydrostatic_density()
_ = _.subs(x, calc_eruption_depth())
_ = sympy.expand_power_base(_, force=True)
_ = _.simplify()
return _
def calc_eruption_mass():
_ = calc_eruption_density()
_ = _*x**3
_ = _.subs(x, calc_eruption_depth())
_ = sympy.expand_power_base(_, force=True)
_ = _.simplify()
return _
def derive_breakup_lorentz_factor():
_ = calc_asymptotic_gamma()
_ = _.subs({nu:2, eta:sympy.Rational(4,3)})
_ = _.subs(t, derive_fluid_frame_breakup_time())
_ = _.subs(p_i, rho_b0*c**2*gamma_i**2)
_ = sympy.expand_power_base(_, force=True)
_ = _.simplify()
return _
def test_issue_19627():
# if you use force the user must verify
assert powdenest(sqrt(sin(x)**2), force=True) == sin(x)
assert powdenest((x**(S.Half / y))**(2 * y), force=True) == x
from sympy import expand_power_base
e = 1 - a
expr = (exp(z / e) * x**(b / e) * y**((1 - b) / e))**e
assert powdenest(expand_power_base(expr, force=True),
force=True) == x**b * y**(1 - b) * exp(z)
def calc_asymptotic_gamma():
xi = sympy.Symbol('xi', positive=True)
Gamma = sympy.Symbol('Gamma', positive=True)
ri = prepare_ri()
_ = make_equation_of_motion()
_ = _.subs(A, (alpha*t*c)**2)
_ = _.subs(M, M_1*(p/p_i)**(1-1/eta))
_ = _.subs(sympy.solve(ri-1, p,dict=True)[0])
_ = _.subs(gamma(t).diff(t), gamma(t)/t)
_ = _.subs(gamma(t), Gamma)
_ = sympy.expand_power_base(_,force=True)
_ = _.subs(eta, xi**2+1)
_ = sympy.solve(_, Gamma)[0]
_ = sympy.expand_power_base(_, force=True).simplify()
_ = _.subs(xi,sympy.sqrt(eta-1))
return _
def derive_fluid_frame_breakup_time():
_ = derive_instability_growth()
_ = _.subs(p_i, rho_b0*c**2*gamma_i**2)
_ = _.subs({eta:sympy.Rational(4,3),
nu:2})
_ = _.simplify()
_ = sympy.solve(_ - 1, t)[0]
_ = sympy.expand_power_base(_, force=True)
_ = _.simplify()
return _
def calc_pressure_history():
xi = sympy.Symbol('xi', positive=True)
gamma_sol = calc_asymptotic_gamma()
ri = prepare_ri()
_ = sympy.solve(ri-1, p)[0]
_ = _.subs(gamma(t), gamma_sol)
_ = _.subs(eta, xi**2+1)
_ = sympy.expand_power_base(_, force=True)
_ = _.simplify()
_ = _.subs(xi, sympy.sqrt(eta-1))
return _
def extra_simple(mul):
"""Simplification of pysb rates
Arguments:
mul: psyb reaction (sympy.Mul object)
Returns:
a simplified version denested of exponents.
TODO: make sure it fully simplifies!
"""
return sp.powsimp((sp.expand_power_base(sp.powdenest(sp.logcombine(
sp.expand_log(mul.simplify(), force=True), force=True),
force=True),
force=True)),
force=True)
def define_z_eqns(self) -> None:
r"""
Form a polynomial-type ODE in $\hat{z}(\hat{x})$.
Attributes:
dzdx_Ci_polylike_eqn
"""
logging.info("gme.core.profile.define_z_eqns")
tmp_eqn = Eq(
(xiv / varphi_r(rvec)),
solve(
simplify(
self.pz_varphi_beta_eqn.subs(e2d(self.pz_xiv_eqn)).subs(
{Abs(sin(beta)): sin(beta)})),
xiv,
)[0] / varphi_r(rvec),
)
self.xiv_eqn = tmp_eqn.subs({
sin(beta): sqrt(1 - cos(beta)**2)
}).subs({cos(beta): sqrt(1 / (1 + tan(beta)**2))})
self.xvi_abs_eqn = factor(
Eq(self.xiv_eqn.lhs**4, (self.xiv_eqn.rhs)**4))
self.dzdx_polylike_eqn = self.xvi_abs_eqn.subs({tan(beta): dzdx})
self.dzdx_Ci_polylike_prelim_eqn = (expand_power_base(
self.dzdx_polylike_eqn.subs(e2d(self.varphi_rx_eqn)).subs(
e2d(self.varphi0_Lc_xiv0_Ci_eqn)).subs({
xiv: xiv_0
}).subs({
varepsilon: varepsilonhat * Lc
}).subs({
rx: xhat * Lc
}).subs({
(Lc * varepsilonhat - Lc * xhat + Lc):
(Lc * (varepsilonhat - xhat + 1))
}))).subs({Abs(dzdx): dzdx})
self.dzdx_Ci_polylike_eqn = Eq(
self.dzdx_Ci_polylike_prelim_eqn.rhs * ((dzdx**2 + 1)**(2 * eta)) -
self.dzdx_Ci_polylike_prelim_eqn.lhs * ((dzdx**2 + 1)**(2 * eta)),
0,
).subs({pzhat_0: 1 / xivhat_0})
def test_integrate_pole_part(self):
for j in (0, 1):
for i, I_j_before in enumerate([
Product(self.monomial_product1, self.regulator_poles,
self.cal_I),
Product(self.monomial_product2, self.regulator_poles,
self.cal_I)
]):
I_j_after = integrate_pole_part(I_j_before, j)
I_j_pole_part = Sum(*I_j_after[:-1])
I_j_numerically_integrable_part = I_j_after[-1]
if j == 0:
expected_pole_part = sympify_expression('''
( 1/(-2 + 0 + 1 - eps0 - 3*eps1) * (A + B*x0 + C*x0**2*x1 + D*x1) +
1/(-2 + 1 + 1 - eps0 - 3*eps1) * (B + 2*C*x0*x1) ) * (x1**2)**(-2 - eps0 - 3*eps1)
''').subs('x0', 0)
expected_numerical_integrand = (
sympify_expression(str(self.cal_I)) -
sympify_expression('A + D*x1 + x0*B')
) * sympify_expression(str(self.exponentiated_monomial1))
if i == 0:
for summand in I_j_pole_part.summands:
np.testing.assert_array_equal(
summand.factors[0].factors[0].expolist,
[(0, 2, 0, 0)])
np.testing.assert_array_equal(
summand.factors[0].factors[0].coeffs, [1])
self.assertEqual(
sympify_expression(
str(summand.factors[0].factors[0].exponent)
) - sympify_expression('-2 - eps0 - 3*eps1'),
0)
elif i == 1:
for summand in I_j_pole_part.summands:
np.testing.assert_array_equal(
summand.factors[0].factors[0].expolist,
[(0, 2, 0, 0)])
np.testing.assert_array_equal(
summand.factors[0].factors[1].expolist,
[(0, 4, 0, 0)])
np.testing.assert_array_equal(
summand.factors[0].factors[0].coeffs, [1])
np.testing.assert_array_equal(
summand.factors[0].factors[1].coeffs, [1])
self.assertEqual(
sympify_expression(
str(summand.factors[0].factors[0].exponent)
) - sympify_expression(-2), 0)
self.assertEqual(
sympify_expression(
str(summand.factors[0].factors[1].exponent)
) - sympify_expression('- eps0 - 3*eps1') / 2,
0)
else:
raise IndexError('`i` should only run over `(0,1)`!')
elif j == 1:
expected_pole_part = sympify_expression('''
( 1/( 2*(-2 - eps0 - 3*eps1) + 0 + 1 ) * (A + B*x0 + C*x0**2*x1 + D*x1) +
1/( 2*(-2 - eps0 - 3*eps1) + 1 + 1 ) * (C*x0**2 + D) +
0 ) * x0**(-2 - eps0 - 3*eps1)
''').subs('x1', 0)
expected_numerical_integrand = (
sympify_expression(str(self.cal_I)) -
sympify_expression('A + B*x0 + C*x0**2*x1 + D*x1')
) * sympify_expression(str(self.exponentiated_monomial1))
else:
raise IndexError('`j` should only run over `(0,1)`!')
self.assertEqual(
sp.powdenest((sympify_expression(str(I_j_pole_part)) -
expected_pole_part),
force=True).simplify(), 0)
for summand in I_j_pole_part.summands:
self.assertEqual(type(summand), Product)
self.assertEqual(type(summand.factors[0]), Product)
for factor in summand.factors[0].factors:
self.assertEqual(type(factor), ExponentiatedPolynomial)
self.assertEqual(type(I_j_numerically_integrable_part),
Product)
self.assertEqual(
type(I_j_numerically_integrable_part.factors[0]), Product)
for factor in I_j_numerically_integrable_part.factors[
0].factors:
self.assertEqual(type(factor), ExponentiatedPolynomial)
should_be_zero = (
sympify_expression(I_j_numerically_integrable_part) -
expected_numerical_integrand).simplify()
if i == 1:
# need some simplifications that are not generally true for all complex numbers
# see http://docs.sympy.org/dev/tutorial/simplification.html#powers for a discussion
should_be_zero = sp.expand_power_base(should_be_zero,
force=True)
should_be_zero = sp.powdenest(should_be_zero, force=True)
self.assertEqual(should_be_zero, 0)