def test_n_link_pendulum_on_cart_higher_order():
l0, m0 = symbols("l0 m0")
l1, m1 = symbols("l1 m1")
m2 = symbols("m2")
g = symbols("g")
q0, q1, q2 = dynamicsymbols("q0 q1 q2")
u0, u1, u2 = dynamicsymbols("u0 u1 u2")
F, T1 = dynamicsymbols("F T1")
kane1 = models.n_link_pendulum_on_cart(2)
massmatrix1 = Matrix([[m0 + m1 + m2, -l0*m1*cos(q1) - l0*m2*cos(q1),
-l1*m2*cos(q2)],
[-l0*m1*cos(q1) - l0*m2*cos(q1), l0**2*m1 + l0**2*m2,
l0*l1*m2*(sin(q1)*sin(q2) + cos(q1)*cos(q2))],
[-l1*m2*cos(q2),
l0*l1*m2*(sin(q1)*sin(q2) + cos(q1)*cos(q2)),
l1**2*m2]])
forcing1 = Matrix([[-l0*m1*u1**2*sin(q1) - l0*m2*u1**2*sin(q1) -
l1*m2*u2**2*sin(q2) + F],
[g*l0*m1*sin(q1) + g*l0*m2*sin(q1) -
l0*l1*m2*(sin(q1)*cos(q2) - sin(q2)*cos(q1))*u2**2],
[g*l1*m2*sin(q2) - l0*l1*m2*(-sin(q1)*cos(q2) +
sin(q2)*cos(q1))*u1**2]])
assert simplify(massmatrix1 - kane1.mass_matrix) == zeros(3)
assert simplify(forcing1 - kane1.forcing) == Matrix([0, 0, 0])
def test_commutation():
c = commutator(B(0), Bd(0))
e = simplify(apply_operators(c*Ket([n])))
assert e == Ket([n])
c = commutator(B(0), B(1))
e = simplify(apply_operators(c*Ket([n,m])))
assert e == 0
def calc(g):
nm = 1
for i, ae in enumerate(g.en):
nm = nm * simplify(g.bn[i]) ** ae
for i, ae in enumerate(g.ed):
nm = nm / simplify(g.bd[i]) ** ae
return [sstr(simplify(nm))]
def _check_orthogonality(equations):
"""
Helper method for _connect_to_cartesian. It checks if
set of transformation equations create orthogonal curvilinear
coordinate system
Parameters
==========
equations : Lambda
Lambda of transformation equations
"""
x1, x2, x3 = symbols("x1, x2, x3", cls=Dummy)
equations = equations(x1, x2, x3)
v1 = Matrix([diff(equations[0], x1),
diff(equations[1], x1), diff(equations[2], x1)])
v2 = Matrix([diff(equations[0], x2),
diff(equations[1], x2), diff(equations[2], x2)])
v3 = Matrix([diff(equations[0], x3),
diff(equations[1], x3), diff(equations[2], x3)])
if any(simplify(i[0] + i[1] + i[2]) == 0 for i in (v1, v2, v3)):
return False
else:
if simplify(v1.dot(v2)) == 0 and simplify(v2.dot(v3)) == 0 \
and simplify(v3.dot(v1)) == 0:
return True
else:
return False
def test_factorial_simplify():
# There are more tests in test_factorials.py. These are just to
# ensure that simplify() calls factorial_simplify correctly
from sympy.specfun.factorials import factorial
x = Symbol('x')
assert simplify(factorial(x)/x) == factorial(x - 1)
assert simplify(factorial(factorial(x))) == factorial(factorial(x))
def test_combsimp_gamma():
from sympy.abc import x, y
assert combsimp(gamma(x)) == gamma(x)
assert combsimp(gamma(x+1)/x) == gamma(x)
assert combsimp(gamma(x)/(x-1)) == gamma(x-1)
assert combsimp(x*gamma(x)) == gamma(x + 1)
assert combsimp((x+1)*gamma(x+1)) == gamma(x + 2)
assert combsimp(gamma(x+y)*(x+y)) == gamma(x + y + 1)
assert combsimp(x/gamma(x+1)) == 1/gamma(x)
assert combsimp((x+1)**2/gamma(x+2)) == (x + 1)/gamma(x + 1)
assert combsimp(x*gamma(x) + gamma(x+3)/(x+2)) == gamma(x+1) + gamma(x+2)
assert combsimp(gamma(2*x)*x) == gamma(2*x + 1)/2
assert combsimp(gamma(2*x)/(x - S(1)/2)) == 2*gamma(2*x - 1)
assert combsimp(gamma(x)*gamma(1-x)) == pi/sin(pi*x)
assert combsimp(gamma(x)*gamma(-x)) == -pi/(x*sin(pi*x))
assert combsimp(1/gamma(x+3)/gamma(1-x)) == sin(pi*x)/(pi*x*(x+1)*(x+2))
assert simplify(combsimp(gamma(x)*gamma(x+S(1)/2)*gamma(y)/gamma(x+y))) \
== 2**(-2*x + 1)*sqrt(pi)*gamma(2*x)*gamma(y)/gamma(x + y)
assert combsimp(1/gamma(x)/gamma(x-S(1)/3)/gamma(x+S(1)/3)) == \
3**(3*x + S(1)/2)/(18*pi*gamma(3*x - 1))
assert simplify(gamma(S(1)/2 + x/2)*gamma(1 + x/2)/gamma(1+x)/sqrt(pi)*2**x) \
== 1
assert combsimp(gamma(S(-1)/4)*gamma(S(-3)/4)) == 16*sqrt(2)*pi/3
assert simplify(combsimp(gamma(2*x)/gamma(x))) == \
2**(2*x - 1)*gamma(x + S(1)/2)/sqrt(pi)
def test_atan2():
assert atan2(0, 0) == S.NaN
assert atan2(0, 1) == 0
assert atan2(1, 1) == pi/4
assert atan2(1, 0) == pi/2
assert atan2(1, -1) == 3*pi/4
assert atan2(0, -1) == pi
assert atan2(-1, -1) == -3*pi/4
assert atan2(-1, 0) == -pi/2
assert atan2(-1, 1) == -pi/4
u = Symbol("u", positive=True)
assert atan2(0, u) == 0
u = Symbol("u", negative=True)
assert atan2(0, u) == pi
assert atan2(y, oo) == 0
assert atan2(y, -oo)== 2*pi*Heaviside(re(y)) - pi
assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
assert atan2(y, x).rewrite(atan) == 2*atan(y/(x + sqrt(x**2 + y**2)))
assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
assert diff(atan2(y, x), y) == x/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2)
assert isinstance(atan2(2, 3*I).n(), atan2)
def calculate(str_expr, return_float=False):
"""
Evaluate a string STR_EXPR.
if STR_EXPR is a mathematical expression, simplify it with sympy.simplify
Otherwise, search WolframAlpha for answers
"""
try:
result = simplify(str_expr)
return result
except Exception:
# use Wolfram Alpha
payload = {'input': str_expr, 'appid': 'UAGAWR-3X6Y8W777Q'}
r = requests.get('http://api.wolframalpha.com/v2/query', params=payload)
soup = BeautifulSoup(r.content,"lxml")
if soup.queryresult['success'] == 'true':
pods = soup.queryresult.findAll('pod')
assert len(pods) >= 2
num_string = pods[1].plaintext.string.split()[0]
if return_float:
return float(simplify(num_string.replace(u"\u00D7", '*')))
else:
return num_string
else:
return "I don't know the answer"
def test_symbolic_twoport():
circuit.default_toolkit = symbolic
cir = SubCircuit()
k = symbolic.kboltzmann
var('R1 R0 C1 w T', real=True, positive=True)
s = 1j*w
cir['R0'] = R(1, gnd, r=R0)
cir['R1'] = R(1, 2, r=R1)
# cir['C1'] = C(2, gnd, c=C1)
## Add an AC source to verify that the source will not affect results
# cir['IS'] = IS(1, gnd, iac=1)
## Run symbolic 2-port analysis
twoport_ana = TwoPortAnalysis(cir, Node('1'), gnd, Node('2'), gnd,
noise = True, toolkit=symbolic,
noise_outquantity = 'v')
result = twoport_ana.solve(freqs=s, complexfreq=True)
ABCD = Matrix(result['twoport'].A)
ABCD.simplify()
assert_array_equal(ABCD, np.array([[1 + 0*R1*C1*s, R1],
[(1 + 0*R0*C1*s + 0*R1*C1*s) / R0, (R0 + R1)/R0]]))
assert_array_equal(simplify(result['Sin'] - (4*k*T/R0 + 4*R1*k*T/R0**2)), 0)
assert_array_equal(simplify(result['Svn']), 4*k*T*R1)
def get_solid_harmonics(shell, xyz):
x, y, z = xyz
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
r = Symbol("r", real=True)
result = []
for m in range(shell+1):
if m > 0:
part_plus = (Ylm(shell,m,theta,phi)*r**shell).expand()
part_plus = part_plus.subs(exp(I*phi),(x+I*y)/sin(theta)/r)
part_min = (Ylm(shell,-m,theta,phi)*r**shell).expand()
part_min = part_min.subs(exp(-I*phi),(x-I*y)/sin(theta)/r)
sym = simplify(((-1)**m*part_plus+part_min)/sqrt(2))
sym = sym.subs(r*cos(theta),z)
sym = sym.subs(r**2,x**2+y**2+z**2)
sym = sym.subs(cos(theta)**2,1-sin(theta)**2)
sym = simplify(sym)
asym = simplify(((-1)**m*part_plus-part_min)/I/sqrt(2))
asym = asym.subs(r*cos(theta),z)
asym = asym.subs(r**2,x**2+y**2+z**2)
asym = asym.subs(cos(theta)**2,1-sin(theta)**2)
asym = simplify(asym)
result.append(sym)
result.append(asym)
else:
part = (Ylm(shell,0,theta,phi)*r**shell).expand()
part = part.subs(r*cos(theta),z)
part = part.subs(r**2,x**2+y**2+z**2)
part = simplify(part)
result.append(part)
return result
def check_sign_algebraically(expr, param_names, species_id, initial_values):
import sympy
from sympy import Symbol
# create variables for each param and species
for parameter_name in param_names:
exec("%s = Symbol('%s', positive=True)" % (parameter_name, parameter_name))
# create variables for time and avogadro's constant: N_A
exec("t = Symbol('t', positive=True)")
exec("N_A = Symbol('N_A', positive=True)")
exec("query_var = %s" % species_id)
converted_rate_law = convert_rate_law(expr, output_type="sympy")
if converted_rate_law == "piecewise":
return "?"
exec("rate = %s" % converted_rate_law)
if sympy.simplify(rate.diff(query_var)).is_positive:
return "monotonic_increasing"
elif sympy.simplify(rate.diff(query_var)).is_negative:
return "monotonic_decreasing"
else:
return "?"
def test_gosper_sum_AeqB_part2():
f2a = n**2*a**n
f2b = (n - r/2)*binomial(r, n)
f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
g2a = -a*(a + 1)/(a - 1)**3 + a**(
m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
g2b = (m - r)*binomial(r, m)/2
ff = factorial(1 - x)*factorial(1 + x)
g2c = 1/ff*(
1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None and simplify(g - g2c) == 0
# delete these lines and unXFAIL the nan test below when it passes
f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
g2d = 1/(factorial(a - 1)*factorial(
b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
assert simplify(
sum(f2d.subs(n, i) for i in range(3)) - g2d.subs(m, 2)) == 0
def test_gosper_sum_AeqB_part3():
f3a = 1/n**4
f3b = (6*n + 3)/(4*n**4 + 8*n**3 + 8*n**2 + 4*n + 3)
f3c = 2**n*(n**2 - 2*n - 1)/(n**2*(n + 1)**2)
f3d = n**2*4**n/((n + 1)*(n + 2))
f3e = 2**n/(n + 1)
f3f = 4*(n - 1)*(n**2 - 2*n - 1)/(n**2*(n + 1)**2*(n - 2)**2*(n - 3)**2)
f3g = (n**4 - 14*n**2 - 24*n - 9)*2**n/(n**2*(n + 1)**2*(n + 2)**2*
(n + 3)**2)
# g3a -> no closed form
g3b = m*(m + 2)/(2*m**2 + 4*m + 3)
g3c = 2**m/m**2 - 2
g3d = S(2)/3 + 4**(m + 1)*(m - 1)/(m + 2)/3
# g3e -> no closed form
g3f = -(-S(1)/16 + 1/((m - 2)**2*(m + 1)**2)) # the AeqB key is wrong
g3g = -S(2)/9 + 2**(m + 1)/((m + 1)**2*(m + 3)**2)
g = gosper_sum(f3a, (n, 1, m))
assert g is None
g = gosper_sum(f3b, (n, 1, m))
assert g is not None and simplify(g - g3b) == 0
g = gosper_sum(f3c, (n, 1, m - 1))
assert g is not None and simplify(g - g3c) == 0
g = gosper_sum(f3d, (n, 1, m))
assert g is not None and simplify(g - g3d) == 0
g = gosper_sum(f3e, (n, 0, m - 1))
assert g is None
g = gosper_sum(f3f, (n, 4, m))
assert g is not None and simplify(g - g3f) == 0
g = gosper_sum(f3g, (n, 1, m))
assert g is not None and simplify(g - g3g) == 0
def test_equation():
p1 = Point(0, 0)
p2 = Point(1, 1)
l1 = Line(p1, p2)
l3 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert Line(p1, Point(1, 0)).equation(x=x, y=y) == y
assert Line(p1, Point(0, 1)).equation() == x
assert Line(Point(2, 0), Point(2, 1)).equation() == x - 2
assert Line(p2, Point(2, 1)).equation() == y - 1
assert Line3D(Point(x1, x1, x1), Point(y1, y1, y1)
).equation() == (-x + y, -x + z)
assert Line3D(Point(1, 2, 3), Point(2, 3, 4)
).equation() == (-x + y - 1, -x + z - 2)
assert Line3D(Point(1, 2, 3), Point(1, 3, 4)
).equation() == (x - 1, -y + z - 1)
assert Line3D(Point(1, 2, 3), Point(2, 2, 4)
).equation() == (y - 2, -x + z - 2)
assert Line3D(Point(1, 2, 3), Point(2, 3, 3)
).equation() == (-x + y - 1, z - 3)
assert Line3D(Point(1, 2, 3), Point(1, 2, 4)
).equation() == (x - 1, y - 2)
assert Line3D(Point(1, 2, 3), Point(1, 3, 3)
).equation() == (x - 1, z - 3)
assert Line3D(Point(1, 2, 3), Point(2, 2, 3)
).equation() == (y - 2, z - 3)
def test_weibull():
a, b = symbols('a b', positive=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2/b) - E(X)**2)
assert simplify(skewness(X)) == (2*gamma(1 + 1/b)**3 - 3*gamma(1 + 1/b)*gamma(1 + 2/b) + gamma(1 + 3/b))/(-gamma(1 + 1/b)**2 + gamma(1 + 2/b))**(S(3)/2)
def test_commutation():
n, m = symbols("n,m", above_fermi=True)
c = Commutator(B(0), Bd(0))
assert c == 1
c = Commutator(Bd(0), B(0))
assert c == -1
c = Commutator(B(n), Bd(0))
assert c == KroneckerDelta(n, 0)
c = Commutator(B(0), Bd(0))
e = simplify(apply_operators(c*BKet([n])))
assert e == BKet([n])
c = Commutator(B(0), B(1))
e = simplify(apply_operators(c*BKet([n, m])))
assert e == 0
c = Commutator(F(m), Fd(m))
assert c == +1 - 2*NO(Fd(m)*F(m))
c = Commutator(Fd(m), F(m))
assert c == -1 + 2*NO(Fd(m)*F(m))
C = Commutator
X, Y, Z = symbols('X,Y,Z', commutative=False)
assert C(C(X, Y), Z) != 0
assert C(C(X, Z), Y) != 0
assert C(Y, C(X, Z)) != 0
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
D = KroneckerDelta
assert C(Fd(a), F(i)) == -2*NO(F(i)*Fd(a))
assert C(Fd(j), NO(Fd(a)*F(i))).doit(wicks=True) == -D(j, i)*Fd(a)
assert C(Fd(a)*F(i), Fd(b)*F(j)).doit(wicks=True) == 0
def interpolation(f, psi, points):
"""
Given a function f(x), return the approximation to
f(x) in the space V, spanned by psi, that interpolates
f at the given points. Must have len(points) = len(psi)
"""
N = len(psi) - 1
A = sym.zeros((N+1, N+1))
b = sym.zeros((N+1, 1))
# Wrap psi and f in Python functions rather than expressions
# so that we can evaluate psi at points[i] (alternative to subs?)
x = sym.Symbol('x')
psi = [sym.lambdify([x], psi[i]) for i in range(N+1)]
f = sym.lambdify([x], f)
print '...evaluating matrix...'
for i in range(N+1):
for j in range(N+1):
print '(%d,%d)' % (i, j)
A[i,j] = psi[j](points[i])
b[i,0] = f(points[i])
print
print 'A:\n', A, '\nb:\n', b
c = A.LUsolve(b)
# c is a sympy Matrix object, turn to list
c = [sym.simplify(c[i,0]) for i in range(c.shape[0])]
print 'coeff:', c
u = 0
for i in range(len(psi)):
u += c[i]*psi[i](x)
# Alternative:
# u = sum(c[i,0]*psi[i] for i in range(len(psi)))
print 'approximation:', sym.simplify(u)
return u, c
def test_simplify_other():
assert simplify(sin(x)**2 + cos(x)**2) == 1
assert simplify(gamma(x + 1)/gamma(x)) == x
assert simplify(sin(x)**2 + cos(x)**2 + factorial(x)/gamma(x)) == 1 + x
assert simplify(Eq(sin(x)**2 + cos(x)**2, factorial(x)/gamma(x))) == Eq(1, x)
nc = symbols('nc', commutative=False)
assert simplify(x + x*nc) == x*(1 + nc)
def check_param(x, y, a, t):
"""
Check if there is a number modulo a such that x and y are both
integers. If exist, then find a parametric representation for x and y.
"""
k, m, n = symbols("k, m, n", Integer=True)
p = Wild("p", exclude=[k])
q = Wild("q", exclude=[k])
ok = False
for i in range(a):
z_x = simplify(Subs(x, t, a*k + i).doit()).match(p*k + q)
z_y = simplify(Subs(y, t, a*k + i).doit()).match(p*k + q)
if (isinstance(z_x[p], Integer) and isinstance(z_x[q], Integer) and
isinstance(z_y[p], Integer) and isinstance(z_y[q], Integer)):
ok = True
break
if ok == True:
x_param = x.match(p*t + q)
y_param = y.match(p*t + q)
eq = S(m -x_param[q])/x_param[p] - S(n - y_param[q])/y_param[p]
lcm_denom, junk = Poly(eq).clear_denoms()
eq = eq * lcm_denom
return diop_solve(eq, t)[m], diop_solve(eq, t)[n]
else:
return (None, None)
def _check_orthogonality(self):
"""
Helper method for _connect_to_cartesian. It checks if
set of transformation equations create orthogonal curvilinear
coordinate system
Parameters
==========
equations : tuple
Tuple of transformation equations
"""
eq = self._transformation_equations()
v1 = Matrix([diff(eq[0], self.x), diff(eq[1], self.x), diff(eq[2], self.x)])
v2 = Matrix([diff(eq[0], self.y), diff(eq[1], self.y), diff(eq[2], self.y)])
v3 = Matrix([diff(eq[0], self.z), diff(eq[1], self.z), diff(eq[2], self.z)])
if any(simplify(i[0] + i[1] + i[2]) == 0 for i in (v1, v2, v3)):
return False
else:
if simplify(v1.dot(v2)) == 0 and simplify(v2.dot(v3)) == 0 and simplify(v3.dot(v1)) == 0:
return True
else:
return False
def test_M23():
x = symbols('x', complex=True)
assert solve(x - 1/sqrt(1 + x**2)) == [
simplify(-I*sqrt((sqrt(5) + 1)/2)),
simplify( sqrt((sqrt(5) - 1)/2)),
]
def __pow__(self, e, z=None):
"""
Perform multinomial expansion and return terms of order less than
`self.order`.
"""
data = {}
exps,coeffs = zip(*(self.d.items()))
m = len(exps)
print "EXPONENT:",e
multinoms = sympy.multinomial.multinomial_coefficients_iterator(m,int(e))
for k,mcoeff in multinoms:
exp = sum( expi*ki for expi,ki in zip(exps,k) )
if exp < self.order:
try:
coeff = sympy.simplify(
mcoeff * sympy.prod(ci**ki for ci,ki in zip(coeffs,k))
)
coeff = sympy.simplify(coeff + data[exp])
except KeyError:
data[exp] = coeff
return PuiseuxSeries(data, self.alpha, self.order, self.var)
def reciprocal_frame_test():
Print_Function()
metric = "1 # #," + "# 1 #," + "# # 1,"
(e1, e2, e3) = MV.setup("e1 e2 e3", metric)
print("g_{ij} =\n", MV.metric)
E = e1 ^ e2 ^ e3
Esq = (E * E).scalar()
print("E =", E)
print("E**2 =", Esq)
Esq_inv = 1 / Esq
E1 = (e2 ^ e3) * E
E2 = (-1) * (e1 ^ e3) * E
E3 = (e1 ^ e2) * E
print("E1 = (e2^e3)*E =", E1)
print("E2 =-(e1^e3)*E =", E2)
print("E3 = (e1^e2)*E =", E3)
w = E1 | e2
w = w.expand()
print("E1|e2 =", w)
w = E1 | e3
w = w.expand()
print("E1|e3 =", w)
w = E2 | e1
w = w.expand()
print("E2|e1 =", w)
w = E2 | e3
w = w.expand()
print("E2|e3 =", w)
w = E3 | e1
w = w.expand()
print("E3|e1 =", w)
w = E3 | e2
w = w.expand()
print("E3|e2 =", w)
w = E1 | e1
w = (w.expand()).scalar()
Esq = expand(Esq)
print("(E1|e1)/E**2 =", simplify(w / Esq))
w = E2 | e2
w = (w.expand()).scalar()
print("(E2|e2)/E**2 =", simplify(w / Esq))
w = E3 | e3
w = (w.expand()).scalar()
print("(E3|e3)/E**2 =", simplify(w / Esq))
return
def test_medium():
m1 = Medium('m1')
assert m1.intrinsic_impedance == sqrt(u0/e0)
assert m1.speed == 1/sqrt(e0*u0)
assert m1.refractive_index == c*sqrt(e0*u0)
assert m1.permittivity == e0
assert m1.permeability == u0
m2 = Medium('m2', epsilon, mu)
assert m2.intrinsic_impedance == sqrt(mu/epsilon)
assert m2.speed == 1/sqrt(epsilon*mu)
assert m2.refractive_index == c*sqrt(epsilon*mu)
assert m2.permittivity == epsilon
assert m2.permeability == mu
# Increasing electric permittivity and magnetic permeability
# by small amount from its value in vacuum.
m3 = Medium('m3', 9.0*10**(-12)*s**4*A**2/(m**3*kg), 1.45*10**(-6)*kg*m/(A**2*s**2))
assert m3.refractive_index > m1.refractive_index
assert m3 > m1
# Decreasing electric permittivity and magnetic permeability
# by small amount from its value in vacuum.
m4 = Medium('m4', 7.0*10**(-12)*s**4*A**2/(m**3*kg), 1.15*10**(-6)*kg*m/(A**2*s**2))
assert m4.refractive_index < m1.refractive_index
assert m4 < m1
m5 = Medium('m5', permittivity=710*10**(-12)*s**4*A**2/(m**3*kg), n=1.33)
assert simplify(m5.intrinsic_impedance - 6.24845417765552*kg*m**2/(A**2*s**3)) == 0
assert abs(m5.speed - 225407863.157895*m/s) < 1e-6*m/s
assert simplify(m5.refractive_index - 1.33000000000000) == 0
assert simplify(m5.permittivity - 7.1e-10*A**2*s**4/(kg*m**3)) == 0
assert simplify(m5.permeability - 2.77206575232851e-8*kg*m/(A**2*s**2)) == 0
def get_S(self, theta=0., strain_type="Engineering"):
"""
Calculates the compliance matrix for the ply oriented at angle theta
in the specified strain units
Parameters
----------
theta : 'Int' or 'Float'
Angle in degrees of ply orientation.
strain_type : 'String'
Specifies 'Engineering' or 'Tensorial' strain.
"""
if strain_type.lower().startswith('e'):
strain_multiplier = 1
else:
strain_multiplier = 2
compliance = sp.Matrix([[1 / self.E11, -self.v21 / self.E22, 0],
[-self.v12 / self.E11, 1 / self.E22, 0],
[0, 0, 1 / (strain_multiplier * self.G12)]])
if theta == 0.:
return compliance
else:
T = T_Matrix(theta)
if strain_type.lower().startswith('e'):
R = R_Matrix()
TI = sp.simplify(R * T.inv() * R.inv())
else:
TI = T.inv()
return sp.simplify(sp.N(TI * compliance * T, chop=1e-10))
def test_nullor_vva():
"""Test nullor element by building a V-V amplifier"""
pycircuit.circuit.circuit.default_toolkit = symbolic
c = SubCircuit()
Vin = Symbol('Vin')
R1 =Symbol('R1')
R2 = Symbol('R2')
nin = c.add_node('in')
n1 = c.add_node('n1')
nout = c.add_node('out')
c['vin'] = VS(nin, gnd, vac=Vin)
c['R1'] = R(n1, gnd, r=R1)
c['R2'] = R(nout, n1, r=R2)
c['nullor'] = Nullor(n1, nin, gnd, nout)
result = AC(c, toolkit=symbolic).solve(Symbol('s'))
vout = result.v(nout)
assert simplify(vout - Vin * (R1 + R2) / R1) == 0, \
'Did not get the expected result, %s != 0'% \
str(simplify(vout - Vin * (R1 + R2) / R1))
def test_SVCVS_laplace_d3_n1():
"""Test VCCS with a laplace defined transfer function with second order
numerator and third order denominator
"""
pycircuit.circuit.circuit.default_toolkit = symbolic
cir = SubCircuit()
n1,n2 = cir.add_nodes('1','2')
b0,a0,a1,a2,a3,Gdc = [sympy.Symbol(symname, real=True) for
symname in 'b0,a0,a1,a2,a3,Gdc'
.split(',')]
s = sympy.Symbol('s', complex=True)
cir['VS'] = VS( n1, gnd, vac=1)
cir['VCVS'] = SVCVS( n1, gnd, n2, gnd,
denominator = [a0, a1, a2, a3],
numerator = [b0, 0, 0])
res = AC(cir, toolkit=symbolic).solve(s, complexfreq=True)
assert_equal(sympy.simplify(res.v(n2,gnd)),
sympy.simplify((b0*s*s)/(a0*s*s*s+a1*s*s+a2*s+a3)))
def hamiltonian(Lagrangian, t = Symbol('t'), delta = False):
"""
Returns the Hamiltonian of the Lagrangian.
Examples
========
>>> from sympy import *
>>> t, k = symbols('t k')
>>> x = symbols('x', cls=Function)
>>> hamiltonian(diff(x(t),t)**2/2 - k*x(t)**2/2)
k*x**2/2 + v_x**2/2
"""
Lagrangian = simplify(Lagrangian)
var_list = [list(x.atoms(Function))[0] for x in Lagrangian.atoms(Derivative)]
nvar = len(var_list)
# New variables.
str_list = [ str(variable).replace("("+str(t)+")","") for variable in var_list ]
v_subs = {diff(var_list[i],t): Symbol('v_' + str_list[i]) for i in range(nvar)}
x_subs = {var_list[i]: Symbol(str_list[i]) for i in range(nvar)}
# Hamiltonian calculus.
dxdLv = 0
for variable in var_list:
dxdLv += diff(variable,t)*diff(Lagrangian, diff(variable,t))
result = simplify((dxdLv - Lagrangian).subs(v_subs).subs(x_subs))
if delta:
v0_subs = {Symbol('v_' + str_list[i]): Symbol('v_' + str_list[i] + "0") for i in range(nvar)}
x0_subs = {Symbol(str_list[i]): Symbol(str_list[i] + "0") for i in range(nvar)}
return result - result.subs(v0_subs).subs(x0_subs)
else:
return result
def calculate_eigenvalues(self):
r"""Calculate the two eigenvalues :math:`\lambda_i(x)` of the potential :math:`V(x)`.
We can do this by symbolic calculations. The multiplicities are taken into account.
Note: This function is idempotent and the eigenvalues are memoized for later reuse.
"""
if self._eigenvalues_s is not None:
return
# Symbolic formula for the eigenvalues of a general 2x2 matrix
T = self._potential_s.trace()
D = self._potential_s.det()
l1 = (T + sympy.sqrt(T**2 - 4*D)) * sympy.Rational(1,2)
l2 = (T - sympy.sqrt(T**2 - 4*D)) * sympy.Rational(1,2)
# Symbolic simplification may fail
if self._try_simplify:
try:
l1 = sympy.simplify(l1)
l2 = sympy.simplify(l2)
except:
pass
# The symbolic expressions for the eigenvalues
self._eigenvalues_s = (l1, l2)
# The numerical functions for the eigenvalues
self._eigenvalues_n = tuple([ sympy.lambdify(self._variables, item, "numpy") for item in self._eigenvalues_s ])
def test_form_2():
symsystem2 = SymbolicSystem(coordinates, comb_implicit_rhs, speeds=speeds,
mass_matrix=comb_implicit_mat,
alg_con=alg_con_full, output_eqns=out_eqns,
bodies=bodies, loads=loads)
assert symsystem2.coordinates == Matrix([x, y, lam])
assert symsystem2.speeds == Matrix([u, v])
assert symsystem2.states == Matrix([x, y, lam, u, v])
assert symsystem2.alg_con == [4]
inter = comb_implicit_rhs
assert simplify(symsystem2.comb_implicit_rhs - inter) == zeros(5, 1)
assert simplify(symsystem2.comb_implicit_mat-comb_implicit_mat) == zeros(5)
assert set(symsystem2.dynamic_symbols()) == set([y, v, lam, u, x])
assert type(symsystem2.dynamic_symbols()) == tuple
assert set(symsystem2.constant_symbols()) == set([l, g, m])
assert type(symsystem2.constant_symbols()) == tuple
inter = comb_explicit_rhs
symsystem2.compute_explicit_form()
assert simplify(symsystem2.comb_explicit_rhs - inter) == zeros(5, 1)
assert symsystem2.output_eqns == out_eqns
assert symsystem2.bodies == (Pa,)
assert symsystem2.loads == ((P, g * m * N.x),)
def test_simplify_rational():
expr = 2**x * 2.**y
assert simplify(expr, rational=True) == 2**(x + y)
assert simplify(expr, rational=None) == 2.0**(x + y)
assert simplify(expr, rational=False) == expr
def test_simplify_float_vs_integer():
# Test for issue 4473:
# https://github.com/sympy/sympy/issues/4473
assert simplify(x**2.0 - x**2) == 0
assert simplify(x**2 - x**2.0) == 0
def test_issue_4194():
# simplify should call cancel
from sympy.abc import x, y
f = Function('f')
assert simplify((4 * x + 6 * f(y)) / (2 * x + 3 * f(y))) == 2
def test_simplify_expr():
x, y, z, k, n, m, w, f, s, A = symbols('x,y,z,k,n,m,w,f,s,A')
assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I])
e = 1 / x + 1 / y
assert e != (x + y) / (x * y)
assert simplify(e) == (x + y) / (x * y)
e = A**2 * s**4 / (4 * pi * k * m**3)
assert simplify(e) == e
e = (4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)
assert simplify(e) == 0
e = (-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2
assert simplify(e) == -2 * y
e = -x - y - (x + y)**(-1) * y**2 + (x + y)**(-1) * x**2
assert simplify(e) == -2 * y
e = (x + x * y) / x
assert simplify(e) == 1 + y
e = (f(x) + y * f(x)) / f(x)
assert simplify(e) == 1 + y
e = (2 * (1 / n - cos(n * pi) / n)) / pi
assert simplify(e) == (-cos(pi * n) + 1) / (pi * n) * 2
e = integrate(1 / (x**3 + 1), x).diff(x)
assert simplify(e) == 1 / (x**3 + 1)
e = integrate(x / (x**2 + 3 * x + 1), x).diff(x)
assert simplify(e) == x / (x**2 + 3 * x + 1)
A = Matrix([[2 * k - m * w**2, -k], [-k, k - m * w**2]]).inv()
assert simplify((A*Matrix([0, f]))[1]) == \
-f*(2*k - m*w**2)/(k**2 - (k - m*w**2)*(2*k - m*w**2))
f = -x + y / (z + t) + z * x / (z + t) + z * a / (z + t) + t * x / (z + t)
assert simplify(f) == (y + a * z) / (z + t)
# issue 10347
expr = -x * (y**2 - 1) * (
2 * y**2 * (x**2 - 1) / (a * (x**2 - y**2)**2) + (x**2 - 1) /
(a * (x**2 - y**2))) / (a * (x**2 - y**2)) + x * (
-2 * x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2)**2) -
x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - 1) * (x**2 - y**2)) +
(x**2 * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) / (x**2 - 1) + sqrt(
(-x**2 + 1) * (y**2 - 1)) *
(x * (-x * y**2 + x) / sqrt(-x**2 * y**2 + x**2 + y**2 - 1) +
sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) * sin(z)) / (a * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) / (
a * (x**2 - y**2)) + x * (
-2 * x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a * (x**2 - y**2)**2) -
x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a * (x**2 - 1) * (x**2 - y**2)) +
(x**2 * sqrt((-x**2 + 1) *
(y**2 - 1)) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1)
* cos(z) / (x**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x * y**2 + x) * cos(z) /
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) + sqrt(
(-x**2 + 1) *
(y**2 - 1)) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) *
cos(z)) / (a * sqrt((-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) / (
a *
(x**2 - y**2)) - y * sqrt((-x**2 + 1) * (y**2 - 1)) * (
-x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2) *
(y**2 - 1)) +
2 * x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2)**2) +
(x * y * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(y**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x**2 * y + y) * sin(z) /
sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) / (a * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sin(z) / (a * (x**2 - y**2)) + y * (x**2 - 1) * (
-2 * x * y *
(x**2 - 1) /
(a * (x**2 - y**2)**2) + 2 * x * y / (a * (x**2 - y**2))
) / (a *
(x**2 - y**2)) + y * (x**2 - 1) * (y**2 - 1) * (
-x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) *
cos(z) / (a * (x**2 - y**2) *
(y**2 - 1)) + 2 * x * y *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a *
(x**2 - y**2)**2) +
(x * y * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(y**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x**2 * y + y) *
cos(z) / sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) /
(a * sqrt((-x**2 + 1) * (y**2 - 1)) *
(x**2 - y**2))) * cos(z) / (a * sqrt(
(-x**2 + 1) *
(y**2 - 1)) * (x**2 - y**2)) - x * sqrt(
(-x**2 + 1) * (y**2 - 1)
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(
z)**2 / (a**2 * (x**2 - 1) * (x**2 - y**2) *
(y**2 - 1)) - x * sqrt(
(-x**2 + 1) *
(y**2 - 1)) * sqrt(
-x**2 * y**2 + x**2 + y**2 -
1) * cos(z)**2 / (
a**2 * (x**2 - 1) *
(x**2 - y**2) *
(y**2 - 1))
assert simplify(expr) == 2 * x / (a**2 * (x**2 - y**2))
A, B = symbols('A,B', commutative=False)
assert simplify(A * B - B * A) == A * B - B * A
assert simplify(A / (1 + y / x)) == x * A / (x + y)
assert simplify(A * (1 / x + 1 / y)) == A / x + A / y #(x + y)*A/(x*y)
assert simplify(log(2) + log(3)) == log(6)
assert simplify(log(2 * x) - log(2)) == log(x)
assert simplify(hyper([], [], x)) == exp(x)
def test_simplify_fail1():
x = Symbol('x')
y = Symbol('y')
e = (x + y)**2 / (-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y)
assert simplify(e) == 1 / (-2 * y)
def test_issue_5652():
assert simplify(E + exp(-E)) == exp(-E) + E
n = symbols('n', commutative=False)
assert simplify(n + n**(-n)) == n + n**(-n)
def test_simplify_issue_1308():
assert simplify(exp(-Rational(1, 2)) + exp(-Rational(3, 2))) == \
(1 + E)*exp(-Rational(3, 2))
def test_coordinate_vars():
"""
Tests the coordinate variables functionality with respect to
reorientation of coordinate systems.
"""
A = CoordSys3D('A')
# Note that the name given on the lhs is different from A.x._name
assert BaseScalar('A.x', 0, A, 'A_x', r'\mathbf{{x}_{A}}') == A.x
assert BaseScalar('A.y', 1, A, 'A_y', r'\mathbf{{y}_{A}}') == A.y
assert BaseScalar('A.z', 2, A, 'A_z', r'\mathbf{{z}_{A}}') == A.z
assert BaseScalar('A.x', 0, A, 'A_x', r'\mathbf{{x}_{A}}').__hash__() == A.x.__hash__()
assert isinstance(A.x, BaseScalar) and \
isinstance(A.y, BaseScalar) and \
isinstance(A.z, BaseScalar)
assert A.x*A.y == A.y*A.x
assert A.scalar_map(A) == {A.x: A.x, A.y: A.y, A.z: A.z}
assert A.x.system == A
assert A.x.diff(A.x) == 1
B = A.orient_new_axis('B', q, A.k)
assert B.scalar_map(A) == {B.z: A.z, B.y: -A.x*sin(q) + A.y*cos(q),
B.x: A.x*cos(q) + A.y*sin(q)}
assert A.scalar_map(B) == {A.x: B.x*cos(q) - B.y*sin(q),
A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z}
assert express(B.x, A, variables=True) == A.x*cos(q) + A.y*sin(q)
assert express(B.y, A, variables=True) == -A.x*sin(q) + A.y*cos(q)
assert express(B.z, A, variables=True) == A.z
assert expand(express(B.x*B.y*B.z, A, variables=True)) == \
expand(A.z*(-A.x*sin(q) + A.y*cos(q))*(A.x*cos(q) + A.y*sin(q)))
assert express(B.x*B.i + B.y*B.j + B.z*B.k, A) == \
(B.x*cos(q) - B.y*sin(q))*A.i + (B.x*sin(q) + \
B.y*cos(q))*A.j + B.z*A.k
assert simplify(express(B.x*B.i + B.y*B.j + B.z*B.k, A, \
variables=True)) == \
A.x*A.i + A.y*A.j + A.z*A.k
assert express(A.x*A.i + A.y*A.j + A.z*A.k, B) == \
(A.x*cos(q) + A.y*sin(q))*B.i + \
(-A.x*sin(q) + A.y*cos(q))*B.j + A.z*B.k
assert simplify(express(A.x*A.i + A.y*A.j + A.z*A.k, B, \
variables=True)) == \
B.x*B.i + B.y*B.j + B.z*B.k
N = B.orient_new_axis('N', -q, B.k)
assert N.scalar_map(A) == \
{N.x: A.x, N.z: A.z, N.y: A.y}
C = A.orient_new_axis('C', q, A.i + A.j + A.k)
mapping = A.scalar_map(C)
assert mapping[A.x].equals(C.x*(2*cos(q) + 1)/3 +
C.y*(-2*sin(q + pi/6) + 1)/3 +
C.z*(-2*cos(q + pi/3) + 1)/3)
assert mapping[A.y].equals(C.x*(-2*cos(q + pi/3) + 1)/3 +
C.y*(2*cos(q) + 1)/3 +
C.z*(-2*sin(q + pi/6) + 1)/3)
assert mapping[A.z].equals(C.x*(-2*sin(q + pi/6) + 1)/3 +
C.y*(-2*cos(q + pi/3) + 1)/3 +
C.z*(2*cos(q) + 1)/3)
D = A.locate_new('D', a*A.i + b*A.j + c*A.k)
assert D.scalar_map(A) == {D.z: A.z - c, D.x: A.x - a, D.y: A.y - b}
E = A.orient_new_axis('E', a, A.k, a*A.i + b*A.j + c*A.k)
assert A.scalar_map(E) == {A.z: E.z + c,
A.x: E.x*cos(a) - E.y*sin(a) + a,
A.y: E.x*sin(a) + E.y*cos(a) + b}
assert E.scalar_map(A) == {E.x: (A.x - a)*cos(a) + (A.y - b)*sin(a),
E.y: (-A.x + a)*sin(a) + (A.y - b)*cos(a),
E.z: A.z - c}
F = A.locate_new('F', Vector.zero)
assert A.scalar_map(F) == {A.z: F.z, A.x: F.x, A.y: F.y}
def test_issue1893():
from sympy import simplify, expand_func, polygamma, gamma
a = Symbol('a', positive=True)
assert simplify(expand_func(integrate(exp(-x)*log(x)*x**a, (x, 0, oo)))) == \
(a*polygamma(0, a) + 1)*gamma(a)
def test_issue_7263():
assert abs((simplify(30.8**2 - 82.5**2 * sin(rad(11.6))**2)).evalf() - \
673.447451402970) < 1e-12
def test_issue_1277():
n = Symbol('n', integer=True, positive=True)
assert simplify(integrate(n*(x**(1/n) - 1), (x, 0, S.Half)) -
(n**2 - 2**(1/n)*n**2 - n*2**(1/n))/(2**(1 + 1/n) + n*2**(1 + 1/n))) == 0
def test_issue1388():
from sympy import lowergamma, simplify
assert simplify(integrate(exp(-x)*x**y, x)) == lowergamma(y + 1, x)
def test_aux_dep():
# This test is about rolling disc dynamics, comparing the results found
# with KanesMethod to those found when deriving the equations "manually"
# with SymPy.
# The terms Fr, Fr*, and Fr*_steady are all compared between the two
# methods. Here, Fr*_steady refers to the generalized inertia forces for an
# equilibrium configuration.
# Note: comparing to the test of test_rolling_disc() in test_kane.py, this
# test also tests auxiliary speeds and configuration and motion constraints
#, seen in the generalized dependent coordinates q[3], and depend speeds
# u[3], u[4] and u[5].
# First, mannual derivation of Fr, Fr_star, Fr_star_steady.
# Symbols for time and constant parameters.
# Symbols for contact forces: Fx, Fy, Fz.
t, r, m, g, I, J = symbols('t r m g I J')
Fx, Fy, Fz = symbols('Fx Fy Fz')
# Configuration variables and their time derivatives:
# q[0] -- yaw
# q[1] -- lean
# q[2] -- spin
# q[3] -- dot(-r*B.z, A.z) -- distance from ground plane to disc center in
# A.z direction
# Generalized speeds and their time derivatives:
# u[0] -- disc angular velocity component, disc fixed x direction
# u[1] -- disc angular velocity component, disc fixed y direction
# u[2] -- disc angular velocity component, disc fixed z direction
# u[3] -- disc velocity component, A.x direction
# u[4] -- disc velocity component, A.y direction
# u[5] -- disc velocity component, A.z direction
# Auxiliary generalized speeds:
# ua[0] -- contact point auxiliary generalized speed, A.x direction
# ua[1] -- contact point auxiliary generalized speed, A.y direction
# ua[2] -- contact point auxiliary generalized speed, A.z direction
q = dynamicsymbols('q:4')
qd = [qi.diff(t) for qi in q]
u = dynamicsymbols('u:6')
ud = [ui.diff(t) for ui in u]
ud_zero = dict(zip(ud, [0.]*len(ud)))
ua = dynamicsymbols('ua:3')
ua_zero = dict(zip(ua, [0.]*len(ua)))
# Reference frames:
# Yaw intermediate frame: A.
# Lean intermediate frame: B.
# Disc fixed frame: C.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q[0], N.z])
B = A.orientnew('B', 'Axis', [q[1], A.x])
C = B.orientnew('C', 'Axis', [q[2], B.y])
# Angular velocity and angular acceleration of disc fixed frame
# u[0], u[1] and u[2] are generalized independent speeds.
C.set_ang_vel(N, u[0]*B.x + u[1]*B.y + u[2]*B.z)
C.set_ang_acc(N, C.ang_vel_in(N).diff(t, B)
+ cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Velocity and acceleration of points:
# Disc-ground contact point: P.
# Center of disc: O, defined from point P with depend coordinate: q[3]
# u[3], u[4] and u[5] are generalized dependent speeds.
P = Point('P')
P.set_vel(N, ua[0]*A.x + ua[1]*A.y + ua[2]*A.z)
O = P.locatenew('O', q[3]*A.z + r*sin(q[1])*A.y)
O.set_vel(N, u[3]*A.x + u[4]*A.y + u[5]*A.z)
O.set_acc(N, O.vel(N).diff(t, A) + cross(A.ang_vel_in(N), O.vel(N)))
# Kinematic differential equations:
# Two equalities: one is w_c_n_qd = C.ang_vel_in(N) in three coordinates
# directions of B, for qd0, qd1 and qd2.
# the other is v_o_n_qd = O.vel(N) in A.z direction for qd3.
# Then, solve for dq/dt's in terms of u's: qd_kd.
w_c_n_qd = qd[0]*A.z + qd[1]*B.x + qd[2]*B.y
v_o_n_qd = O.pos_from(P).diff(t, A) + cross(A.ang_vel_in(N), O.pos_from(P))
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_o_n_qd - O.vel(N), A.z)])
qd_kd = solve(kindiffs, qd)
# Values of generalized speeds during a steady turn for later substitution
# into the Fr_star_steady.
steady_conditions = solve(kindiffs.subs({qd[1] : 0, qd[3] : 0}), u)
steady_conditions.update({qd[1] : 0, qd[3] : 0})
# Partial angular velocities and velocities.
partial_w_C = [C.ang_vel_in(N).diff(ui, N) for ui in u + ua]
partial_v_O = [O.vel(N).diff(ui, N) for ui in u + ua]
partial_v_P = [P.vel(N).diff(ui, N) for ui in u + ua]
# Configuration constraint: f_c, the projection of radius r in A.z direction
# is q[3].
# Velocity constraints: f_v, for u3, u4 and u5.
# Acceleration constraints: f_a.
f_c = Matrix([dot(-r*B.z, A.z) - q[3]])
f_v = Matrix([dot(O.vel(N) - (P.vel(N) + cross(C.ang_vel_in(N),
O.pos_from(P))), ai).expand() for ai in A])
v_o_n = cross(C.ang_vel_in(N), O.pos_from(P))
a_o_n = v_o_n.diff(t, A) + cross(A.ang_vel_in(N), v_o_n)
f_a = Matrix([dot(O.acc(N) - a_o_n, ai) for ai in A])
# Solve for constraint equations in the form of
# u_dependent = A_rs * [u_i; u_aux].
# First, obtain constraint coefficient matrix: M_v * [u; ua] = 0;
# Second, taking u[0], u[1], u[2] as independent,
# taking u[3], u[4], u[5] as dependent,
# rearranging the matrix of M_v to be A_rs for u_dependent.
# Third, u_aux ==0 for u_dep, and resulting dictionary of u_dep_dict.
M_v = zeros(3, 9)
for i in range(3):
for j, ui in enumerate(u + ua):
M_v[i, j] = f_v[i].diff(ui)
M_v_i = M_v[:, :3]
M_v_d = M_v[:, 3:6]
M_v_aux = M_v[:, 6:]
M_v_i_aux = M_v_i.row_join(M_v_aux)
A_rs = - M_v_d.inv() * M_v_i_aux
u_dep = A_rs[:, :3] * Matrix(u[:3])
u_dep_dict = dict(zip(u[3:], u_dep))
# Active forces: F_O acting on point O; F_P acting on point P.
# Generalized active forces (unconstrained): Fr_u = F_point * pv_point.
F_O = m*g*A.z
F_P = Fx * A.x + Fy * A.y + Fz * A.z
Fr_u = Matrix([dot(F_O, pv_o) + dot(F_P, pv_p) for pv_o, pv_p in
zip(partial_v_O, partial_v_P)])
# Inertia force: R_star_O.
# Inertia of disc: I_C_O, where J is a inertia component about principal axis.
# Inertia torque: T_star_C.
# Generalized inertia forces (unconstrained): Fr_star_u.
R_star_O = -m*O.acc(N)
I_C_O = inertia(B, I, J, I)
T_star_C = -(dot(I_C_O, C.ang_acc_in(N)) \
+ cross(C.ang_vel_in(N), dot(I_C_O, C.ang_vel_in(N))))
Fr_star_u = Matrix([dot(R_star_O, pv) + dot(T_star_C, pav) for pv, pav in
zip(partial_v_O, partial_w_C)])
# Form nonholonomic Fr: Fr_c, and nonholonomic Fr_star: Fr_star_c.
# Also, nonholonomic Fr_star in steady turning condition: Fr_star_steady.
Fr_c = Fr_u[:3, :].col_join(Fr_u[6:, :]) + A_rs.T * Fr_u[3:6, :]
Fr_star_c = Fr_star_u[:3, :].col_join(Fr_star_u[6:, :])\
+ A_rs.T * Fr_star_u[3:6, :]
Fr_star_steady = Fr_star_c.subs(ud_zero).subs(u_dep_dict)\
.subs(steady_conditions).subs({q[3]: -r*cos(q[1])}).expand()
# Second, using KaneMethod in mechanics for fr, frstar and frstar_steady.
# Rigid Bodies: disc, with inertia I_C_O.
iner_tuple = (I_C_O, O)
disc = RigidBody('disc', O, C, m, iner_tuple)
bodyList = [disc]
# Generalized forces: Gravity: F_o; Auxiliary forces: F_p.
F_o = (O, F_O)
F_p = (P, F_P)
forceList = [F_o, F_p]
# KanesMethod.
kane = KanesMethod(
N, q_ind= q[:3], u_ind= u[:3], kd_eqs=kindiffs,
q_dependent=q[3:], configuration_constraints = f_c,
u_dependent=u[3:], velocity_constraints= f_v,
u_auxiliary=ua
)
# fr, frstar, frstar_steady and kdd(kinematic differential equations).
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
(fr, frstar)= kane.kanes_equations(forceList, bodyList)
frstar_steady = frstar.subs(ud_zero).subs(u_dep_dict).subs(steady_conditions)\
.subs({q[3]: -r*cos(q[1])}).expand()
kdd = kane.kindiffdict()
assert Matrix(Fr_c).expand() == fr.expand()
assert Matrix(Fr_star_c.subs(kdd)).expand() == frstar.expand()
assert (simplify(Matrix(Fr_star_steady).expand()) ==
simplify(frstar_steady.expand()))
def test_issue1394():
from sympy import simplify
assert simplify(integrate(x*sqrt(1 + 2*x), x)) == \
sqrt(2*x + 1)*(6*x**2 + x - 1)/15
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
# print(sym.pretty(equ.coeff(eps**3)))
# A1 = a * sym.exp(-3 * alpha2 / (2 * omega0**5) * T1)
# # Equation for eps0
X0 = A(T1) * sym.exp(sym.I * omega0 * T0) + sym.conjugate(
A(T1) * sym.exp(sym.I * omega0 * T0))
# X0 = A1(T1) * A2(T2) * sym.exp(sym.I * omega0 * T0) + g / omega0**2
# X0 = A1 * A2(T2) * sym.exp(sym.I * omega0 * T0) + g / omega0**2
# equEps0 = sym.diff(sym.diff(X0, T0), T0) - g + omega0**2 * X0
# equEps0 = equEps0.expand()
# print(sym.pretty(equEps0))
# # Equation for eps1
equEps1RHS = -(2 * sym.diff(sym.diff(X0, T1), T0) + alpha * X0**2 +
beta * X0**3)
equEps1RHS = sym.simplify(equEps1RHS.expand())
# print(sym.pretty(equEps1RHS))
print(sym.pretty(equEps1RHS.coeff(sym.exp(sym.I * omega0 * T0))))
#
# fid = open('equation.txt', 'w')
# fid.write(sym.latex(equEps1RHS, mode='equation'))
# fid.close()
# A1 = a * sym.exp(-3 * alpha2 / (2 * omega0**5) * T1)
# X1 = -alpha2 * g**3 / omega0**8 - \
# 3 * alpha2 * g * A1**2 * A2**2 / (omega0**2 * (omega0**2 - 4 * omega0**2)) * \
# sym.exp(2 * sym.I * T0 * omega0) - \
# alpha2 * A1**3 + A2**3 / (omega0**2 - 9 * omega0**2) * \
# sym.exp(3 * sym.I * T0 * omega0)
#
# # # Equation for eps2
def test_inverse_mellin_transform():
from sympy import (sin, simplify, Max, Min, expand, powsimp, exp_polar,
cos, cot)
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1 / x)
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
(x**2 + 1)*Heaviside(1 - x)/(4*x)
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
# test expansion of sums
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1) * exp(-x) / x
# test factorisation of polys
r = symbols('r', real=True)
assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
).subs(x, r).rewrite(sin).simplify() \
== sin(r)*Heaviside(1 - exp(-r))
# test multiplicative substitution
_a, _b = symbols('a b', positive=True)
assert IMT(_b**(-s / _a) * factorial(s / _a) / s, s, x,
(0, oo)) == exp(-_b * x**_a)
assert IMT(factorial(_a / _b + s / _b) / (_a + s), s, x,
(-_a, oo)) == x**_a * exp(-x**_b)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False),
force=True)).replace(exp_polar, exp)
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
nu = symbols('nu', real=True, finite=True)
assert IMT(-1 / (nu + s), s, x, (-oo, None)) == x**nu * Heaviside(x - 1)
assert IMT(1 / (nu + s), s, x, (None, oo)) == x**nu * Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
== (1 - x)**(beta - 1)*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
s, x, (-oo, None))) \
== (x - 1)**(beta - 1)*Heaviside(x - 1)
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
== (1/(x + 1))**rho
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
*gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
*gamma(-c/2 - s)/gamma(1 - c - s),
s, x, (0, -re(c)/2))) == \
(1 + sqrt(x + 1))**c
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
/gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
b**2 + x)/(b**2 + x)
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
b**c*(sqrt(1 + x/b**2) + 1)**c
# Section 8.4.5
assert IMT(24 / s**5, s, x, (0, oo)) == log(x)**4 * Heaviside(1 - x)
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
log(x)**3*Heaviside(x - 1)
assert IMT(pi / (s * sin(pi * s)), s, x, (-1, 0)) == log(x + 1)
assert IMT(pi / (s * sin(pi * s / 2)), s, x, (-2, 0)) == log(x**2 + 1)
assert IMT(pi / (s * sin(2 * pi * s)), s, x,
(-S(1) / 2, 0)) == log(sqrt(x) + 1)
assert IMT(pi / (s * sin(pi * s)), s, x, (0, 1)) == log(1 + 1 / x)
# TODO
def mysimp(expr):
from sympy import expand, logcombine, powsimp
return expand(powsimp(logcombine(expr, force=True),
force=True,
deep=True),
force=True).replace(exp_polar, exp)
assert mysimp(mysimp(IMT(pi / (s * tan(pi * s)), s, x, (-1, 0)))) in [
log(1 - x) * Heaviside(1 - x) + log(x - 1) * Heaviside(x - 1),
log(x) * Heaviside(x - 1) + log(1 - 1 / x) * Heaviside(x - 1) +
log(-x + 1) * Heaviside(-x + 1)
]
# test passing cot
assert mysimp(IMT(pi * cot(pi * s) / s, s, x, (0, 1))) in [
log(1 / x - 1) * Heaviside(1 - x) + log(1 - 1 / x) * Heaviside(x - 1),
-log(x) * Heaviside(-x + 1) + log(1 - 1 / x) * Heaviside(x - 1) +
log(-x + 1) * Heaviside(-x + 1),
]
# 8.4.14
assert IMT(-gamma(s + S(1)/2)/(sqrt(pi)*s), s, x, (-S(1)/2, 0)) == \
erf(sqrt(x))
# 8.4.19
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, S(3)/4))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(S(1)/2 - 2*s)*gamma(s + (a + 1)/2)
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-(re(a) + 1)/2, S(1)/4))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
/ (gamma(S(1)/2 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, S(1)/4))) == \
cos(sqrt(x))*besselj(a, sqrt(x))
# TODO this comes out as an amazing mess, but simplifies nicely
assert simplify(IMT(gamma(a + s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), S(1)/2))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, S(1)/2))) == \
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
*gamma(a/2 + b/2 - s + 1)),
s, x, (-(re(a) + re(b))/2, S(1)/2))) == \
besselj(a, sqrt(x))*besselj(b, sqrt(x))
# Section 8.4.20
# TODO this can be further simplified!
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
(pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
s, x,
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), S(1)/2))) == \
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
# TODO more
# for coverage
assert IMT(pi / cos(pi * s), s, x, (0, S(1) / 2)) == sqrt(x) / (x + 1)
result_pattern = "i" #"i"
kz_zero = False ## True for xy surface
pretty_print = True
try_parallel = True
gname = "D3d"
Info = get_data(TR=True)[gname]
print(Info["genes"])
multi_orbs_with_arr = []
multi_orbs_with_arr.append([['p, 3/2, 1/2', 'p, 3/2, -1/2'], [1, -1, -1]])
multi_orbs_with_arr.append([['p, 3/2, 1/2', 'p, 3/2, -1/2'], [1, 1, 1]])
order_list = [0, 1, 2]
similar_trans = sp.Matrix([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0],
[0, 0, 0, 1]]) * sp.diag(1, sp.simplify("I"), 1,
-sp.simplify("I"))
#-------------------------------------------------------------------------------
if "bandchars" not in locals(): bandchars = []
if "multi_orbs_with_arr" not in locals(): multi_orbs_with_arr = []
if "similar_trans" not in locals(): similar_trans = False
start_time = time.time()
if try_parallel:
cores = multiprocessing.cpu_count()
try:
pool = multiprocessing.Pool(processes=cores)
except:
pool = ""
def test_weibull():
a, b = symbols('a b', positive=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2/b) - E(X)**2)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False),
force=True)).replace(exp_polar, exp)
def actToExpression(act: Activity, extract_activities=None):
"""Computes a symbolic expression of the model, referencing background activities and model parameters as symbols
Returns
-------
(sympy_expr, dict of symbol => activity)
"""
act_symbols = dict() # Dict of act = > symbol
def act_to_symbol(db_name, code):
act = _getDb(db_name).get(code)
name = act['name']
base_slug = _slugify(name)
slug = base_slug
i = 1
while symbols(slug) in act_symbols.values():
slug = f"{base_slug}{i}"
i += 1
return symbols(slug)
def rec_func(act: Activity, in_extract_path):
res = 0
outputAmount = 1
for exch in act.exchanges():
formula = _getAmountOrFormula(exch)
if isinstance(formula, types.FunctionType):
# Some amounts in EIDB are functions ... we ignore them
continue
# Production exchange
if exch['input'] == exch['output']:
# Not 1 ?
if exch['amount'] != 1:
outputAmount = exch['amount']
continue
input_db, input_code = exch['input']
sub_act = _getDb(input_db).get(input_code)
# If list of extract activites requested, we only integrate activites below a tracked one
exch_in_path = in_extract_path
if extract_activities is not None:
if sub_act in extract_activities:
exch_in_path = in_extract_path or (sub_act
in extract_activities)
# Background DB => reference it as a symbol
if input_db in [BIOSPHERE3_DB_NAME, ECOINVENT_DB_NAME()]:
if exch_in_path:
# Add to dict of background symbols
if not (input_db, input_code) in act_symbols:
act_symbols[(input_db, input_code)] = act_to_symbol(
input_db, input_code)
act_expr = act_symbols[(input_db, input_code)]
else:
continue
# Our model : recursively it to a symbolic expression
else:
if input_db == act['database'] and input_code == act['code']:
raise Exception("Recursive exchange : %s" % (act.__dict__))
act_expr = rec_func(sub_act, exch_in_path)
avoidedBurden = 1
if exch['type'] == 'production' and not exch['input'] == exch[
'output']:
avoidedBurden = -1
res += formula * act_expr * avoidedBurden
return res / outputAmount
expr = rec_func(act, extract_activities is None)
if isinstance(expr, float):
expr = simplify(expr)
else:
# Replace fixed params with their default value
expr = _replace_fixed_params(expr, _fixed_params().values())
return (expr, _reverse_dict(act_symbols))
def test_conjugate_priors():
mu = Normal('mu', 2, 3)
x = Normal('x', mu, 1)
assert isinstance(simplify(density(mu, Eq(x, y), evaluate=False)(z)),
Integral)
h = X[1] - X[0]
coeff = []
coeff.append(Y[0])
n = len(Y)
for i in range(1, n):
a = (f_diff[i - 1][0] / (factorial(i) * (h**i)))
coeff.append(a)
# Displaying coefficients
print("\n\n\n\nCoefficients are\n")
for i in range(n):
print("a" + str(i), " = ", coeff[i])
u = (x - X[0]) / h
# Calcualting P(x)
P = 0
for k in range(1, n):
U = u
for i in range(1, k):
U *= (u - i)
P += (U * f_diff[k - 1][0]) / factorial(k)
P = P + Y[0]
print("\n\nThe expression of p(x) is ", s.simplify(P))
print("\n\nThe value of p(x) is ", P.subs(x, Xx))
def test_gumbel():
beta = Symbol("beta", positive=True)
mu = Symbol("mu")
x = Symbol("x")
X = Gumbel("x", beta, mu)
assert simplify(density(X)(x)) == exp((beta*exp((mu - x)/beta) + mu - x)/beta)/beta
def test_simplify():
y, t, v = symbols('y, t, v')
assert simplify(Sum(x*y, (x, n, m), (y, a, k)) + \
Sum(y, (x, n, m), (y, a, k))) == Sum(y * (x + 1), (x, n, m), (y, a, k))
assert simplify(Sum(x, (x, n, m)) + Sum(x, (x, m + 1, a))) == \
Sum(x, (x, n, a))
assert simplify(Sum(x, (x, k + 1, a)) + Sum(x, (x, n, k))) == \
Sum(x, (x, n, a))
assert simplify(Sum(x, (x, k + 1, a)) + Sum(x + 1, (x, n, k))) == \
Sum(x, (x, n, a)) + Sum(1, (x, n, k))
assert simplify(Sum(x, (x, 0, 3)) * 3 + 3 * Sum(x, (x, 4, 6)) + \
4 * Sum(z, (z, 0, 1))) == 4*Sum(z, (z, 0, 1)) + 3*Sum(x, (x, 0, 6))
assert simplify(3*Sum(x**2, (x, a, b)) + Sum(x, (x, a, b))) == \
Sum(x*(3*x + 1), (x, a, b))
assert simplify(Sum(x**3, (x, n, k)) * 3 + 3 * Sum(x, (x, n, k)) + \
4 * y * Sum(z, (z, n, k))) + 1 == \
4*y*Sum(z, (z, n, k)) + 3*Sum(x**3 + x, (x, n, k)) + 1
assert simplify(Sum(x, (x, a, b)) + 1 + Sum(x, (x, b + 1, c))) == \
1 + Sum(x, (x, a, c))
assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + \
Sum(x, (t, b+1, c))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
assert simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + \
Sum(y, (t, a, b))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
assert simplify(Sum(x, (t, a, b)) + 2 * Sum(x, (t, b+1, c))) == \
simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + Sum(x, (t, b+1, c)))
assert simplify(Sum(x, (x, a, b))*Sum(x**2, (x, a, b))) == \
Sum(x, (x, a, b)) * Sum(x**2, (x, a, b))
assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b))) \
== (x + y + z) * Sum(1, (t, a, b)) # issue 8596
assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b)) + \
Sum(v, (t, a, b))) == (x + y + z + v) * Sum(1, (t, a, b)) # issue 8596
assert simplify(Sum(x * y, (x, a, b)) / (3 * y)) == \
(Sum(x, (x, a, b)) / 3)
assert simplify(Sum(Function('f')(x) * y * z, (x, a, b)) / (y * z)) \
== Sum(Function('f')(x), (x, a, b))
assert simplify(Sum(c * x, (x, a, b)) - c * Sum(x, (x, a, b))) == 0
assert simplify(c * (Sum(x, (x, a, b)) + y)) == c * (y + Sum(x, (x, a, b)))
assert simplify(c * (Sum(x, (x, a, b)) + y * Sum(x, (x, a, b)))) == \
c * (y + 1) * Sum(x, (x, a, b))
assert simplify(Sum(Sum(c * x, (x, a, b)), (y, a, b))) == \
c * Sum(x, (x, a, b), (y, a, b))
assert simplify(Sum((3 + y) * Sum(c * x, (x, a, b)), (y, a, b))) == \
c * Sum((3 + y), (y, a, b)) * Sum(x, (x, a, b))
assert simplify(Sum((3 + t) * Sum(c * t, (x, a, b)), (y, a, b))) == \
c*t*(t + 3)*Sum(1, (x, a, b))*Sum(1, (y, a, b))
assert simplify(Sum(Sum(d * t, (x, a, b - 1)) + \
Sum(d * t, (x, b, c)), (t, a, b))) == \
d * Sum(1, (x, a, c)) * Sum(t, (t, a, b))
# Define :math:`y=A\exp(-t/\tau)` and the gaussian IRF. # %% y = step(t) * A * exp(-t / tau) y # %% irf = 1 / sqrt(2 * pi * sigma**2) * exp(-t**2 / (2 * sigma**2)) irf # %% # Next, we will calculate the covolution integral. # %% func = integrate((irf.subs(t, t - ti) * y.subs(t, ti)), (ti, -oo, oo)) func = simplify(func) func # %% # Rewirte the `erf` with the `erfc` function: #erfc, erf = special.error_functions.erfc, special.error_functions.erf func2 = func.rewrite(erfc) func2 # %% [markdown] # Plot the result to it makes sense: plot(func2.subs(sigma, 0.2).subs(tau, 2).subs(A, 1), (t, -1, 10)) # %% # Normalized derivatives of a gaussian # ------------------------------------
#We first need the three equations we are going to try and set to zero, i.e. the first partial derivitives wrt e h and F.
print "/*This code was automatically generated by "+str(sys.argv[0])+"*/\n"
print "#include \"allele.h\""
print "#include \"quartet.h\""
print "#include \"typedef.h\""
print "#include \"constants.h\""
print
#numpy.set_printoptions(precission=18)
for x in range(0, 7):
system_eq.append(sympy.diff(lnL, params[x]) )
print "inline float_t H"+str(x)+" (const Genotype_pair &pair, const Constants <float_t, const std::pair<const Genotype_pair &, const Relatedness &> > &con) {"
sys.stdout.write("\treturn ")
out=(system_eq[-1])
out=sympy.simplify(out)
pre(out)
keys=dag_sort(keys)
for key in keys:
out=exact_sub(out, constants[key], key)
string=sympy.printing.ccode(out)
print string, ";\n}\n"
#Then we need to make the Jacobian, which is a matrix with ...
for x in range(0, 7):
for y in range(0, 7):
print "inline float_t J"+str(x)+str(y)+" (const Genotype_pair &pair, const Constants <float_t, const std::pair <const Genotype_pair &, const Relatedness &> > &con) {"
sys.stdout.write("\treturn ")
out=(sympy.diff(system_eq[x], params[y]))
#out=sympy.simplify(out)
pre(out)
# ASSEMBLE THE LAGRANGIAN!!! #################################################### ## Calculate the Kinetic Energy of each body ################################# # KE = .5 * V.T * I * V Vp1 = invertTrans(g_wp1) * g_wp1.diff(t) Vp1 = se3ToVec(Vp1) Vp2 = invertTrans(g_wp2) * g_wp2.diff(t) Vp2 = se3ToVec(Vp2) Vx = invertTrans(g_wb) * g_wb.diff(t) Vx = se3ToVec(Vx) KEp1 = sym.simplify(0.5 * Vp1.T * Ipend1 * Vp1) KEp2 = sym.simplify(0.5 * Vp2.T * Ipend2 * Vp2) KEx = sym.simplify(0.5 * Vx.T * Icart * Vx) KE = sym.simplify(KEp1[0] + KEp2[0] + KEx[0]) ## Calculate the Potential Energy of each body ############################### # PE = m * g * h PEp2 = m * g * g_wp2[1, 3] PE = sym.simplify(PEp2) ## Calculate the Lagrangian!!! ############################################### L = sym.simplify(KE - PE) ####################################################
#coding:utf-8
'''
特征值问题
'''
from __future__ import division
from scipy import linalg as la
from scipy import optimize
import sympy
import numpy as np
sympy.init_printing()
import matplotlib.pyplot as plt
# 使用符号方式求解矩阵的特征值
eps, delta = sympy.symbols("epsilon, Delta")
H = sympy.Matrix([[eps, delta], [delta, -eps]])
eigenvalue = H.eigenvals()
results = H.eigenvects()
(eval1, _, evec1), (eval2, _, evec2) = H.eigenvects()
result = sympy.simplify(evec1[0].T * evec2[0])
print('result = ', result)
# 使用scipy求解矩阵特征值
A = np.array([[1, 3, 5], [3, 5, 3], [5, 3, 9]])
evals, evecs = la.eig(A)
eigvalues = la.eigvalsh(A)
def test_karr_convention():
# Test the Karr summation convention that we want to hold.
# See his paper "Summation in Finite Terms" for a detailed
# reasoning why we really want exactly this definition.
# The convention is described on page 309 and essentially
# in section 1.4, definition 3:
#
# \sum_{m <= i < n} f(i) 'has the obvious meaning' for m < n
# \sum_{m <= i < n} f(i) = 0 for m = n
# \sum_{m <= i < n} f(i) = - \sum_{n <= i < m} f(i) for m > n
#
# It is important to note that he defines all sums with
# the upper limit being *exclusive*.
# In contrast, sympy and the usual mathematical notation has:
#
# sum_{i = a}^b f(i) = f(a) + f(a+1) + ... + f(b-1) + f(b)
#
# with the upper limit *inclusive*. So translating between
# the two we find that:
#
# \sum_{m <= i < n} f(i) = \sum_{i = m}^{n-1} f(i)
#
# where we intentionally used two different ways to typeset the
# sum and its limits.
i = Symbol("i", integer=True)
k = Symbol("k", integer=True)
j = Symbol("j", integer=True)
# A simple example with a concrete summand and symbolic limits.
# The normal sum: m = k and n = k + j and therefore m < n:
m = k
n = k + j
a = m
b = n - 1
S1 = Sum(i**2, (i, a, b)).doit()
# The reversed sum: m = k + j and n = k and therefore m > n:
m = k + j
n = k
a = m
b = n - 1
S2 = Sum(i**2, (i, a, b)).doit()
assert simplify(S1 + S2) == 0
# Test the empty sum: m = k and n = k and therefore m = n:
m = k
n = k
a = m
b = n - 1
Sz = Sum(i**2, (i, a, b)).doit()
assert Sz == 0
# Another example this time with an unspecified summand and
# numeric limits. (We can not do both tests in the same example.)
f = Function("f")
# The normal sum with m < n:
m = 2
n = 11
a = m
b = n - 1
S1 = Sum(f(i), (i, a, b)).doit()
# The reversed sum with m > n:
m = 11
n = 2
a = m
b = n - 1
S2 = Sum(f(i), (i, a, b)).doit()
assert simplify(S1 + S2) == 0
# Test the empty sum with m = n:
m = 5
n = 5
a = m
b = n - 1
Sz = Sum(f(i), (i, a, b)).doit()
assert Sz == 0
e = Piecewise((exp(-i), Mod(i, 2) > 0), (0, True))
s = Sum(e, (i, 0, 11))
assert s.n(3) == s.doit().n(3)
