def test_D():
s = symbols('s')
C = Rational(3, 2)
L = Rational(1, 3)
r = Rational(1, 2)
# series L, shunt C
C0 = Cascade.Series(L * s)
C1 = Cascade.Series(1 / (C * s))
C2 = Cascade.Shunt(1 / r)
print("test_D")
print(C0)
print(C1)
print(C2)
print(C1.hit(C0))
print(C2.hit(C1.hit(C0)))
Y = ratsimp(1 / C2.hit(C1.hit(C0)).terminate(0))
print(Y)
Z = ratsimp(1 / (Y - 2))
print(Z)
Y = ratsimp(Z - s / 3)
print(Y)
def heuristics(e, z, z0, dir):
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1 / z), z, S.Zero,
"+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
def heuristics(e, z, z0, dir):
"""Computes the limit of an expression term-wise.
Parameters are the same as for the ``limit`` function.
Works with the arguments of expression ``e`` one by one, computing
the limit of each and then combining the results. This approach
works only for simple limits, but it is fast.
"""
from sympy.calculus.util import AccumBounds
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
if isinstance(e, Add):
m = factor_terms(e)
if not isinstance(m, Mul): # try together
m = together(m)
if not isinstance(m, Mul): # try factor if the previous methods failed
m = factor(e)
if isinstance(m, Mul):
return heuristics(m, z, z0, dir)
return
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
r2 = []
e2 = []
for ii in range(len(r)):
if isinstance(r[ii], AccumBounds):
r2.append(r[ii])
else:
e2.append(e.args[ii])
if len(e2) > 0:
e3 = Mul(*e2).simplify()
l = limit(e3, z, z0, dir)
rv = l * Mul(*r2)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
def heuristics(e, z, z0, dir):
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
def heuristics(e, z, z0, dir):
"""Computes the limit of an expression term-wise.
Parameters are the same as for the ``limit`` function.
Works with the arguments of expression ``e`` one by one, computing
the limit of each and then combining the results. This approach
works only for simple limits, but it is fast.
"""
from sympy.calculus.util import AccumBounds
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
if isinstance(e, Add):
m = factor_terms(e)
if not isinstance(m, Mul): # try together
m = together(m)
if not isinstance(m, Mul): # try factor if the previous methods failed
m = factor(e)
if isinstance(m, Mul):
return heuristics(m, z, z0, dir)
return
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
r2 = []
e2 = []
for ii in range(len(r)):
if isinstance(r[ii], AccumBounds):
r2.append(r[ii])
else:
e2.append(e.args[ii])
if len(e2) > 0:
e3 = Mul(*e2).simplify()
l = limit(e3, z, z0, dir)
rv = l * Mul(*r2)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
def test_C():
s = symbols('s')
C = Rational(3, 2)
L = Rational(1, 3)
r = Rational(1, 2)
# series L, shunt C
C0 = Cascade.Shunt(1 / (1 / (C * s) + L * s))
print(ratsimp(C0.terminate(r)))
X = ratsimp(1 / C0.terminate(r))
print(X)
Y = ratsimp(1 / (X - 2))
print(Y)
Z = ratsimp(Y - s / 3)
print(Z)
def test_pmint_logexp():
if ON_TRAVIS:
# See https://github.com/sympy/sympy/pull/12795
skip("Too slow for travis.")
f = (1 + x + x * exp(x)) * (x + log(x) + exp(x) - 1) / (x + log(x) +
exp(x))**2 / x
g = log(x + exp(x) + log(x)) + 1 / (x + exp(x) + log(x))
assert ratsimp(heurisch(f, x)) == g
def test_F():
"Hazony example 5.2.2"
s = symbols('s')
print("test_F")
Z = (s**2 + s + 1) / (s**2 + 2 * s + 2)
print(f"Z: {Z}")
min_r = (3 - sympy.sqrt(2)) / 4
Z1 = ratsimp(Z - min_r)
print(f"Z1: {Z1}")
#plot_real_part( sympy.lambdify(s, Z1, "numpy"))
Y1 = ratsimp(1 / Z - 1)
print(f"Y1: {Y1}")
C = Cascade.Shunt(1)
Z2 = ratsimp(1 / Y1 - s)
print(f"Z2: {Z2}")
C = C.hit(Cascade.Series(s))
Y3 = ratsimp(1 / Z2 - s - 1)
print(f"Y3: {Y3}")
Ytotal = C.hit(Cascade.Shunt(1).hit(
Cascade.Shunt(s))).terminate_with_admittance(0)
assert sympy.Eq(0, ratsimp(1 / Ytotal - Z))
assert sympy.Eq(1, ratsimp(Ytotal * Z))
Ytotal = C.hit(Cascade.Shunt(s)).terminate_with_admittance(1)
assert sympy.Eq(0, ratsimp(1 / Ytotal - Z))
assert sympy.Eq(1, ratsimp(Ytotal * Z))
Ytotal = C.terminate_with_admittance(1 + s)
assert sympy.Eq(0, ratsimp(1 / Ytotal - Z))
assert sympy.Eq(1, ratsimp(Ytotal * Z))
def find_riccati_ode(ratfunc, x, yf):
y = ratfunc
yp = y.diff(x)
q1 = rand_rational_function(x, 1, 3)
q2 = rand_rational_function(x, 1, 3)
while q2 == 0:
q2 = rand_rational_function(x, 1, 3)
q0 = ratsimp(yp - q1 * y - q2 * y**2)
eq = Eq(yf.diff(), q0 + q1 * yf + q2 * yf**2)
sol = Eq(yf, y)
assert checkodesol(eq, sol) == (True, 0)
return eq, q0, q1, q2
def heuristics(e, z, z0, dir):
from sympy.calculus.util import AccumBounds
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
if isinstance(e, Add):
m = factor_terms(e)
if not isinstance(m, Mul): # try together
m = together(m)
if not isinstance(m, Mul): # try factor if the previous methods failed
m = factor(e)
if isinstance(m, Mul):
return heuristics(m, z, z0, dir)
return
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
r2 = []
e2 = []
for ii in range(len(r)):
if isinstance(r[ii], AccumBounds):
r2.append(r[ii])
else:
e2.append(e.args[ii])
if len(e2) > 0:
e3 = Mul(*e2).simplify()
l = limit(e3, z, z0, dir)
rv = l * Mul(*r2)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
def test_pmint_rat():
# TODO: heurisch() is off by a constant: -3/4. Possibly different permutation
# would give the optimal result?
def drop_const(expr, x):
if expr.is_Add:
return Add(*[arg for arg in expr.args if arg.has(x)])
else:
return expr
f = (x**7 - 24 * x**4 - 4 * x**2 + 8 * x - 8) / (x**8 + 6 * x**6 +
12 * x**4 + 8 * x**2)
g = (4 + 8 * x**2 + 6 * x + 3 * x**3) / (x**5 + 4 * x**3 + 4 * x) + log(x)
assert drop_const(ratsimp(heurisch(f, x)), x) == g
def test_E():
"Chop Chop example"
s = symbols('s')
print("test_E")
Z = (2 * s**2 + 2 * s + 1) / (s * (s**2 + s + 1))
print(f"Z: {Z}")
Z = ratsimp(Z)
print(f"Z: {Z}")
Z = ratsimp(Z - 1 / s)
print(f"Z-1/s: {Z}")
C = Cascade.Series(1 / s)
Y = ratsimp(1 / Z - s)
print(f"Y: {Y}")
C = C.hit(Cascade.Shunt(s))
Z = ratsimp(1 / Y)
print(f"Z: {Z-1-s}")
C = C.hit(Cascade.Series(1 + s)).terminate(0)
print(ratsimp(C))
def heuristics(e, z, z0, dir):
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1 / z), z, S.Zero,
"+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
if isinstance(e, Add):
m = factor_terms(e)
if not isinstance(m, Mul): # try together
m = together(m)
if not isinstance(
m,
Mul): # try factor if the previous methods failed
m = factor(e)
if isinstance(m, Mul):
return heuristics(m, z, z0, dir)
return
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
def heuristics(e, z, z0, dir):
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
if isinstance(e, Add):
m = factor_terms(e)
if not isinstance(m, Mul): # try together
m = together(m)
if not isinstance(m, Mul): # try factor if the previous methods failed
m = factor(e)
if isinstance(m, Mul):
return heuristics(m, z, z0, dir)
return
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
def test_ratsimp():
f, g = 1 / x + 1 / y, (x + y) / (x * y)
assert f != g and ratsimp(f) == g
f, g = 1 / (1 + 1 / x), 1 - 1 / (x + 1)
assert f != g and ratsimp(f) == g
f, g = x / (x + y) + y / (x + y), 1
assert f != g and ratsimp(f) == g
f, g = -x - y - y**2 / (x + y) + x**2 / (x + y), -2 * y
assert f != g and ratsimp(f) == g
f = (a * c * x * y + a * c * z - b * d * x * y - b * d * z -
b * t * x * y - b * t * x - b * t * z + e * x) / (x * y + z)
G = [
a * c - b * d - b * t + (-b * t * x + e * x) / (x * y + z),
a * c - b * d - b * t - (b * t * x - e * x) / (x * y + z)
]
assert f != g and ratsimp(f) in G
A = sqrt(pi)
B = log(erf(x) - 1)
C = log(erf(x) + 1)
D = 8 - 8 * erf(x)
f = A * B / D - A * C / D + A * C * erf(x) / D - A * B * erf(
x) / D + 2 * A / D
assert ratsimp(f) == A * B / 8 - A * C / 8 - A / (4 * erf(x) - 4)
def test_pmint_erf():
f = exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1)
g = sqrt(pi) * log(erf(x) - 1) / 8 - sqrt(pi) * log(erf(x) + 1) / 8 - sqrt(
pi) / (4 * erf(x) - 4)
assert ratsimp(heurisch(f, x)) == g
def test_I():
"Hazony problem 5.3.a"
s, k = symbols('s k')
w = symbols('w', real=True)
pprint("test_I")
Z = (s**3 + 3 * s**2 + s + 1) / (s**3 + s**2 + 3 * s + 1)
pprint(f"Z: {Z}")
#plot_real_part( sympy.lambdify(s, Z, "numpy"))
real_part = cancel(sympy.re(Z.subs({s: sympy.I * w})))
print(f"real_part: {real_part}")
roots = sympy.solveset(real_part, w)
print(f"roots for w: {roots}")
#plot( sympy.lambdify(w, real_part, "numpy"))
w0 = 1
target0 = radsimp(Z.subs({s: sympy.I * w0}) / (sympy.I * w0))
print(f"target: {target0}")
target1 = radsimp(Z.subs({s: sympy.I * w0}) * (sympy.I * w0))
print(f"target: {target1}")
assert target0 > 0
eq = sympy.Eq(Z.subs({s: k}) / k, target0)
#assert target1 > 0
#eq = sympy.Eq( Z.subs({s:k})*k, target1)
roots = sympy.solveset(eq, k)
print(f"roots for k: {roots}")
k0 = Rational(1, 1)
Z_k0 = Z.subs({s: k0})
print(k0, Z_k0)
print(k0.evalf(), Z_k0.evalf())
den = cancel((k0 * Z_k0 - s * Z))
print(f"den factored: {sympy.factor(den)}")
num = cancel((k0 * Z - s * Z_k0))
print(f"num factored: {sympy.factor(num)}")
eta = cancel(num / den)
print(k0, Z_k0, eta)
print("normal")
Z0 = eta * Z_k0
print(f"Z0: {Z0}")
Y1 = ratsimp(1 / Z0 - 1)
print(f"Y1: {Y1}")
C = Cascade.Shunt(1)
Z2 = ratsimp(1 / Y1 - s / 2 - 1 / (2 * s))
print(f"Z2: {Z2}")
C = C.hit(Cascade.Series(s / 2))
C = C.hit(Cascade.Series(1 / (2 * s)))
eta_Z_k0 = cancel(C.terminate(0))
print(f"eta_Z_k0: {eta_Z_k0}")
assert sympy.Eq(cancel(eta_Z_k0 - Z0), 0)
print("recip")
Z0 = cancel(Z_k0 / eta)
print(f"Z0: {Z0}")
Z1 = ratsimp(Z0 - 1)
print(f"Z1: {Z1}")
C = Cascade.Series(1)
Y2 = ratsimp(1 / Z1 - s / 2 - 1 / (2 * s))
print(f"Y2: {Y2}")
C = C.hit(Cascade.Shunt(s / 2))
C = C.hit(Cascade.Shunt(1 / (2 * s)))
eta_over_Z_k0 = cancel(1 / C.terminate_with_admittance(0))
print(f"eta_over_Z_k0: {eta_over_Z_k0}")
assert sympy.Eq(cancel(eta_over_Z_k0 - Z0), 0)
def p(a, b):
return a * b / (a + b)
constructed_Z = cancel(
p(eta_Z_k0, (k0 * Z_k0) / s) + p(eta_over_Z_k0, (Z_k0 * s) / k0))
print(f"constructed_Z: {constructed_Z}")
assert sympy.Eq(cancel(constructed_Z - Z), 0)
def test_J():
"Second problem in Guillemin"
s, k = symbols('s k')
w = symbols('w', real=True)
pprint("test_I")
Z = (s**2 + s + 8) / (s**2 + 2 * s + 2)
pprint(f"Z: {Z}")
Y = 1 / Z
#plot_real_part( sympy.lambdify(s, Y, "numpy"))
real_part = cancel(sympy.re(Y.subs({s: sympy.I * w})))
print(f"real_part: {real_part}")
roots = sympy.solveset(real_part, w)
print(f"roots for w: {roots}")
#plot( sympy.lambdify(w, real_part, "numpy"))
w0 = 2
target0 = radsimp(Y.subs({s: sympy.I * w0}) / (sympy.I * w0))
print(f"target: {target0.evalf()}")
target0 = Rational(1, 2)
target1 = radsimp(Y.subs({s: sympy.I * w0}) * (sympy.I * w0))
print(f"target: {target1.evalf()}")
target1 = Rational(2, 1)
assert target0 > 0
eq = sympy.Eq(Y.subs({s: k}) / k, target0)
#assert target1 > 0
#eq = sympy.Eq( Z.subs({s:k})*k, target1)
roots = sympy.solveset(eq, k)
print(f"roots for k: {roots}")
k0 = Rational(1, 1)
Y_k0 = Y.subs({s: k0})
print(k0, Y_k0)
print(k0.evalf(), Y_k0.evalf())
den = cancel((k0 * Y_k0 - s * Y))
print(f"den factored: {sympy.factor(den)}")
num = cancel((k0 * Y - s * Y_k0))
print(f"num factored: {sympy.factor(num)}")
eta = cancel(num / den)
print(k0, Y_k0, eta)
print("normal")
Y0 = eta * Y_k0
print(f"Y0: {Y0}")
Z1 = ratsimp(1 / Y0 - 4)
print(f"Z1: {Z1}")
C = Cascade.Series(4)
Y2 = ratsimp(1 / Z1)
print(f"Y2: {Y2}")
C = C.hit(Cascade.Shunt(s / 10))
C = C.hit(Cascade.Shunt(2 / (5 * s)))
eta_Y_k0 = cancel(C.terminate_with_admittance(0))
print(f"eta_Y_k0: {eta_Y_k0}")
assert sympy.Eq(cancel(eta_Y_k0 - Y0), 0)
print("recip")
Y0 = ratsimp(Y_k0 / eta)
print(f"Y0: {Y0}")
Y1 = ratsimp(Y0 - 1)
print(f"Y1: {Y1}")
C = Cascade.Shunt(1)
Z2 = ratsimp(1 / Y1 - 2 * s / 5 - 8 / (5 * s))
print(f"Z2: {Z2}")
C = C.hit(Cascade.Series(2 * s / 5))
C = C.hit(Cascade.Series(8 / (5 * s)))
eta_over_Y_k0 = cancel(1 / C.terminate(0))
print(f"eta_over_Y_k0: {eta_over_Y_k0}")
assert sympy.Eq(cancel(eta_over_Y_k0 - Y0), 0)
def p(a, b):
return a * b / (a + b)
constructed_Y = cancel(
p(eta_Y_k0, (k0 * Y_k0) / s) + p(eta_over_Y_k0, (Y_k0 * s) / k0))
print(f"constructed_Y: {constructed_Y}")
assert sympy.Eq(cancel(constructed_Y - Y), 0)
def test_ratsimp():
assert ratsimp(A*B - B*A) == A*B - B*A
