def test_sympy_interaction():
pytest.importorskip("sympy")
import sympy as sp
x, y = sp.symbols("x y")
f = sp.symbols("f")
s1_expr = 1/f(x/sp.sqrt(x**2+y**2)).diff(x, 5)
from pymbolic.sympy_interface import (
SympyToPymbolicMapper,
PymbolicToSympyMapper)
s2p = SympyToPymbolicMapper()
p2s = PymbolicToSympyMapper()
p1_expr = s2p(s1_expr)
s2_expr = p2s(p1_expr)
assert sp.ratsimp(s1_expr - s2_expr) == 0
p2_expr = s2p(s2_expr)
s3_expr = p2s(p2_expr)
assert sp.ratsimp(s1_expr - s3_expr) == 0
def test_sympy_interaction():
pytest.importorskip("sympy")
import sympy as sp
x, y = sp.symbols("x y")
f = sp.Function("f")
s1_expr = 1/f(x/sp.sqrt(x**2+y**2)).diff(x, 5) # pylint:disable=not-callable
from pymbolic.interop.sympy import (
SympyToPymbolicMapper,
PymbolicToSympyMapper)
s2p = SympyToPymbolicMapper()
p2s = PymbolicToSympyMapper()
p1_expr = s2p(s1_expr)
s2_expr = p2s(p1_expr)
assert sp.ratsimp(s1_expr - s2_expr) == 0
p2_expr = s2p(s2_expr)
s3_expr = p2s(p2_expr)
assert sp.ratsimp(s1_expr - s3_expr) == 0
def test_ratsimp():
x = Symbol("x")
y = Symbol("y")
e = 1/x+1/y
assert e != (x+y)/(x*y)
assert ratsimp(e) == (x+y)/(x*y)
e = 1/(1+1/x)
assert ratsimp(e) == x/(x+1)
assert ratsimp(exp(e)) == exp(x/(x+1))
def test_ratsimp():
x = Symbol("x")
y = Symbol("y")
e = 1/x+1/y
assert e != (x+y)/(x*y)
assert ratsimp(e) == (x+y)/(x*y)
e = 1/(1+1/x)
assert ratsimp(e) == x/(x+1)
assert ratsimp(exp(e)) == exp(x/(x+1))
def print_Div(self, rule):
with self.new_step():
f, g = rule.numerator, rule.denominator
fp, gp = f.diff(rule.symbol), g.diff(rule.symbol)
x = rule.symbol
ff = sympy.Function("f")(x)
gg = sympy.Function("g")(x)
qrule_left = sympy.Derivative(ff / gg, rule.symbol)
qrule_right = sympy.ratsimp(
sympy.diff(sympy.Function("f")(x) / sympy.Function("g")(x)))
qrule = sympy.Eq(qrule_left, qrule_right)
self.append("Aplicando la regla del cociente que es:")
self.append(self.format_math_display(qrule))
self.append("{} y {}.".format(self.format_math(sympy.Eq(ff, f)),
self.format_math(sympy.Eq(gg, g))))
self.append("Para hallar {}:".format(
self.format_math(ff.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.numerstep)
self.append("Para hallar {}:".format(
self.format_math(gg.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.denomstep)
self.append("Sutituyendo en la regla del cociente:")
self.append(self.format_math(diff(rule)))
def print_Div(self, rule):
with self.new_step():
f, g = rule.numerator, rule.denominator
x = rule.symbol
ff = sympy.Function("f")(x)
gg = sympy.Function("g")(x)
qrule_left = sympy.Derivative(ff / gg, rule.symbol)
qrule_right = sympy.ratsimp(
sympy.diff(sympy.Function("f")(x) / sympy.Function("g")(x)))
qrule = sympy.Eq(qrule_left, qrule_right)
self.append("Apply the quotient rule, which is:")
self.append(self.format_math_display(qrule))
self.append("{} and {}.".format(self.format_math(sympy.Eq(ff, f)),
self.format_math(sympy.Eq(gg, g))))
self.append("Find {}:".format(
self.format_math(ff.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.numerstep)
self.append("Find {}:".format(
self.format_math(gg.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.denomstep)
self.append(
'<div class="collapsible"><h2>open answer</h2><ol class="content">Now plug in to the quotient rule to get:'
)
self.append(self.format_math(diff(rule)) + '</ol></div>')
def _solsimp(e, t):
no_t, has_t = powsimp(expand_mul(e)).as_independent(t)
no_t = ratsimp(no_t)
has_t = has_t.replace(exp, lambda a: exp(factor_terms(a)))
return no_t + has_t
def print_Div(self, rule):
with self.new_step():
f, g = rule.numerator, rule.denominator
fp, gp = f.diff(rule.symbol), g.diff(rule.symbol)
x = rule.symbol
ff = sympy.Function("f")(x)
gg = sympy.Function("g")(x)
qrule_left = sympy.Derivative(ff / gg, rule.symbol)
qrule_right = sympy.ratsimp(
sympy.diff(sympy.Function("f")(x) / sympy.Function("g")(x)))
qrule = sympy.Eq(qrule_left, qrule_right)
self.append("Apply the quotient rule, which is:")
self.append(self.format_math_display(qrule))
self.append("{} and {}.".format(self.format_math(sympy.Eq(ff, f)),
self.format_math(sympy.Eq(gg, g))))
self.append("To find {}:".format(
self.format_math(ff.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.numerstep)
self.append("To find {}:".format(
self.format_math(gg.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.denomstep)
self.append("Now plug in to the quotient rule:")
self.append(self.format_math(diff(rule)))
def test_pmint_logexp():
f = (1 + x + x*exp(x))*(x + log(x) + exp(x) - 1)/(x + log(x) + exp(x))**2/x
g = log(x**2 + 2*x*exp(x) + 2*x*log(x) + exp(2*x) + 2*exp(x)*log(x) + log(x)**2)/2 + 1/(x + exp(x) + log(x))
# TODO: Optimal solution is g = 1/(x + log(x) + exp(x)) + log(x + log(x) + exp(x)),
# but SymPy requires a lot of guidance to properly simplify heurisch() output.
assert ratsimp(heurisch(f, x)) == g
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_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_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_pmint_logexp():
f = (1 + x + x * exp(x)) * (x + log(x) + exp(x) - 1) / (x + log(x) +
exp(x))**2 / x
g = log(x**2 + 2 * x * exp(x) + 2 * x * log(x) + exp(2 * x) +
2 * exp(x) * log(x) + log(x)**2) / 2 + 1 / (x + exp(x) + log(x))
# TODO: Optimal solution is g = 1/(x + log(x) + exp(x)) + log(x + log(x) + exp(x)),
# but SymPy requires a lot of guidance to properly simplify heurisch() output.
assert ratsimp(heurisch(f, x)) == g
def capacitance_matrix_variables(self, symbolic=False):
"""
Calculates the capacitance matrix for the energy term of the qubit Lagrangian in the variable respresentation.
"""
if symbolic:
C = self.linear_coordinate_transform.T*self.capacitance_matrix(symbolic)*self.linear_coordinate_transform
C = sympy.Matrix([sympy.nsimplify(sympy.ratsimp(x)) for x in C]).reshape(*(C.shape))
else:
C = np.einsum('ji,jk,kl->il', self.linear_coordinate_transform,self.capacitance_matrix(symbolic),self.linear_coordinate_transform)
return C
def lagrange():
polynom = 0 # Polynomial
for i in range(n):
element = Ys[i] # Element of one sum
for k in range(n):
if i != k:
element *= (x - Xs[k]) / (Xs[i] - Xs[k])
polynom += element
polynom = ratsimp(polynom) # Simplify the function
print(polynom)
return polynom
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 _simpsol(soleq):
lhs = soleq.lhs
sol = soleq.rhs
sol = powsimp(sol)
gens = list(sol.atoms(exp))
p = Poly(sol, *gens, expand=False)
gens = [factor_terms(g) for g in gens]
if not gens:
gens = p.gens
syms = [Symbol('C1'), Symbol('C2')]
terms = []
for coeff, monom in zip(p.coeffs(), p.monoms()):
coeff = piecewise_fold(coeff)
if type(coeff) is Piecewise:
coeff = Piecewise(*((ratsimp(coef).collect(syms), cond)
for coef, cond in coeff.args))
else:
coeff = ratsimp(coeff).collect(syms)
monom = Mul(*(g**i for g, i in zip(gens, monom)))
terms.append(coeff * monom)
return Eq(lhs, Add(*terms))
def test_trigsimp1():
x, y = symbols('x,y')
assert trigsimp(1 - sin(x)**2) == cos(x)**2
assert trigsimp(1 - cos(x)**2) == sin(x)**2
assert trigsimp(sin(x)**2 + cos(x)**2) == 1
assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2
assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2
assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1
assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2
assert trigsimp(1/sin(x)**2 - 1) == cot(x)**2
assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1
assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5
assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) in \
[2 + 3*cos(x/2)**2, 5 - 3*sin(x/2)**2]
assert trigsimp(sin(x)/cos(x)) == tan(x)
assert trigsimp(2*tan(x)*cos(x)) == 2*sin(x)
assert trigsimp(cot(x)**3*sin(x)**3) == cos(x)**3
assert trigsimp(y*tan(x)**2/sin(x)**2) == y/cos(x)**2
assert trigsimp(cot(x)/cos(x)) == 1/sin(x)
assert trigsimp(sin(x + y) + sin(x - y)) == 2*sin(x)*cos(y)
assert trigsimp(sin(x + y) - sin(x - y)) == 2*sin(y)*cos(x)
assert trigsimp(cos(x + y) + cos(x - y)) == 2*cos(x)*cos(y)
assert trigsimp(cos(x + y) - cos(x - y)) == -2*sin(x)*sin(y)
assert ratsimp(trigsimp(tan(x + y) - tan(x)/(1 - tan(x)*tan(y)))) == \
-tan(y)/(tan(x)*tan(y) -1)
assert trigsimp(sinh(x + y) + sinh(x - y)) == 2*sinh(x)*cosh(y)
assert trigsimp(sinh(x + y) - sinh(x - y)) == 2*sinh(y)*cosh(x)
assert trigsimp(cosh(x + y) + cosh(x - y)) == 2*cosh(x)*cosh(y)
assert trigsimp(cosh(x + y) - cosh(x - y)) == 2*sinh(x)*sinh(y)
assert ratsimp(trigsimp(tanh(x + y) - tanh(x)/(1 + tanh(x)*tanh(y)))) == \
tanh(y)/(tanh(x)*tanh(y) + 1)
assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1
e = 2*sin(x)**2 + 2*cos(x)**2
assert trigsimp(log(e), deep=True) == log(2)
def test_trigsimp1():
x, y = symbols('x,y')
assert trigsimp(1 - sin(x)**2) == cos(x)**2
assert trigsimp(1 - cos(x)**2) == sin(x)**2
assert trigsimp(sin(x)**2 + cos(x)**2) == 1
assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2
assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2
assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1
assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2
assert trigsimp(1/sin(x)**2 - 1) == cot(x)**2
assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1
assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5
assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) in \
[2 + 3*cos(x/2)**2, 5 - 3*sin(x/2)**2]
assert trigsimp(sin(x)/cos(x)) == tan(x)
assert trigsimp(2*tan(x)*cos(x)) == 2*sin(x)
assert trigsimp(cot(x)**3*sin(x)**3) == cos(x)**3
assert trigsimp(y*tan(x)**2/sin(x)**2) == y/cos(x)**2
assert trigsimp(cot(x)/cos(x)) == 1/sin(x)
assert trigsimp(sin(x + y) + sin(x - y)) == 2*sin(x)*cos(y)
assert trigsimp(sin(x + y) - sin(x - y)) == 2*sin(y)*cos(x)
assert trigsimp(cos(x + y) + cos(x - y)) == 2*cos(x)*cos(y)
assert trigsimp(cos(x + y) - cos(x - y)) == -2*sin(x)*sin(y)
assert ratsimp(trigsimp(tan(x + y) - tan(x)/(1 - tan(x)*tan(y)))) == \
-tan(y)/(tan(x)*tan(y) -1)
assert trigsimp(sinh(x + y) + sinh(x - y)) == 2*sinh(x)*cosh(y)
assert trigsimp(sinh(x + y) - sinh(x - y)) == 2*sinh(y)*cosh(x)
assert trigsimp(cosh(x + y) + cosh(x - y)) == 2*cosh(x)*cosh(y)
assert trigsimp(cosh(x + y) - cosh(x - y)) == 2*sinh(x)*sinh(y)
assert ratsimp(trigsimp(tanh(x + y) - tanh(x)/(1 + tanh(x)*tanh(y)))) == \
tanh(y)/(tanh(x)*tanh(y) + 1)
assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1
e = 2*sin(x)**2 + 2*cos(x)**2
assert trigsimp(log(e), deep=True) == log(2)
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_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_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_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 spline():
# Define Xs and Ys as numpy array and assign 'n' a local value
xn = np.asarray(Xs)
yn = np.asarray(Ys)
n = len(xn) - 1
# Define h
hn = np.array([xn[i + 1] - xn[i] for i in range(n)])
# Create matrix (A) and a vector (b) to solve the system
A = np.zeros((n, n))
b = np.zeros(n)
# Calculate the matrix
A[0, 0] = 2 * (hn[0] + hn[n - 1])
A[0, 1] = hn[0]
A[0, -1] = hn[n - 1]
b[0] = 6 * ((yn[1] - yn[0]) / hn[0] - (yn[n] - yn[n - 1]) / hn[n - 1])
for i in range(1, n):
A[i, i] = 2 * (hn[i - 1] + hn[i])
A[i, i - 1] = hn[i - 1]
if i < n - 1:
A[i, i + 1] = hn[i]
b[i] = 6 * ((yn[i + 1] - yn[i]) / hn[i] -
(yn[i] - yn[i - 1]) / hn[i - 1])
A[-1, 0] = hn[n - 1]
# Solve the system that we got (it's not the original that we've got as final result)
solv = np.linalg.solve(A, b)
solv = np.append(solv, solv[0])
# Calculate a, b, c, d
a = yn
b = np.array([
(yn[i + 1] - yn[i]) / hn[i] - (2. * solv[i] + solv[i + 1]) * hn[i] / 6
for i in range(len(hn))
])
c = solv * 0.5
d = np.array([(solv[i + 1] - solv[i]) / hn[i] / 6 for i in range(len(hn))])
print(' Polynomials: ')
sx = []
for i in range(n):
# Create a polynomial and print it
sx.append(
ratsimp(a[i] + b[i] * (x - xn[i]) + c[i] * (x - xn[i])**2 + d[i] *
(x - xn[i])**3))
print('[{}, {}]: {}'.format(Xs[i], Xs[i + 1], sx[i]))
# Show a plot of all calculations
plot(sx)
def capacitance_matrix_variables(self, symbolic=False):
"""
Calculates the capacitance matrix for the energy term of the qubit Lagrangian in the variable representation.
"""
if symbolic:
C = self.linear_coordinate_transform.T * self.capacitance_matrix(
symbolic) * self.linear_coordinate_transform
C = sympy.Matrix([sympy.nsimplify(sympy.ratsimp(x))
for x in C]).reshape(*(C.shape))
else:
C = np.einsum('ji,jk,kl->il', self.linear_coordinate_transform,
self.capacitance_matrix(symbolic),
self.linear_coordinate_transform)
return C
def test_action_verbs():
assert nsimplify((1/(exp(3*pi*x/5)+1))) == (1/(exp(3*pi*x/5)+1)).nsimplify()
assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp()
assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep = True)
assert radsimp(1/(2+sqrt(2))) == (1/(2+sqrt(2))).radsimp()
assert powsimp(x**y*x**z*y**z, combine='all') == (x**y*x**z*y**z).powsimp(combine='all')
assert simplify(x**y*x**z*y**z) == (x**y*x**z*y**z).simplify()
assert together(1/x + 1/y) == (1/x + 1/y).together()
assert separate((x*(y*z)**3)**2) == ((x*(y*z)**3)**2).separate()
assert collect(a*x**2 + b*x**2 + a*x - b*x + c, x) == (a*x**2 + b*x**2 + a*x - b*x + c).collect(x)
assert apart(y/(y+2)/(y+1), y) == (y/(y+2)/(y+1)).apart(y)
assert combsimp(y/(x+2)/(x+1)) == (y/(x+2)/(x+1)).combsimp()
assert factor(x**2+5*x+6) == (x**2+5*x+6).factor()
assert refine(sqrt(x**2)) == sqrt(x**2).refine()
assert cancel((x**2+5*x+6)/(x+2)) == ((x**2+5*x+6)/(x+2)).cancel()
def test_action_verbs():
assert nsimplify((1/(exp(3*pi*x/5)+1))) == (1/(exp(3*pi*x/5)+1)).nsimplify()
assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp()
assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep = True)
assert radsimp(1/(2+sqrt(2))) == (1/(2+sqrt(2))).radsimp()
assert powsimp(x**y*x**z*y**z, combine='all') == (x**y*x**z*y**z).powsimp(combine='all')
assert simplify(x**y*x**z*y**z) == (x**y*x**z*y**z).simplify()
assert together(1/x + 1/y) == (1/x + 1/y).together()
assert separate((x*(y*z)**3)**2) == ((x*(y*z)**3)**2).separate()
assert collect(a*x**2 + b*x**2 + a*x - b*x + c, x) == (a*x**2 + b*x**2 + a*x - b*x + c).collect(x)
assert apart(y/(y+2)/(y+1), y) == (y/(y+2)/(y+1)).apart(y)
assert combsimp(y/(x+2)/(x+1)) == (y/(x+2)/(x+1)).combsimp()
assert factor(x**2+5*x+6) == (x**2+5*x+6).factor()
assert refine(sqrt(x**2)) == sqrt(x**2).refine()
assert cancel((x**2+5*x+6)/(x+2)) == ((x**2+5*x+6)/(x+2)).cancel()
def newton():
polynom = 0 # Polynomial
for i in range(n):
if i == 0:
element = f.subs(
x, Xs[0]
) # If it's the first element, which can't be multiplied like others
else:
asd = [Xs[j] for j in range(i + 1)]
element = findmult(asd)
for k in range(i):
element *= (x - Xs[k])
polynom += element
polynom = ratsimp(polynom) # Simplify the function
print(polynom)
return polynom
def print_Div(self, rule):
with self.new_step():
f, g = rule.numerator, rule.denominator
fp, gp = f.diff(rule.symbol), g.diff(rule.symbol)
x = rule.symbol
ff = sympy.Function("f")(x)
gg = sympy.Function("g")(x)
qrule_left = sympy.Derivative(ff / gg, rule.symbol)
qrule_right = sympy.ratsimp(sympy.diff(sympy.Function("f")(x) /
sympy.Function("g")(x)))
qrule = sympy.Eq(qrule_left, qrule_right)
self.append("Apply the quotient rule, which is:")
self.append(self.format_math_display(qrule))
self.append("{} and {}.".format(self.format_math(sympy.Eq(ff, f)),
self.format_math(sympy.Eq(gg, g))))
self.append("To find {}:".format(self.format_math(ff.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.numerstep)
self.append("To find {}:".format(self.format_math(gg.diff(rule.symbol))))
with self.new_level():
self.print_rule(rule.denomstep)
self.append("Now plug in to the quotient rule:")
self.append(self.format_math(diff(rule)))
def test_pmint_logexp():
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
formula = recursion(m,n,base)
formula = (formula.subs([(alpha,1/a),(beta,1/b)]))
functions.append(("%s_%d_%d"%(name,m,n), formula))
# Analytical expressions are singular for alpha=beta.
# When alpha \approx beta, we therefore compute the expression
# by writing beta=alpha+epsilon and writing a Taylor expansion
# in epsilon.
if m==0 and n==0:
# The Taylor expansion in epsilon is computed only for the
# base integral...
form = base.subs(beta,alpha+epsilon)
series_exp = 1
for itaylor in xrange(1,4*mmax+2):
series_exp += (-epsilon*R)**itaylor/factorial(itaylor)
form = form.subs(exp(-R*(alpha+epsilon)),exp(-R*alpha)*series_exp)
form = ratsimp(form)
taylor_expansion = simplify(form.subs(epsilon,0))
for itaylor in xrange(4*mmax+1):
form = ratsimp(diff(form,epsilon))
taylor_expansion += simplify(form.subs(epsilon, 0)*epsilon**(itaylor+1)/factorial(itaylor+1))
#print taylor_expansion
# ...other Taylor expansions are derived using a modified recursion scheme.
formula_taylor = simplify(recursiontaylor(m,n,taylor_expansion))
functions.append(("%s_taylor_%d_%d"%(name,m,n), formula_taylor.subs([(alpha,1/a),(epsilon,b)])))
# Write routines in Fortran source code
print "Writing source code..."
codegen(functions, "F95", "../src/libslater",to_files=True,
argument_sequence=(a,b,R), project='libslater')
# Write python wrapper
with open('../src/libslater.py','w') as f:
f.write('''#!/usr/bin/env python\n\n''')
n=500
x0,x1,y0,y1=sp.symbols('x0 x1 y0 y1',real=True)
sx=[x0,x1]
sy=[y0,y1]
#Kx=1
#Py=1
Kx=x1**2
Py=x1**2
#Phi1NL=-0.25/sp.pi*sp.log((x0-y0)**2+(x1-y1)**2)
Phi1NL = sp.log(((x0-y0)**2+(x1+y1)**2)/((x0-y0)**2+(x1-y1)**2)) / (sp.pi*2*x1*y1)
omgNL=[ Py*sp.diff(Phi1NL,sy[j]) for j in range(2) ]
#Phi2NL=-0.5/sp.pi*sp.atan((x1-y1)/(x0-y0))
Phi2NL = sp.log(((x0-y0)**2+(x1+y1)**2)/((x0-y0)**2+(x1-y1)**2)) * (x0-y0)/(2*sp.pi*x1)
+ (sp.atan((x1-y1)/(x0-y0)) - sp.atan((x1+y1)/(x0-y0))) / (2*sp.pi)
V2NL=[sp.ratsimp(Kx*sp.diff(Phi2NL,sx[j])) for j in range(2)]
VSNL=[sp.ratsimp(Kx*sp.diff(Phi1NL,sx[j])) for j in range(2)]
def Normal(x):
n=x[0].shape[0]-1
return np.array([-x[1,1:]+x[1,:n],x[0,1:]-x[0,:n]])
def normal(x):
nr=Normal(x)
return nr/np.sqrt(nr[0]**2+nr[1]**2)
def withoutZero(sFunct):
def wrapped(sExpress,M,minDist=myeps):
gd=np.sum([(M[i,1]-M[i,0])**2 for i in range(2)],axis=0)>minDist**2
AA=M[0,0].copy()
AA[np.logical_not(gd)]+=1
def integral_basis(f,x,y):
"""
Compute the integral basis {b1, ..., bg} of the algebraic function
field C[x,y] / (f).
"""
# If the curve is not monic then map y |-> y/lc(x) where lc(x)
# is the leading coefficient of f
T = sympy.Dummy('T')
d = sympy.degree(f,y)
lc = sympy.LC(f,y)
if x in lc:
f = sympy.ratsimp( f.subs(y,y/lc)*lc**(d-1) )
else:
f = f/lc
lc = 1
#
# Compute df
#
p = sympy.Poly(f,[x,y])
n = p.degree(y)
res = sympy.resultant(p,p.diff(y),y)
factors = sympy.factor_list(res)[1]
df = [k for k,deg in factors if (deg > 1) and (sympy.LC(k) == 1)]
#
# Compute series truncations at appropriate x points
#
alpha = []
r = []
for l in range(len(df)):
k = df[l]
alphak = sympy.roots(k).keys()[0]
rk = compute_series_truncations(f,x,y,alphak,T)
alpha.append(alphak)
r.append(rk)
#
# Main Loop
#
a = [sympy.Dummy('a%d'%k) for k in xrange(n)]
b = [1]
for d in range(1,n):
bd = y*b[-1]
for l in range(len(df)):
k = df[l]
alphak = alpha[l]
rk = r[l]
found_something = True
while found_something:
# construct system of equations consisting of the coefficients
# of negative powers of (x-alphak) in the substitutions
# A(r_{k,1}),...,A(r_{k,n})
A = (sum(ak*bk for ak,bk in zip(a,b)) + bd) / (x - alphak)
coeffs = []
for rki in rk:
# substitute and extract coefficients
A_rki = A.subs(y,rki)
coeffs.extend(_negative_power_coeffs(A_rki, x, alphak))
# solve the coefficient equations for a0,...,a_{d-1}
coeffs = [coeff.as_numer_denom()[0] for coeff in coeffs]
sols = sympy.solve_poly_system(coeffs, a[:d])
if sols is None or sols == []:
found_something = False
else:
sol = sols[0]
bdm1 = sum( sol[i]*bk for i,bk in zip(range(d),b) )
bd = (bdm1 + bd) / k
# bd found. Append to list of basis elements
b.append( bd )
# finally, convert back to singularized curve if necessary
for i in xrange(1,len(b)):
b[i] = b[i].subs(y,y*lc).ratsimp()
return b
def test_ratsimp2():
x = Symbol("x")
e = 1/(1+1/x)
assert (x+1)*ratsimp(e)/x == 1
def test_ratsimp_X1():
e = -x-y-(x+y)**(-1)*y**2+(x+y)**(-1)*x**2
assert e != -2*y
assert ratsimp(e) == -2*y
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 getCumulants(model, chords, param=[], simp=[], unsimp=[]):
doSimplify=not(simp==[] and unsimp == [])
start_time = time.time()
if( not isConsistent(model,chords) ):
print(" ERROR: Model and chords are not correct or inconsistent. ")
return( False )
### number of states
N = len(getStateSpace(model))
### The first Betti number (cyclomatic number) of the model, equals the number of chords
B = len(chords)
### Generate transition matrix W from model
W = zeros(N)
for edge in model:
W[edge] = (model[edge])
for j in range(N):
W[j,j] = -sum(W[j,i] for i in range(N))
#display(W)
### Generate tilted matrix, from copy of W
Wq = W[:,:]
q=zeros(B,1)
for i in range(B):
name = 'q_{{{0}}}'.format(i)
q[i] = symbols(name)
Wq[chords[i]] = W[chords[i]]*exp(q[i])
Wq[chords[i][::-1]] = W[chords[i][::-1]]*exp(-q[i])
#display(Wq)
### Find coefficients of characteristic polynomial
print("--- %s seconds ---" % (time.time() - start_time))
print("Start calculating characteristic polynomial")
a = simplify(Wq.berkowitz()[-1][::-1]) # symbolic simplification is *crucial* here!
### Initialize current vector and covariance matrix
c = zeros(B,1)
C = zeros(B,B)
### Calculate current vector
print("--- %s seconds ---" % (time.time() - start_time))
print("Start calculating current vector")
for i in range(B):
c[i] = -(diff(a[0],q[i])/a[1]).ratsimp() ## populate current vector
for i in range(B):
c=c.subs(q[i],0) ## subsitute q=0
if(doSimplify):
c = c.subs(param)
print("--- %s seconds ---" % (time.time() - start_time))
print("Start simplifying current vector")
c = simplify(c.subs(simp)) #simplify cancel
### Calculate co-variance matrix
print("--- %s seconds ---" % (time.time() - start_time))
print("Start calculating covariance matrix")
### Do in-place parametrization, before simplification, if latter is demanded
if(doSimplify):
for i in range(B):
for j in range(i+1):
t1 = ratsimp( diff(a[0],q[i],q[j]).subs(param).subs(simp) )
t2 = ratsimp( (diff(a[1],q[i])*c[j]).subs(param).subs(simp) )
t3 = ratsimp( (diff(a[1],q[j])*c[i]).subs(param).subs(simp) )
t4 = ratsimp( (2*a[2]*c[i]*c[j]).subs(param).subs(simp) )
t5 = ratsimp( a[1].subs(param).subs(simp) )
C[i,j] = -(t1 + t2 + t3 + t4)/t5
else:
for i in range(B):
for j in range(i+1):
t1 = ( diff(a[0],q[i],q[j]) )#.ratsimp()
t2 = ( (diff(a[1],q[i])*c[j]) )#.ratsimp()
t3 = ( (diff(a[1],q[j])*c[i]) )#.ratsimp()
t4 = ( (2*a[2]*c[i]*c[j]) )#.ratsimp()
t5 = ( a[1] )#.ratsimp()
C[i,j] = -(t1 + t2 + t3 + t4)/t5
### Populate Covariance Matrix
## If no simplification, perform parametrization now
if(not doSimplify):
c = c.subs(param)
### simplification of the expectation should be safe to do, in any case
c = simplify(c)
### simplification of covariance is possibly very time consuming
C = C.subs(param)
if(doSimplify):
print("--- %s seconds ---" % (time.time() - start_time))
print("Start simplifying covariance matrix")
C = simplify(C)
### subsitute q=0 into covariance
for i in range(B):
C = C.subs(q[i],0)
print("--- %s seconds ---" % (time.time() - start_time))
print("All Done")
### Symmetrize covariance matrix:
for i in range(B):
for j in range(i):
C[j,i] = C[i,j]
### return unsimplified expecation and covariance
return( [c.subs(unsimp),C.subs(unsimp)])
def test_ratsimp_X2():
e = x/(x+y)+y/(x+y)
assert e != 1
assert ratsimp(e) == 1
def getCumulants(model, chords, param=[], simp=[], unsimp=[]):
doSimplify = not (simp == [] and unsimp == [])
start_time = time.time()
if (not isConsistent(model, chords)):
print(" ERROR: Model and chords are not correct or inconsistent. ")
return (False)
### number of states
N = len(getStateSpace(model))
### The first Betti number (cyclomatic number) of the model, equals the number of chords
B = len(chords)
### Generate transition matrix W from model
W = zeros(N)
for edge in model:
W[edge] = (model[edge])
for j in range(N):
W[j, j] = -sum(W[j, i] for i in range(N))
#display(W)
### Generate tilted matrix, from copy of W
Wq = W[:, :]
q = zeros(B, 1)
for i in range(B):
name = 'q_{{{0}}}'.format(i)
q[i] = symbols(name)
Wq[chords[i]] = W[chords[i]] * exp(q[i])
Wq[chords[i][::-1]] = W[chords[i][::-1]] * exp(-q[i])
#display(Wq)
### Find coefficients of characteristic polynomial
print("--- %s seconds ---" % (time.time() - start_time))
print("Start calculating characteristic polynomial")
a = simplify(
Wq.berkowitz()[-1][::-1]) # symbolic simplification is *crucial* here!
### Initialize current vector and covariance matrix
c = zeros(B, 1)
C = zeros(B, B)
### Calculate current vector
print("--- %s seconds ---" % (time.time() - start_time))
print("Start calculating current vector")
for i in range(B):
c[i] = -(diff(a[0], q[i]) / a[1]).ratsimp() ## populate current vector
for i in range(B):
c = c.subs(q[i], 0) ## subsitute q=0
if (doSimplify):
c = c.subs(param)
print("--- %s seconds ---" % (time.time() - start_time))
print("Start simplifying current vector")
c = simplify(c.subs(simp)) #simplify cancel
### Calculate co-variance matrix
print("--- %s seconds ---" % (time.time() - start_time))
print("Start calculating covariance matrix")
### Do in-place parametrization, before simplification, if latter is demanded
if (doSimplify):
for i in range(B):
for j in range(i + 1):
t1 = ratsimp(diff(a[0], q[i], q[j]).subs(param).subs(simp))
t2 = ratsimp((diff(a[1], q[i]) * c[j]).subs(param).subs(simp))
t3 = ratsimp((diff(a[1], q[j]) * c[i]).subs(param).subs(simp))
t4 = ratsimp((2 * a[2] * c[i] * c[j]).subs(param).subs(simp))
t5 = ratsimp(a[1].subs(param).subs(simp))
C[i, j] = -(t1 + t2 + t3 + t4) / t5
else:
for i in range(B):
for j in range(i + 1):
t1 = (diff(a[0], q[i], q[j])) #.ratsimp()
t2 = ((diff(a[1], q[i]) * c[j])) #.ratsimp()
t3 = ((diff(a[1], q[j]) * c[i])) #.ratsimp()
t4 = ((2 * a[2] * c[i] * c[j])) #.ratsimp()
t5 = (a[1]) #.ratsimp()
C[i, j] = -(t1 + t2 + t3 + t4) / t5
### Populate Covariance Matrix
## If no simplification, perform parametrization now
if (not doSimplify):
c = c.subs(param)
### simplification of the expectation should be safe to do, in any case
c = simplify(c)
### simplification of covariance is possibly very time consuming
C = C.subs(param)
if (doSimplify):
print("--- %s seconds ---" % (time.time() - start_time))
print("Start simplifying covariance matrix")
C = simplify(C)
### subsitute q=0 into covariance
for i in range(B):
C = C.subs(q[i], 0)
print("--- %s seconds ---" % (time.time() - start_time))
print("All Done")
### Symmetrize covariance matrix:
for i in range(B):
for j in range(i):
C[j, i] = C[i, j]
### return unsimplified expecation and covariance
return ([c.subs(unsimp), C.subs(unsimp)])
def emit(name, func):
for x in sorted(exparg):
test(name,
x,
sp.ratsimp(sp.radsimp(func(x).rewrite(sp.exp))),
no_trigh=True)
def test_ratsimp_X2():
e = x/(x+y)+y/(x+y)
assert e != 1
assert ratsimp(e) == 1
def test_pmint_logexp():
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
from sympy import Symbol, simplify, ratsimp, sympify, factor, limit
from numpy import array, zeros
from copy import copy
from pycircuit.circuit.symbolicapprox import *
## Create regulated cascode circuit
c = SubCircuit()
nin, nout, n1 = c.add_nodes('in', 'out', 'n1')
gm1, gm2, gds1, gds2, Cgs1, Cgs2= [Symbol(symname, real=True) for symname in 'gm1,gm2,gds1,gds2,Cgs1,Cgs2'.split(',')]
c['M2'] = MOS(nin, n1, gnd, gnd, gm = gm2, gds = gds2, Cgs=0*Cgs1)
c['M1'] = MOS(n1, nout, nin, nin, gm = gm1, gds = gds1, Cgs=0*Cgs2)
#c['r'] = R(nin, gnd, r = Symbol('Rs', real=True))
## Perform twoport analysis with noise
twoportana = TwoPortAnalysis(c, nin, gnd, nout, gnd, noise=True, noise_outquantity='i', toolkit=symbolic)
res2port = twoportana.solve(Symbol('s'), complexfreq=True)
y11 = res2port['twoport'].Y[0,0]
print 'Input impedance:', 1/y11
#print 'Approx. input impedance', approx(1/y11, ['gds'], n = 1)
print 'Input referred current noise PSD, Sin:', ratsimp(res2port['Sin'])
print 'Approx. input referred current noise PSD, Sin:', approx(res2port['Sin'], ['gds'], n=1)
print 'Input referred voltage noise PSD, Svn:', ratsimp(res2port['Svn'])
def test_ratsimp():
assert ratsimp(A*B - B*A) == A*B - B*A
def emit(name, func):
for x in sorted(exparg):
test(name, x, sp.ratsimp(sp.radsimp(func(x).rewrite(sp.exp))),
no_trigh=True)
print(S.simplify((x**3 + x**2 - x - 1) / (x**2 + 2 * x + 1)))
#分解--表达式进行部分分式分解，它将一个有理函数变为数个分子及分母次数较小的有理函数
print("分解")
print(S.apart(1 / ((x + 2) * (x + 1)), x))
#三角函数
print(S.sin(x + y).expand(trig=True))
#化简-去括号
f = (a - b + c) * (a + b - c)
print(S.expand(f))
#因式分解
print(S.factor(a**2 - b**2 + 2 * b * c - c**2))
#通分运算 ratsimp()对表达式中的分母进行通分运算，即将表达式转换为分子除分母的形式
print(S.ratsimp(x / (x + y) + y / (x - y)))
print(S.ratsimp(x / a + x / b))
#分母有理化
#radsimp()对表达式的分母进行有理化，结果中的分母部分不含无理数.
print(S.ratsimp(1 / (S.sqrt(5))))
#fraction()返回包含表达式的分子与分母的元组， 用它可以获得ratsimp()通分之后的分子或分母
print(S.fraction(S.ratsimp(1 / x + 1 / y)))
#化简三角函数
#trigsimp()化简表达式中的三角函数，通过method参数可以选择化简算法.
print(S.trigsimp(S.sin(x)**2 + S.cos(x)**2 + 2 * S.cos(x)))
def matratsimp(mat):
return sympy.matrices.Matrix(mat.rows, mat.cols, lambda i, j: sympy.ratsimp(mat[i, j]))
def test_ratsimp2():
x = Symbol("x")
e = 1/(1+1/x)
assert (x+1)*ratsimp(e)/x == 1
def test_ratsimp_X1():
e = -x-y-(x+y)**(-1)*y**2+(x+y)**(-1)*x**2
assert e != -2*y
assert ratsimp(e) == -2*y
Symbol(symname, real=True)
for symname in 'gm1,gm2,gds1,gds2,Cgs1,Cgs2'.split(',')
]
c['M2'] = MOS(nin, n1, gnd, gnd, gm=gm2, gds=gds2, Cgs=0 * Cgs1)
c['M1'] = MOS(n1, nout, nin, nin, gm=gm1, gds=gds1, Cgs=0 * Cgs2)
#c['r'] = R(nin, gnd, r = Symbol('Rs', real=True))
## Perform twoport analysis with noise
twoportana = TwoPortAnalysis(c,
nin,
gnd,
nout,
gnd,
noise=True,
noise_outquantity='i',
toolkit=symbolic)
res2port = twoportana.solve(Symbol('s'), complexfreq=True)
y11 = res2port['twoport'].Y[0, 0]
print 'Input impedance:', 1 / y11
#print 'Approx. input impedance', approx(1/y11, ['gds'], n = 1)
print 'Input referred current noise PSD, Sin:', ratsimp(res2port['Sin'])
print 'Approx. input referred current noise PSD, Sin:', approx(res2port['Sin'],
['gds'],
n=1)
print 'Input referred voltage noise PSD, Svn:', ratsimp(res2port['Svn'])
#!/usr/bin/python
import numpy as np
import sympy as sp
myeps=0.000001
x0,x1,y0,y1=sp.symbols('x0 x1 y0 y1',real=True)
sx=[x0,x1]
sy=[y0,y1]
_Kx_=1
_Py_=1
_Phi1_=-0.25/sp.pi*sp.log((x0-y0)**2+(x1-y1)**2)
_sOmega_=[_Py_*sp.diff(_Phi1_,sy[j]) for j in range(2)]
_Phi2_=-0.5/sp.pi*sp.atan((x1-y1)/(x0-y0))
_sV2_=[sp.ratsimp(_Kx_*sp.diff(_Phi2_,sx[j])) for j in range(2)]
_sVS_=[_Kx_*sp.diff(_Phi1_,sx[j]) for j in range(2)]
def Normal(x):
n=x[0].shape[0]-1
return np.array([-x[1,1:]+x[1,:n],x[0,1:]-x[0,:n]])
def normal(x):
nr=Normal(x)
return nr/np.sqrt(nr[0]**2+nr[1]**2)
def withoutZero(sFunct):
def wrapped(sExpress,M,minDist=myeps):
gd=np.sum([(M[i,1]-M[i,0])**2 for i in range(2)],axis=0)>minDist**2
AA=M[0,0].copy()
AA[np.logical_not(gd)]+=1
result=np.zeros_like(M[0,0])
result[gd]=sFunct(sExpress,np.array([[AA,M[0,1]],[M[1,0],M[1,1]]]),minDist)[gd]