def main():
x=Symbol("x")
s = Poly(A(x), x)
num = [s.coeff(n) for n in range(11)]
print s.as_basic()
print num
def main():
x = Symbol("x")
s = Poly(A(x), x)
num = list(reversed(s.coeffs()))[:11]
print(s.as_expr())
print(num)
def gauss_lobatto_points(p):
"""
Returns the list of Gauss-Lobatto points of the order 'p'.
"""
x = Symbol("x")
print "creating"
e = legendre(p, x).diff(x)
e = Poly(e, x)
print "polydone"
if e == 0:
return []
print "roots"
if e == 1:
r = []
else:
with workdps(40):
r, err = polyroots(e.all_coeffs(), error=True)
#if err > 1e-40:
# raise Exception("Internal Error: Root is not precise")
print "done"
p = []
p.append("-1.0")
for x in r:
if abs(x) < 1e-40: x = 0
p.append(str(x))
p.append("1.0")
return p
def _eval_expand_func(self, **hints):
from sympy import exp, I, floor, Add, Poly, Dummy, exp_polar, unpolarify
z, s, a = self.args
if z == 1:
return zeta(s, a)
if s.is_Integer and s <= 0:
t = Dummy('t')
p = Poly((t + a)**(-s), t)
start = 1/(1 - t)
res = S(0)
for c in reversed(p.all_coeffs()):
res += c*start
start = t*start.diff(t)
return res.subs(t, z)
if a.is_Rational:
# See section 18 of
# Kelly B. Roach. Hypergeometric Function Representations.
# In: Proceedings of the 1997 International Symposium on Symbolic and
# Algebraic Computation, pages 205-211, New York, 1997. ACM.
# TODO should something be polarified here?
add = S(0)
mul = S(1)
# First reduce a to the interaval (0, 1]
if a > 1:
n = floor(a)
if n == a:
n -= 1
a -= n
mul = z**(-n)
add = Add(*[-z**(k - n)/(a + k)**s for k in xrange(n)])
elif a <= 0:
n = floor(-a) + 1
a += n
mul = z**n
add = Add(*[z**(n - 1 - k)/(a - k - 1)**s for k in xrange(n)])
m, n = S([a.p, a.q])
zet = exp_polar(2*pi*I/n)
root = z**(1/n)
return add + mul*n**(s - 1)*Add(
*[polylog(s, zet**k*root)._eval_expand_func(**hints)
/ (unpolarify(zet)**k*root)**m for k in xrange(n)])
# TODO use minpoly instead of ad-hoc methods when issue 2789 is fixed
if z.func is exp and (z.args[0]/(pi*I)).is_Rational or z in [-1, I, -I]:
# TODO reference?
if z == -1:
p, q = S([1, 2])
elif z == I:
p, q = S([1, 4])
elif z == -I:
p, q = S([-1, 4])
else:
arg = z.args[0]/(2*pi*I)
p, q = S([arg.p, arg.q])
return Add(*[exp(2*pi*I*k*p/q)/q**s*zeta(s, (k + a)/q)
for k in xrange(q)])
return lerchphi(z, s, a)
def test_squarefree():
assert Poly(x-1, x).is_squarefree == True
assert Poly((x-1)**2, x).is_squarefree == False
assert Poly(3*x**2, x).as_squarefree() == Poly(3*x, x)
assert Poly(x**2+2*x+1, x).as_squarefree() == Poly(x+1, x)
assert Poly(x**5-x**4-x+1, x).as_squarefree() == Poly(x**4-1, x)
assert poly_sqf(1, x) == [Poly(1, x)]
assert poly_sqf(x, x) == [Poly(x, x)]
assert poly_sqf(3*x**2, x) == [Poly(3, x), Poly(x, x)]
assert poly_sqf(x**2+2*x+1, x) == [Poly(1, x), Poly(x+1, x)]
assert poly_sqf(x**5-x**4-x+1, x) == \
[Poly(x**3 + x**2 + x + 1, x), Poly(x-1, x)]
assert poly_sqf(x**8+6*x**6+12*x**4+8*x**2, x) == \
[Poly(1, x), Poly(x, x), Poly(x**2+2, x)]
# Bronstein, Symbolic Integration, pp. 52
A = Poly(x**4 - 3*x**2 + 6, x)
D = Poly(x**6 - 5*x**4 + 5*x**2 + 4, x)
f, g = D, A - D.diff(x).mul_term(t)
res, R = poly_subresultants(f, g)
S = poly_sqf(Poly(res, t))
assert S == [Poly(45796, t), Poly(1, t), Poly(4*t**2 + 1, t)]
def main():
x=Symbol("x")
s = Poly(A(x), x)
num = list(reversed(s.coeffs()))[:11]
print s.as_basic()
print num
def linear_arg(arg):
""" Test if arg is of form a*s+b, raise exception if not. """
if not arg.is_polynomial(s):
raise exception(fact)
p = Poly(arg, s)
if p.degree() != 1:
raise exception(fact)
return p.all_coeffs()
def test_coeff():
p = Poly(3*x**2*y + 4*x*y**2 + 1, x, y)
assert p.coeff(0, 0) == p.coeff() == 1
assert p.coeff(2, 1) == 3
assert p.coeff(1, 2) == 4
assert p.coeff(1, 1) == 0
def test_has_any():
x,y,z,t,u = symbols('x y z t u')
f = Function("f")
g = Function("g")
p = Wild('p')
assert sin(x).has(x)
assert sin(x).has(sin)
assert not sin(x).has(y)
assert not sin(x).has(cos)
assert f(x).has(x)
assert f(x).has(f)
assert not f(x).has(y)
assert not f(x).has(g)
assert f(x).diff(x).has(x)
assert f(x).diff(x).has(f)
assert f(x).diff(x).has(Derivative)
assert not f(x).diff(x).has(y)
assert not f(x).diff(x).has(g)
assert not f(x).diff(x).has(sin)
assert (x**2).has(Symbol)
assert not (x**2).has(Wild)
assert (2*p).has(Wild)
i = Integer(4400)
assert i.has(x) is False
assert (i*x**i).has(x)
assert (i*y**i).has(x) is False
assert (i*y**i).has(x, y)
expr = x**2*y + sin(2**t + log(z))
assert expr.has(u) is False
assert expr.has(x)
assert expr.has(y)
assert expr.has(z)
assert expr.has(t)
assert expr.has(x, y, z, t)
assert expr.has(x, y, z, t, u)
from sympy.physics.units import m, s
assert (x*m/s).has(x)
assert (x*m/s).has(y, z) is False
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has(x)
assert poly.has(x, y, z)
assert poly.has(x, y, z, t)
assert FockState((x, y)).has(x)
def test_add_sub_term():
f = Poly((), x, y, z)
assert Poly(2, x) - Poly(1, x) == Poly(1, x)
assert Poly(1, x) - Poly(1, x) == Poly(0, x)
assert Poly(1, x) - Poly(2, x) == Poly(-1, x)
assert f.add_term(12, (1,2,3)) == Poly(((12,), ((1,2,3),)), x,y,z)
assert f.sub_term(12, (1,2,3)) == Poly(((-12,), ((1,2,3),)), x,y,z)
def LM(self,poly):
from sympy import S,Poly
if poly==0:return poly
poly = Poly(poly,self.variables)
polyDict = poly.as_dict()
exponents = polyDict.keys()
largest = exponents[0]
for i in xrange(len(polyDict)):
if self.compare(exponents[i],largest): largest = exponents[i]
return Poly({largest:S(1)},self.variables)
def poly_factorize(poly):
"""Factorize multivariate polynomials into a sum of products of monomials.
This function can be used to decompose polynomials into a form which
minimizes the number of additions and multiplications, and which thus
can be evaluated efficently."""
max_deg = {}
if 'horner' in dir(sympy):
return sympy.horner(poly)
if not isinstance(poly, Poly):
poly = Poly(sympy.expand(poly), *poly.atoms(Symbol))
denom, poly = poly.as_integer()
# Determine the order of factorization. We proceed through the
# symbols, starting with the one present in the highest order
# in the polynomial.
for i, sym in enumerate(poly.symbols):
max_deg[i] = 0
for monom in poly.monoms:
for i, symvar in enumerate(monom):
max_deg[i] = max(max_deg[i], symvar)
ret_poly = 0
monoms = list(poly.monoms)
for isym, maxdeg in sorted(max_deg.items(), key=itemgetter(1), reverse=True):
drop_idx = []
new_poly = []
for i, monom in enumerate(monoms):
if monom[isym] > 0:
drop_idx.append(i)
new_poly.append((poly.coeff(*monom), monom))
if not new_poly:
continue
ret_poly += sympy.factor(Poly(new_poly, *poly.symbols))
for idx in reversed(drop_idx):
del monoms[idx]
# Add any remaining O(1) terms.
new_poly = []
for i, monom in enumerate(monoms):
new_poly.append((poly.coeff(*monom), monom))
if new_poly:
ret_poly += Poly(new_poly, *poly.symbols)
return ret_poly / denom
def test_iterators():
f = Poly(x**5 + x, x)
assert list(f.iter_all_coeffs()) == \
[1, 0, 0, 0, 1, 0]
assert list(f.iter_all_monoms()) == \
[(5,), (4,), (3,), (2,), (1,), (0,)]
assert list(f.iter_all_terms()) == \
[(1, (5,)), (0, (4,)), (0, (3,)), (0, (2,)), (1, (1,)), (0, (0,))]
def test_as_poly_as_expr():
f = x ** 2 + 2 * x * y
assert f.as_poly().as_expr() == f
assert f.as_poly(x, y).as_expr() == f
assert (f + sin(x)).as_poly(x, y) is None
p = Poly(f, x, y)
assert p.as_poly() == p
def test_map_coeffs():
p = Poly(x**2 + 2*x*y, x, y)
q = p.map_coeffs(lambda c: 2*c)
assert q.as_basic() == 2*x**2 + 4*x*y
p = Poly(u*x**2 + v*x*y, x, y)
q = p.map_coeffs(expand, complex=True)
assert q.as_basic() == x**2*(I*im(u) + re(u)) + x*y*(I*im(v) + re(v))
raises(PolynomialError, "p.map_coeffs(lambda c: x*c)")
def test_as_poly_basic():
x, y = symbols('xy')
f = x**2 + 2*x*y
assert f.as_poly().as_basic() == f
assert f.as_poly(x, y).as_basic() == f
assert (f + sin(x)).as_poly(x, y) is None
p = Poly(f, x, y)
assert p.as_poly() == p
def reduce(self, debug=False, inputFilename=None, outputFilename=None):
previousLeftBlockRegion = -1
previousRightBlockRegion = -1
equationCount = 0
partIndex = 0
equationInPartIndex = 0
output = None
if debug:
input = open(inputFilename, 'r')
output = open(outputFilename, 'w')
else:
input = sys.stdin
for strEquation in input:
equationCount += 1
index, strEquation = strEquation.strip().split('\t')
equation, symbols = decodeExpression(strEquation)
equation = Poly(equation, symbols)
equationCoefficients = [0] * len(symbols)
for monom, coeff in equation.terms():
try:
i = list(monom).index(1)
equationCoefficients[i] = 1 if coeff != 0 else 0
except ValueError:
continue
equationLen = len(equationCoefficients)
rightColumnIndex = equationLen - 1
leftColumnIndex = 0
while leftColumnIndex < equationLen and equationCoefficients[leftColumnIndex] != 1:
leftColumnIndex += 1
while rightColumnIndex > 0 and equationCoefficients[rightColumnIndex] != 1:
rightColumnIndex -= 1
if previousLeftBlockRegion != leftColumnIndex or previousRightBlockRegion != rightColumnIndex:
if equationInPartIndex > 0:
partIndex += 1
equationInPartIndex = 0
previousLeftBlockRegion = leftColumnIndex
previousRightBlockRegion = rightColumnIndex
if output is not None:
output.write('{0}\t{1}\t{2}\n'.format(str(partIndex), index, strEquation))
else:
print('{0}\t{1}\t{2}'.format(str(partIndex), index, strEquation))
equationInPartIndex += 1
if debug:
input.close()
if output is not None:
output.close()
def _simplify(expr):
u"""Simplifie une expression.
Alias de simplify (sympy 0.6.4).
Mais simplify n'est pas garanti d'être stable dans le temps.
(Cf. simplify.__doc__)."""
return together(expand(Poly.cancel(powsimp(expr))))
def gauss_lobatto_points(p):
"""
Returns the list of Gauss-Lobatto points of the order 'p'.
"""
x = Symbol("x")
e = (1-x**2)*legendre(p, x).diff(x)
e = Poly(e, x)
if e == 0:
return []
with workdps(40):
r, err = polyroots(e.all_coeffs(), error=True)
if err > 1e-40:
raise Exception("Internal Error: Root is not precise")
p = []
for x in r:
if abs(x) < 1e-40: x = 0
p.append(str(x))
return p
def test_mul_div_term():
f = Poly(x*y**2 + 2*y, x, y)
assert f.mul_term(0, (0, 1)) == Poly(0, x, y)
assert f.mul_term(1) == Poly(x*y**2 + 2*y, x, y)
assert f.mul_term(1, (0, 1)) == Poly(x*y**3 + 2*y**2, x, y)
raises(ZeroDivisionError, "f.div_term(0)")
assert f.div_term(1) == Poly(x*y**2 + 2*y, x, y)
assert f.div_term(1, (0, 1)) == Poly(x*y + 2, x, y)
assert f.mul_term(2) == Poly(2*x*y**2 + 4*y, x, y)
assert f.mul_term(2, (0, 1)) == Poly(2*x*y**3 + 4*y**2, x, y)
assert f.div_term(2) == Poly(x*y**2 / 2 + y, x, y)
assert f.div_term(2, (0, 1)) == Poly(x*y / 2 + 1, x, y)
def new_poly(V_s, fields, n_coeffs):
"""Make a new polynomial function that has the same powers as V_s function
but with coefficients C1, C2..."""
from sympy import Poly
from sympy import diff, Symbol, var, simplify, sympify, S
from sympy.core.sympify import SympifyError
P = Poly(V_s,*fields)
d = P.as_dict()
e = {}
for key in d.iterkeys():
#print d[key], str(d[key])
for i in xrange(1, n_coeffs+1):
if 'C' + str(i) in str(d[key]):
e[key] = sympify('C'+str(i))
P2 = Poly(e,*fields)
return str(P2.as_basic())
def get_matrix(self):
"""
Returns
-------
macaulay_matrix: Matrix
The Macaulay's matrix
"""
rows = []
row_coefficients = self.get_row_coefficients()
for i in range(self.n):
for multiplier in row_coefficients[i]:
coefficients = []
poly = Poly(self.polynomials[i] * multiplier,
*self.variables)
for mono in self.monomial_set:
coefficients.append(poly.coeff_monomial(mono))
rows.append(coefficients)
macaulay_matrix = Matrix(rows)
return macaulay_matrix
def discrete_realization_tustin(n0, n1, d0, T):
z = symbols('z')
s = 2/T*(z-1)/(z+1)
num = ((n1*s + n0)*T*(z + 1)).simplify()
den = ((s + d0)*T*(z + 1)).simplify()
num_poly = Poly(num, z)
den_poly = Poly(den, z)
n1_z, n0_z = num_poly.coeffs()
d1_z, d0_z = den_poly.coeffs()
# Make denominator monic and divide numerator appropriately
n1_z /= d1_z
n0_z /= d1_z
d0_z /= d1_z
a = -d0_z
b_times_c = (n0_z - n1_z * d0_z).simplify()
d = n1_z
return a, b_times_c, d
def test_has_all():
x,y,z,t,u = symbols('xyztu')
u = symbols('u')
i = Integer(4400)
assert i.has(x, all=True) is False
assert (i*x**i).has(x, all=True)
assert (i*y**i).has(x, all=True) is False
expr = x**2*y + sin(2**t + log(z))
assert expr.has(y, z, t, all=True)
assert expr.has(x, z, t, all=True)
assert expr.has(x, y, t, all=True)
assert expr.has(x, y, z, all=True)
assert expr.has(y, u, t, all=True) is False
assert expr.has(x, z, u, all=True) is False
assert expr.has(u, y, z, all=True) is False
assert expr.has(x, y, z, t, all=True)
assert expr.has(x, y, z, t, u, all=True) is False
from sympy.physics.units import m, s
assert (x*m/s).has(x, all=True)
assert (x*m/s).has(x, y, all=True) is False
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has(x, y, z, all=True)
assert poly.has(x, y, z, t, all=True) is False
f = FockState((x, y))
assert f.has(x, y, all=True)
assert f.has(x, y, z, all=True) is False
def test_unify():
p = Poly(x**2+x*y, x, y)
q = Poly(x**2+2*x*y+1, x)
r = Poly(x*z+y*z+1, x,y,z)
assert p.unify_with(q) == p.unify_with(x**2+2*x*y+1) == \
(Poly(x**2+x*y, x, y), Poly(x**2+2*x*y+1, x, y))
assert p.unify_with([q, r]) == p.unify_with([x**2+2*x*y+1, r]) == \
(Poly(x**2+x*y, x, y, z), [Poly(x**2+2*x*y+1, x,y,z), Poly(x*z+y*z+1, x,y,z)])
assert p.unify_with((q, r)) == p.unify_with((x**2+2*x*y+1, r)) == \
(Poly(x**2+x*y, x, y, z), (Poly(x**2+2*x*y+1, x,y,z), Poly(x*z+y*z+1, x,y,z)))
def test_diff():
f = Poly(a*x**2 + b*x + 2, x)
assert f.diff(x) == Poly(2*a*x + b, x)
assert f.diff(y) == Poly(0, x)
assert f.diff(a) == Poly(x**2, x)
assert f.diff(b) == Poly(x, x)
assert f.diff() == Poly(2*a*x + b, x)
g = Poly(a*x**2 + b*x*y + 2, x, y)
assert g.diff(x) == Poly(2*a*x + b*y, x, y)
assert g.diff(y) == Poly(b*x, x, y)
assert g.diff(a) == Poly(x**2, x, y)
assert g.diff(b) == Poly(x*y, x, y)
assert g.diff() == g
def test_evaluate():
f = x**2*y*z + 2*x*y*z**3 + 3*x*y + 4*y*z
p = Poly(f, x, y, z)
assert p.evaluate({x: 7}) == Poly(f.subs({x: 7}), y, z)
assert p.evaluate({x: 7, y: 5}) == Poly(f.subs({x: 7, y: 5}), z)
assert p.evaluate({x: 7, y: 5, z: 4}) == f.subs({x: 7, y: 5, z: 4})
p = Poly(x**2 + x*y*sin(z), x, y, order='lex')
assert p.evaluate({x: 3, y: 0}) == 9
assert p.evaluate({x: 0, y: 0}) == 0
q = p.evaluate({x: 0})
assert q == Poly(0, y, order='lex')
assert q.is_zero == True
assert p.evaluate({y: 0}) == \
Poly(x**2, x, order='lex')
raises(PolynomialError, "Poly(x + y, x, y).evaluate({x: y})")
def check_convergence(numer, denom, n):
"""
Returns (h, g, p) where
-- h is:
> 0 for convergence of rate 1/factorial(n)**h
< 0 for divergence of rate factorial(n)**(-h)
= 0 for geometric or polynomial convergence or divergence
-- abs(g) is:
> 1 for geometric convergence of rate 1/h**n
< 1 for geometric divergence of rate h**n
= 1 for polynomial convergence or divergence
(g < 0 indicates an alternating series)
-- p is:
> 1 for polynomial convergence of rate 1/n**h
<= 1 for polynomial divergence of rate n**(-h)
"""
from sympy import Poly
npol = Poly(numer, n)
dpol = Poly(denom, n)
p = npol.degree()
q = dpol.degree()
rate = q - p
if rate:
return rate, None, None
constant = dpol.LC() / npol.LC()
if abs(constant) != 1:
return rate, constant, None
if npol.degree() == dpol.degree() == 0:
return rate, constant, 0
pc = npol.all_coeffs()[1]
qc = dpol.all_coeffs()[1]
return rate, constant, (qc - pc)/dpol.LC()
def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f, g = Poly(f, x), Poly(g, x)
u = poly_gcd(g, g.diff())
v = poly_div(g, u)[0]
n = u.degree - 1
m = v.degree - 1
d = g.degree
A_coeff = [ Symbol('a' + str(n-i), dummy=True) for i in xrange(0, n+1) ]
B_coeff = [ Symbol('b' + str(m-i), dummy=True) for i in xrange(0, m+1) ]
symbols = A_coeff + B_coeff
A = Poly(zip(A_coeff, xrange(n, -1, -1)), x)
B = Poly(zip(B_coeff, xrange(m, -1, -1)), x)
H = f - A.diff()*v + A*poly_div(u.diff()*v, u)[0] - B*u
result = solve(H.coeffs, symbols)
A = A.subs(result)
B = B.subs(result)
rat_part = Poly.cancel((A, u), x)
log_part = Poly.cancel((B, v), x)
return rat_part, log_part
def test_has_any_symbols():
x,y,z,t,u = symbols('xyztu')
i = Integer(4400)
assert i.has_any_symbols(x) == False
assert (i*x**i).has_any_symbols(x) == True
assert (i*y**i).has_any_symbols(x) == False
assert (i*y**i).has_any_symbols(x, y) == True
expr = x**2*y + sin(2**t + log(z))
assert expr.has_any_symbols(u) == False
assert expr.has_any_symbols(x) == True
assert expr.has_any_symbols(y) == True
assert expr.has_any_symbols(z) == True
assert expr.has_any_symbols(t) == True
assert expr.has_any_symbols(x, y, z, t) == True
assert expr.has_any_symbols(x, y, z, t, u) == True
from sympy.physics.units import m, s
assert (x*m/s).has_any_symbols(x) == True
assert (x*m/s).has_all_symbols(x) == True
assert (x*m/s).has_any_symbols(y, z) == False
assert (x*m/s).has_all_symbols(x, y) == False
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has_any_symbols(x) == True
assert poly.has_any_symbols(x, y, z) == True
assert poly.has_any_symbols(x, y, z, t) == True
assert poly.has_all_symbols(x, y, z) == True
assert poly.has_all_symbols(x, y, z, t) == False
def test_weak_normalizer():
a = Poly((1 + x)*t**5 + 4*t**4 + (-1 - 3*x)*t**3 - 4*t**2 + (-2 + 2*x)*t, t)
d = Poly(t**4 - 3*t**2 + 2, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
r = weak_normalizer(a, d, DE, z)
assert r == (Poly(t**5 - t**4 - 4*t**3 + 4*t**2 + 4*t - 4, t),
(Poly((1 + x)*t**2 + x*t, t), Poly(t + 1, t)))
assert weak_normalizer(r[1][0], r[1][1], DE) == (Poly(1, t), r[1])
r = weak_normalizer(Poly(1 + t**2), Poly(t**2 - 1, t), DE, z)
assert r == (Poly(t**4 - 2*t**2 + 1, t), (Poly(-3*t**2 + 1, t), Poly(t**2 - 1, t)))
assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2)]})
r = weak_normalizer(Poly(1 + t**2), Poly(t, t), DE, z)
assert r == (Poly(t, t), (Poly(0, t), Poly(1, t)))
assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
def test_minimal_polynomial():
assert minimal_polynomial(-7, x) == x + 7
assert minimal_polynomial(-1, x) == x + 1
assert minimal_polynomial(0, x) == x
assert minimal_polynomial(1, x) == x - 1
assert minimal_polynomial(7, x) == x - 7
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(5), x) == x**2 - 5
assert minimal_polynomial(sqrt(6), x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2), x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5), x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6), x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3, x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6, x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7, x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3, x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6, x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7, x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)), x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)), x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I + sqrt(2 + I),
x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3), x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) + sqrt(6),
x) == x**4 - 22 * x**2 - 48 * x - 23
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a, x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a), x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
raises(NotAlgebraic, lambda: minimal_polynomial(y, x))
raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
raises(NotAlgebraic, lambda: minimal_polynomial(sin(1), x))
assert minimal_polynomial(sqrt(2)).dummy_eq(x**2 - 2)
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(2), polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True) == Poly(x**2 - 2)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
assert minimal_polynomial(a, x) == x**2 - 2
assert minimal_polynomial(b, x) == x**2 - 3
assert minimal_polynomial(a, x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3)
assert minimal_polynomial(sqrt(a / 2 + 17),
x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 + 17),
x) == 4 * x**4 - 136 * x**2 + 1153
a, b = sqrt(2) / 3 + 7, AlgebraicNumber(sqrt(2) / 3 + 7)
f = 81 * x**8 - 2268 * x**6 - 4536 * x**5 + 22644 * x**4 + 63216 * x**3 - 31608 * x**2 - 189648 * x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)), x) == f
assert minimal_polynomial(sqrt(b) + sqrt(sqrt(b)), x) == f
assert minimal_polynomial(a**Q(3, 2),
x) == 729 * x**4 - 506898 * x**2 + 84604519
def test_DifferentialExtension_exp():
assert DifferentialExtension(exp(x) + exp(x**2), x)._important_attrs == \
(Poly(t1 + t0, t1), Poly(1, t1), [Poly(1, x,), Poly(t0, t0),
Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
assert DifferentialExtension(exp(x) + exp(2*x), x)._important_attrs == \
(Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0, t0)], [x, t0],
[Lambda(i, exp(i))], [], [None, 'exp'], [None, x])
assert DifferentialExtension(exp(x) + exp(x/2), x)._important_attrs == \
(Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)],
[x, t0], [Lambda(i, exp(i/2))], [], [None, 'exp'], [None, x/2])
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x + x**2), x)._important_attrs == \
(Poly((1 + t0)*t1 + t0, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x + x**2 + 1), x)._important_attrs == \
(Poly((1 + S.Exp1*t0)*t1 + t0, t1), Poly(1, t1), [Poly(1, x),
Poly(t0, t0), Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x/2 + x**2), x)._important_attrs == \
(Poly((t0 + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1],
[Lambda(i, exp(i/2)), Lambda(i, exp(i**2))],
[(exp(x/2), sqrt(exp(x)))], [None, 'exp', 'exp'], [None, x/2, x**2])
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x/2 + x**2 + 3), x)._important_attrs == \
(Poly((t0*exp(3) + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i/2)),
Lambda(i, exp(i**2))], [(exp(x/2), sqrt(exp(x)))], [None, 'exp', 'exp'],
[None, x/2, x**2])
assert DifferentialExtension(sqrt(exp(x)), x)._important_attrs == \
(Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
[Lambda(i, exp(i/2))], [(exp(x/2), sqrt(exp(x)))], [None, 'exp'], [None, x/2])
assert DifferentialExtension(exp(x/2), x)._important_attrs == \
(Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
[Lambda(i, exp(i/2))], [], [None, 'exp'], [None, x/2])
def test_DifferentialExtension_symlog():
# See comment on test_risch_integrate below
assert DifferentialExtension(log(x**x), x)._important_attrs == \
(Poly(t0*x, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0), Poly((t0 +
1)*t1, t1)], [x, t0, t1], [Lambda(i, log(i)), Lambda(i, exp(i*t0))],
[(exp(x*log(x)), x**x)], [None, 'log', 'exp'], [None, x, t0*x])
assert DifferentialExtension(log(x**y), x)._important_attrs == \
(Poly(y*t0, t0), Poly(1, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
[Lambda(i, log(i))], [(y*log(x), log(x**y))], [None, 'log'],
[None, x])
assert DifferentialExtension(log(sqrt(x)), x)._important_attrs == \
(Poly(t0, t0), Poly(2, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
[Lambda(i, log(i))], [(log(x)/2, log(sqrt(x)))], [None, 'log'],
[None, x])
def test_order_at():
a = Poly(t**4, t)
b = Poly((t**2 + 1)**3*t, t)
c = Poly((t**2 + 1)**6*t, t)
d = Poly((t**2 + 1)**10*t**10, t)
e = Poly((t**2 + 1)**100*t**37, t)
p1 = Poly(t, t)
p2 = Poly(1 + t**2, t)
assert order_at(a, p1, t) == 4
assert order_at(b, p1, t) == 1
assert order_at(c, p1, t) == 1
assert order_at(d, p1, t) == 10
assert order_at(e, p1, t) == 37
assert order_at(a, p2, t) == 0
assert order_at(b, p2, t) == 3
assert order_at(c, p2, t) == 6
assert order_at(d, p1, t) == 10
assert order_at(e, p2, t) == 100
assert order_at(Poly(0, t), Poly(t, t), t) == oo
assert order_at_oo(Poly(t**2 - 1, t), Poly(t + 1), t) == \
order_at_oo(Poly(t - 1, t), Poly(1, t), t) == -1
assert order_at_oo(Poly(0, t), Poly(1, t), t) == oo
def test_solve_poly_rde_cancel():
# exp
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert cancel_exp(Poly(2*x, t), Poly(2*x, t), 0, DE) == \
Poly(1, t)
assert cancel_exp(Poly(2*x, t), Poly((1 + 2*x)*t, t), 1, DE) == \
Poly(t, t)
# TODO: Add more exp tests, including tests that require is_deriv_in_field()
# primitive
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
# If the DecrementLevel context manager is working correctly, this shouldn't
# cause any problems with the further tests.
raises(NonElementaryIntegralException, lambda: cancel_primitive(Poly(1, t), Poly(t, t), oo, DE))
assert cancel_primitive(Poly(1, t), Poly(t + 1/x, t), 2, DE) == \
Poly(t, t)
assert cancel_primitive(Poly(4*x, t), Poly(4*x*t**2 + 2*t/x, t), 3, DE) == \
Poly(t**2, t)
def test_splitfactor():
p = Poly(4 * x**4 * t**5 + (-4 * x**3 - 4 * x**4) * t**4 +
(-3 * x**2 + 2 * x**3) * t**3 +
(2 * x + 7 * x**2 + 2 * x**3) * t**2 +
(1 - 4 * x - 4 * x**2) * t - 1 + 2 * x,
t,
field=True)
DE = DifferentialExtension(extension={
'D': [Poly(1, x),
Poly(-t**2 - 3 / (2 * x) * t + 1 / (2 * x), t)]
})
assert splitfactor(p, DE) == (Poly(
4 * x**4 * t**3 + (-8 * x**3 - 4 * x**4) * t**2 +
(4 * x**2 + 8 * x**3) * t - 4 * x**2,
t), Poly(t**2 + 1 / x * t + (1 - 2 * x) / (4 * x**2),
t,
domain='ZZ(x)'))
assert splitfactor(Poly(x, t), DE) == (Poly(x, t), Poly(1, t))
r = Poly(
-4 * x**4 * z**2 + 4 * x**6 * z**2 - z * x**3 - 4 * x**5 * z**3 +
4 * x**3 * z**3 + x**4 + z * x**5 - x**6, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 / x, t)]})
assert splitfactor(r, DE, coefficientD=True) == \
(Poly(x*z - x**2 - z*x**3 + x**4, t), Poly(-x**2 + 4*x**2*z**2, t))
assert splitfactor_sqf(r, DE, coefficientD=True) == \
(((Poly(x*z - x**2 - z*x**3 + x**4, t), 1),), ((Poly(-x**2 + 4*x**2*z**2, t), 1),))
assert splitfactor(Poly(0, t), DE) == (Poly(0, t), Poly(1, t))
assert splitfactor_sqf(Poly(0, t), DE) == (((Poly(0, t), 1), ), ())
def test_canonical_representation():
DE = DifferentialExtension(
extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert canonical_representation(Poly(x - t, t), Poly(t**2, t), DE) == \
(Poly(0, t), (Poly(0, t),
Poly(1, t)), (Poly(-t + x, t),
Poly(t**2, t)))
DE = DifferentialExtension(
extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert canonical_representation(Poly(t**5 + t**3 + x**2*t + 1, t),
Poly((t**2 + 1)**3, t), DE) == \
(Poly(0, t), (Poly(t**5 + t**3 + x**2*t + 1, t),
Poly(t**6 + 3*t**4 + 3*t**2 + 1, t)), (Poly(0, t), Poly(1, t)))
from itertools import product
from sympy import Symbol, div, Poly
x = Symbol('x')
gf28_mod = Poly(x ** 8 + x ** 4 + x ** 3 + x + 1, x, modulus=2)
def reduce_gf28(poly):
"""
Reduces the given sympy polynomial under the GF(2**8) polynomial x**8 + x**4 + x**3 + x + 1
Example:
>>> # Example taken from the textbook
>>> f = Poly(x**2 * (x**7 + x**6 + x**3 + x + 1), x, domain='GF(2)')
>>> f
Poly(x**9 + x**8 + x**5 + x**3 + x**2, x, modulus=2)
>>> reduce_gf28(f)
Poly(1, x, modulus=2)
>>> reduce_gf28(Poly((x**2 + x + 1) * (x**8 + x**6 + x**2 + 1), x, modulus=2))
Poly(x**7 + 1, x, modulus=2)
"""
_, remainder = div(poly, gf28_mod, x)
# GF(2) sets coefficients as elements of Z_2
return remainder.as_poly(x, domain='GF(2)')
def coeffs2poly(coeffs):
"""
Given a list of coefficients (mod 2), most significant coefficient first, return a sympy
def test_sympify_poly():
p = Poly(x**2 + x + 1, x)
assert _sympify(p) is p
assert sympify(p) is p
def test_solve_poly_inequality():
assert psolve(Poly(0, x), '==') == [S.Reals]
assert psolve(Poly(1, x), '==') == [S.EmptySet]
assert psolve(PurePoly(x + 1, x), ">") == [Interval(-1, oo, True, False)]
def test_H19():
a = symbols('a')
# The idea is to let a**2 == 2, then solve 1/(a-1). Answer is a+1")
assert Poly(a - 1).invert(Poly(a**2 - 2)) == a + 1
def test_issue_19161():
polynomial = Poly('x**2').simplify()
assert (polynomial - x**2).simplify() == 0
def test_minimal_polynomial():
assert minimal_polynomial(-7, x) == x + 7
assert minimal_polynomial(-1, x) == x + 1
assert minimal_polynomial(0, x) == x
assert minimal_polynomial(1, x) == x - 1
assert minimal_polynomial(7, x) == x - 7
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(5), x) == x**2 - 5
assert minimal_polynomial(sqrt(6), x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2), x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5), x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6), x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3, x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6, x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7, x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3, x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6, x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7, x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)), x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)), x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I + sqrt(2 + I),
x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3), x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) + sqrt(6),
x) == x**4 - 22 * x**2 - 48 * x - 23
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a, x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a), x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
raises(NotAlgebraic, lambda: minimal_polynomial(sin(1), x))
assert minimal_polynomial(sqrt(2)).dummy_eq(x**2 - 2)
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(2), polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True,
compose=False) == Poly(x**2 - 2)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
assert minimal_polynomial(a, x) == x**2 - 2
assert minimal_polynomial(b, x) == x**2 - 3
assert minimal_polynomial(a, x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3)
assert minimal_polynomial(sqrt(a / 2 + 17),
x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 + 17),
x) == 4 * x**4 - 136 * x**2 + 1153
a, b = sqrt(2) / 3 + 7, AlgebraicNumber(sqrt(2) / 3 + 7)
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)), x) == f
assert minimal_polynomial(sqrt(b) + sqrt(sqrt(b)), x) == f
assert minimal_polynomial(a**Q(3, 2),
x) == 729 * x**4 - 506898 * x**2 + 84604519
# issue 5994
eq = S('''
-1/(800*sqrt(-1/240 + 1/(18000*(-1/17280000 +
sqrt(15)*I/28800000)**(1/3)) + 2*(-1/17280000 +
sqrt(15)*I/28800000)**(1/3)))''')
assert minimal_polynomial(eq, x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex, x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex, x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = (expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))**Rational(1, 3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(S("-sqrt(5)/2 - 1/2 + (-sqrt(5)/2 - 1/2)**2"),
x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3, x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I, x,
compose=False) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1, x,
compose=False) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
x,
compose=False) == x**4 + 18 * x**2 + 49
# minimal polynomial of I
assert minimal_polynomial(I, x, domain=QQ.algebraic_field(I)) == x - I
K = QQ.algebraic_field(I * (sqrt(2) + 1))
assert minimal_polynomial(I, x, domain=K) == x - I
assert minimal_polynomial(I, x, domain=QQ) == x**2 + 1
assert minimal_polynomial(I, x, domain='QQ(y)') == x**2 + 1
def optimize_lsq(p, nodes):
# Define phi
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
h1, h2, h3, h4, J12, J13, J23, J24, J34, a, b = symbols(
'h1, h2, h3, h4, J12, J13, J23, J24, J34, a, b')
phi = x1 * h1 + x2 * h2 + x3 * h3 + x4 * h4 + J12 * x1 * x2 + J13 * x1 * x3 + J23 * x2 * x3 + J24 * x2 * x4 + J34 * x3 * x4 + a + b
phi1 = x1 * h1 + x2 * h2 + x3 * h3 + J12 * x1 * x2 + J13 * x1 * x3 + J23 * x2 * x3 + a
phi2 = x2 * h2 + x3 * h3 + x4 * h4 + J23 * x2 * x3 + J24 * x2 * x4 + J34 * x3 * x4 + b
coeff = Poly(phi, (h1, h2, h3, h4, J12, J13, J23, J24, J34, a, b)).coeffs()
coeff1 = Poly(phi1, (h1, h2, h3, J12, J13, J23, a)).coeffs()
coeff2 = Poly(phi2, (h2, h3, h4, J23, J24, J34, b)).coeffs()
A = np.zeros(shape=(2**len(nodes), 11))
A1 = np.zeros(shape=(2**len(nodes), 11))
for i, v in enumerate(list(itertools.product([0, 1], repeat=len(nodes)))):
A[i] = [
ii.subs({
x1: v[0],
x2: v[1],
x3: v[2],
x4: v[3]
}) for ii in coeff
]
for i, v in enumerate(
list(itertools.product([0, 1], repeat=len(nodes) - 1))):
A1[i, [0, 1, 2, 4, 5, 6, 9]] = [
ii.subs({
x1: v[0],
x2: v[1],
x3: v[2]
}) for ii in coeff1
]
A1[i + 8, [1, 2, 3, 6, 7, 8, 10]] = [
ii.subs({
x2: v[0],
x3: v[1],
x4: v[2]
}) for ii in coeff2
]
for i, v in enumerate(
list(itertools.product([0, 1], repeat=len(nodes) - 1))):
A1[i, [0, 1, 2, 4, 5, 6, 9]] = [
ii.subs({
x1: v[0],
x2: v[1],
x3: v[2]
}) for ii in coeff1
]
A1[i + 8, [1, 2, 3, 6, 7, 8, 10]] = [
ii.subs({
x2: v[0],
x3: v[1],
x4: v[2]
}) for ii in coeff2
]
p_true_view = p[[bit2int(v) for v in A1[:, :4]]]
B = np.array(A[:, :9])
## CONST_IN_ISING = False
## if not CONST_IN_ISING:
## A = A[:, :-1]
## plt.suptitle('No constant in QUBO')
## solve y = A*x
Z_lsq = 1.0
y = -np.log(Z_lsq * (p_true_view + 1e-20))
x = np.linalg.lstsq(A1, y, rcond=None)
hJ = x[0][:9]
## Visualize the solution
plt.bar(np.array(range(len(y))), y, width=0.3, label='true y')
plt.bar(np.array(range(len(y))) + 0.3,
np.dot(A1, x[0]),
width=0.3,
label='lsq-fit')
plt.xlabel('x')
plt.ylabel('probability')
plt.legend()
print('Z_lsq', np.sum(np.exp(-np.dot(A, x[0]))))
return hJ
def test_log_to_atan():
f, g = (Poly(x + S.Half, x, domain='QQ'), Poly(sqrt(3) / 2, x,
domain='EX'))
fg_ans = 2 * atan(2 * sqrt(3) * x / 3 + sqrt(3) / 3)
assert log_to_atan(f, g) == fg_ans
assert log_to_atan(g, f) == -fg_ans
def test_DifferentialExtension_misc():
# Odd ends
assert DifferentialExtension(sin(y)*exp(x), x)._important_attrs == \
(Poly(sin(y)*t0, t0, domain='ZZ[sin(y)]'), Poly(1, t0, domain='ZZ'),
[Poly(1, x, domain='ZZ'), Poly(t0, t0, domain='ZZ')], [x, t0],
[Lambda(i, exp(i))], [], [None, 'exp'], [None, x])
raises(NotImplementedError, lambda: DifferentialExtension(sin(x), x))
assert DifferentialExtension(10**x, x)._important_attrs == \
(Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(log(10)*t0, t0)], [x, t0],
[Lambda(i, exp(i*log(10)))], [(exp(x*log(10)), 10**x)], [None, 'exp'],
[None, x*log(10)])
assert DifferentialExtension(log(x) + log(x**2), x)._important_attrs in [
(Poly(3 * t0, t0), Poly(2, t0), [Poly(1, x),
Poly(2 / x, t0)], [x, t0],
[Lambda(i, log(i**2))], [], [
None,
], [], [1], [x**2]),
(Poly(3 * t0, t0), Poly(1, t0), [Poly(1, x),
Poly(1 / x, t0)], [x, t0],
[Lambda(i, log(i))], [], [None, 'log'], [None, x])
]
assert DifferentialExtension(S.Zero, x)._important_attrs == \
(Poly(0, x), Poly(1, x), [Poly(1, x)], [x], [], [], [None], [None])
assert DifferentialExtension(tan(atan(x).rewrite(log)), x)._important_attrs == \
(Poly(x, x), Poly(1, x), [Poly(1, x)], [x], [], [], [None], [None])
def test_normal_denom():
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
raises(NonElementaryIntegralException, lambda: normal_denom(Poly(1, x), Poly(1, x),
Poly(1, x), Poly(x, x), DE))
fa, fd = Poly(t**2 + 1, t), Poly(1, t)
ga, gd = Poly(1, t), Poly(t**2, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert normal_denom(fa, fd, ga, gd, DE) == \
(Poly(t, t), (Poly(t**3 - t**2 + t - 1, t), Poly(1, t)), (Poly(1, t),
Poly(1, t)), Poly(t, t))
def test_solve_poly_rde_no_cancel():
# deg(b) large
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
oo, DE) == Poly(t + x, t)
# deg(b) small
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert solve_poly_rde(Poly(0, x), Poly(x/2 - S(1)/4, x), oo, DE) == \
Poly(x**2/4 - x/4, x)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
Poly(t + 1, t, x)
# deg(b) == deg(D) - 1
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert no_cancel_equal(Poly(1 - t, t),
Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
(Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))
def test_integrate_primitive():
DE = DifferentialExtension(extension={
'D': [Poly(1, x), Poly(1 / x, t)],
'Tfuncs': [log]
})
assert integrate_primitive(Poly(t, t), Poly(1, t),
DE) == (x * log(x), -1, True)
assert integrate_primitive(Poly(x, t), Poly(t, t),
DE) == (0, NonElementaryIntegral(x / log(x),
x), False)
DE = DifferentialExtension(
extension={
'D': [Poly(1, x),
Poly(1 / x, t1),
Poly(1 / (x + 1), t2)],
'Tfuncs': [log, Lambda(i, log(i + 1))]
})
assert integrate_primitive(Poly(t1, t2), Poly(t2, t2), DE) == \
(0, NonElementaryIntegral(log(x)/log(1 + x), x), False)
DE = DifferentialExtension(
extension={
'D': [Poly(1, x),
Poly(1 / x, t1),
Poly(1 / (x * t1), t2)],
'Tfuncs': [log, Lambda(i, log(log(i)))]
})
assert integrate_primitive(Poly(t2, t2), Poly(t1, t2), DE) == \
(0, NonElementaryIntegral(log(log(x))/log(x), x), False)
DE = DifferentialExtension(extension={
'D': [Poly(1, x), Poly(1 / x, t0)],
'Tfuncs': [log]
})
assert integrate_primitive(Poly(x**2*t0**3 + (3*x**2 + x)*t0**2 + (3*x**2
+ 2*x)*t0 + x**2 + x, t0), Poly(x**2*t0**4 + 4*x**2*t0**3 + 6*x**2*t0**2 +
4*x**2*t0 + x**2, t0), DE) == \
(-1/(log(x) + 1), NonElementaryIntegral(1/(log(x) + 1), x), False)
def test_DifferentialExtension_handle_first():
assert DifferentialExtension(exp(x)*log(x), x, handle_first='log')._important_attrs == \
(Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0),
Poly(t1, t1)], [x, t0, t1], [Lambda(i, log(i)), Lambda(i, exp(i))],
[], [None, 'log', 'exp'], [None, x, x])
assert DifferentialExtension(exp(x)*log(x), x, handle_first='exp')._important_attrs == \
(Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
Poly(1/x, t1)], [x, t0, t1], [Lambda(i, exp(i)), Lambda(i, log(i))],
[], [None, 'exp', 'log'], [None, x, x])
# This one must have the log first, regardless of what we set it to
# (because the log is inside of the exponential: x**x == exp(x*log(x)))
assert DifferentialExtension(-x**x*log(x)**2 + x**x - x**x/x, x,
handle_first='exp')._important_attrs == \
DifferentialExtension(-x**x*log(x)**2 + x**x - x**x/x, x,
handle_first='log')._important_attrs == \
(Poly((-1 + x - x*t0**2)*t1, t1), Poly(x, t1),
[Poly(1, x), Poly(1/x, t0), Poly((1 + t0)*t1, t1)], [x, t0, t1],
[Lambda(i, log(i)), Lambda(i, exp(t0*i))], [(exp(x*log(x)), x**x)],
[None, 'log', 'exp'], [None, x, t0*x])
def test_integrate_hyperexponential():
# TODO: Add tests for integrate_hyperexponential() from the book
a = Poly(
(1 + 2 * t1 + t1**2 + 2 * t1**3) * t**2 + (1 + t1**2) * t + 1 + t1**2,
t)
d = Poly(1, t)
DE = DifferentialExtension(
extension={
'D': [Poly(1, x),
Poly(1 + t1**2, t1),
Poly(t * (1 + t1**2), t)],
'Tfuncs': [tan, Lambda(i, exp(tan(i)))]
})
assert integrate_hyperexponential(a, d, DE) == \
(exp(2*tan(x))*tan(x) + exp(tan(x)), 1 + t1**2, True)
a = Poly((t1**3 + (x + 1) * t1**2 + t1 + x + 2) * t, t)
assert integrate_hyperexponential(a, d, DE) == \
((x + tan(x))*exp(tan(x)), 0, True)
a = Poly(t, t)
d = Poly(1, t)
DE = DifferentialExtension(extension={
'D': [Poly(1, x), Poly(2 * x * t, t)],
'Tfuncs': [Lambda(i, exp(x**2))]
})
assert integrate_hyperexponential(a, d, DE) == \
(0, NonElementaryIntegral(exp(x**2), x), False)
DE = DifferentialExtension(extension={
'D': [Poly(1, x), Poly(t, t)],
'Tfuncs': [exp]
})
assert integrate_hyperexponential(a, d, DE) == (exp(x), 0, True)
a = Poly(
25 * t**6 - 10 * t**5 + 7 * t**4 - 8 * t**3 + 13 * t**2 + 2 * t - 1, t)
d = Poly(25 * t**6 + 35 * t**4 + 11 * t**2 + 1, t)
assert integrate_hyperexponential(a, d, DE) == \
(-(11 - 10*exp(x))/(5 + 25*exp(2*x)) + log(1 + exp(2*x)), -1, True)
DE = DifferentialExtension(
extension={
'D': [Poly(1, x), Poly(t0, t0),
Poly(t0 * t, t)],
'Tfuncs': [exp, Lambda(i, exp(exp(i)))]
})
assert integrate_hyperexponential(Poly(2 * t0 * t**2, t), Poly(1, t),
DE) == (exp(2 * exp(x)), 0, True)
DE = DifferentialExtension(
extension={
'D': [Poly(1, x), Poly(t0, t0),
Poly(-t0 * t, t)],
'Tfuncs': [exp, Lambda(i, exp(-exp(i)))]
})
assert integrate_hyperexponential(Poly(-27*exp(9) - 162*t0*exp(9) +
27*x*t0*exp(9), t), Poly((36*exp(18) + x**2*exp(18) - 12*x*exp(18))*t, t), DE) == \
(27*exp(exp(x))/(-6*exp(9) + x*exp(9)), 0, True)
DE = DifferentialExtension(extension={
'D': [Poly(1, x), Poly(t, t)],
'Tfuncs': [exp]
})
assert integrate_hyperexponential(Poly(x**2/2*t, t), Poly(1, t), DE) == \
((2 - 2*x + x**2)*exp(x)/2, 0, True)
assert integrate_hyperexponential(Poly(1 + t, t), Poly(t, t), DE) == \
(-exp(-x), 1, True) # x - exp(-x)
assert integrate_hyperexponential(Poly(x, t), Poly(t + 1, t), DE) == \
(0, NonElementaryIntegral(x/(1 + exp(x)), x), False)
DE = DifferentialExtension(
extension={
'D': [Poly(1, x),
Poly(1 / x, t0),
Poly(2 * x * t1, t1)],
'Tfuncs': [log, Lambda(i, exp(i**2))]
})
elem, nonelem, b = integrate_hyperexponential(
Poly(
(8 * x**7 - 12 * x**5 + 6 * x**3 - x) * t1**4 +
(8 * t0 * x**7 - 8 * t0 * x**6 - 4 * t0 * x**5 + 2 * t0 * x**3 +
2 * t0 * x**2 - t0 * x + 24 * x**8 - 36 * x**6 - 4 * x**5 +
22 * x**4 + 4 * x**3 - 7 * x**2 - x + 1) * t1**3 +
(8 * t0 * x**8 - 4 * t0 * x**6 - 16 * t0 * x**5 - 2 * t0 * x**4 +
12 * t0 * x**3 + t0 * x**2 - 2 * t0 * x + 24 * x**9 - 36 * x**7 -
8 * x**6 + 22 * x**5 + 12 * x**4 - 7 * x**3 - 6 * x**2 + x + 1) *
t1**2 + (8 * t0 * x**8 - 8 * t0 * x**6 - 16 * t0 * x**5 +
6 * t0 * x**4 + 10 * t0 * x**3 - 2 * t0 * x**2 - t0 * x +
8 * x**10 - 12 * x**8 - 4 * x**7 + 2 * x**6 + 12 * x**5 +
3 * x**4 - 9 * x**3 - x**2 + 2 * x) * t1 + 8 * t0 * x**7 -
12 * t0 * x**6 - 4 * t0 * x**5 + 8 * t0 * x**4 - t0 * x**2 -
4 * x**7 + 4 * x**6 + 4 * x**5 - 4 * x**4 - x**3 + x**2, t1),
Poly((8 * x**7 - 12 * x**5 + 6 * x**3 - x) * t1**4 +
(24 * x**8 + 8 * x**7 - 36 * x**6 - 12 * x**5 + 18 * x**4 +
6 * x**3 - 3 * x**2 - x) * t1**3 +
(24 * x**9 + 24 * x**8 - 36 * x**7 - 36 * x**6 + 18 * x**5 +
18 * x**4 - 3 * x**3 - 3 * x**2) * t1**2 +
(8 * x**10 + 24 * x**9 - 12 * x**8 - 36 * x**7 + 6 * x**6 +
18 * x**5 - x**4 - 3 * x**3) * t1 + 8 * x**10 - 12 * x**8 +
6 * x**6 - x**4, t1), DE)
assert factor(elem) == -((x - 1) * log(x) / ((x + exp(x**2)) *
(2 * x**2 - 1)))
assert (nonelem,
b) == (NonElementaryIntegral(exp(x**2) / (exp(x**2) + 1),
x), False)
def test_derivation():
p = Poly(
4 * x**4 * t**5 + (-4 * x**3 - 4 * x**4) * t**4 +
(-3 * x**2 + 2 * x**3) * t**3 + (2 * x + 7 * x**2 + 2 * x**3) * t**2 +
(1 - 4 * x - 4 * x**2) * t - 1 + 2 * x, t)
DE = DifferentialExtension(extension={
'D': [Poly(1, x),
Poly(-t**2 - 3 / (2 * x) * t + 1 / (2 * x), t)]
})
assert derivation(p, DE) == Poly(
-20 * x**4 * t**6 + (2 * x**3 + 16 * x**4) * t**5 +
(21 * x**2 + 12 * x**3) * t**4 +
(7 * x / 2 - 25 * x**2 - 12 * x**3) * t**3 +
(-5 - 15 * x / 2 + 7 * x**2) * t**2 -
(3 - 8 * x - 10 * x**2 - 4 * x**3) / (2 * x) * t + (1 - 4 * x**2) /
(2 * x), t)
assert derivation(Poly(1, t), DE) == Poly(0, t)
assert derivation(Poly(t, t), DE) == DE.d
assert derivation(Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t), DE) == \
Poly(-2*t**3 - 4/x*t**2 - (5 - 2*x)/(2*x**2)*t - (1 - 2*x)/(2*x**3), t, domain='ZZ(x)')
DE = DifferentialExtension(
extension={'D': [Poly(1, x), Poly(1 / x, t1),
Poly(t, t)]})
assert derivation(Poly(x * t * t1, t), DE) == Poly(t * t1 + x * t * t1 + t,
t)
assert derivation(Poly(x*t*t1, t), DE, coefficientD=True) == \
Poly((1 + t1)*t, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert derivation(Poly(x, x), DE) == Poly(1, x)
# Test basic option
assert derivation((x + 1) / (x - 1), DE,
basic=True) == -2 / (1 - 2 * x + x**2)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert derivation((t + 1) / (t - 1), DE,
basic=True) == -2 * t / (1 - 2 * t + t**2)
assert derivation(t + 1, DE, basic=True) == t
def test_spde():
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
raises(NonElementaryIntegralException, lambda: spde(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert spde(Poly(t**2 + x*t*2 + x**2, t), Poly(t**2/x**2 + (2/x - 1)*t, t),
Poly(t**2/x**2 + (2/x - 1)*t, t), 0, DE) == \
(Poly(0, t), Poly(0, t), 0, Poly(0, t), Poly(1, t))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0/x**2, t0), Poly(1/x, t)]})
assert spde(Poly(t**2, t), Poly(-t**2/x**2 - 1/x, t),
Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/(2*x)*t**2 + x*t, t), 3, DE) == \
(Poly(0, t), Poly(0, t), 0, Poly(0, t),
Poly(t0*t**2/2 + x**2*t**2 - x**2*t, t))
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
3*x**4/4 + x**3 - x**2 + 1, x), 4, DE) == \
(Poly(0, x), Poly(x/2 - S(1)/4, x), 2, Poly(x**2 + x + 1, x), Poly(5*x/4, x))
assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
3*x**4/4 + x**3 - x**2 + 1, x), n, DE) == \
(Poly(0, x), Poly(x/2 - S(1)/4, x), -2 + n, Poly(x**2 + x + 1, x), Poly(5*x/4, x))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
raises(NonElementaryIntegralException, lambda: spde(Poly((t - 1)*(t**2 + 1)**2, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert spde(Poly(x**2 - x, x), Poly(1, x), Poly(9*x**4 - 10*x**3 + 2*x**2, x), 4, DE) == (Poly(0, x), Poly(0, x), 0, Poly(0, x), Poly(3*x**3 - 2*x**2, x))
assert spde(Poly(x**2 - x, x), Poly(x**2 - 5*x + 3, x), Poly(x**7 - x**6 - 2*x**4 + 3*x**3 - x**2, x), 5, DE) == \
(Poly(1, x), Poly(x + 1, x), 1, Poly(x**4 - x**3, x), Poly(x**3 - x**2, x))
def test_eigen():
x, y = symbols('x y')
R = Rational
assert eye(3).charpoly(x) == Poly((x - 1)**3, x)
assert eye(3).charpoly(y) == Poly((y - 1)**3, y)
M = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
assert M.eigenvals() == {S.One: 3}
assert canonicalize(M.eigenvects()) == canonicalize([
(1, 3, [Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])
])
M = Matrix([[0, 1, 1], [1, 0, 0], [1, 1, 1]])
assert M.eigenvals() == {2 * S.One: 1, -S.One: 1, S.Zero: 1}
assert canonicalize(M.eigenvects()) == canonicalize([
(2, 1, [Matrix([R(2, 3), R(1, 3), 1])]), (-1, 1, [Matrix([-1, 1, 0])]),
(0, 1, [Matrix([0, -1, 1])])
])
M = Matrix([[1, -1], [1, 3]])
assert canonicalize(M.eigenvects()) == canonicalize(
[[2, 2, [Matrix(1, 2, [-1, 1])]]])
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
a = R(15, 2)
b = 3 * 33**R(1, 2)
c = R(13, 2)
d = (R(33, 8) + 3 * b / 8)
e = (R(33, 8) - 3 * b / 8)
def NS(e, n):
return str(N(e, n))
r = [
(a - b / 2, 1, [
Matrix([
(12 + 24 / (c - b / 2)) / ((c - b / 2) * e) + 3 / (c - b / 2),
(6 + 12 / (c - b / 2)) / e, 1
])
]),
(0, 1, [Matrix([1, -2, 1])]),
(a + b / 2, 1, [
Matrix([
(12 + 24 / (c + b / 2)) / ((c + b / 2) * d) + 3 / (c + b / 2),
(6 + 12 / (c + b / 2)) / d, 1
])
]),
]
r1 = [(NS(r[i][0], 2), NS(r[i][1], 2), [NS(j, 2) for j in r[i][2][0]])
for i in range(len(r))]
r = M.eigenvects()
r2 = [(NS(r[i][0], 2), NS(r[i][1], 2), [NS(j, 2) for j in r[i][2][0]])
for i in range(len(r))]
assert sorted(r1) == sorted(r2)
eps = Symbol('eps', real=True)
M = Matrix([[abs(eps), I * eps], [-I * eps, abs(eps)]])
assert canonicalize(M.eigenvects()) == canonicalize([
(2 * abs(eps), 1, [Matrix([[I * eps / abs(eps)], [1]])]),
(0, 1, [Matrix([[-I * eps / abs(eps)], [1]])])
])
def test_special_denom():
# TODO: add more tests here
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
Poly(t, t), DE) == \
(Poly(1, t), Poly(t**2 - 1, t), Poly(t**2 - 1, t), Poly(t, t))
# assert special_denom(Poly(1, t), Poly(2*x, t), Poly((1 + 2*x)*t, t), DE) == 1
# issue 3940
# Note, this isn't a very good test, because the denominator is just 1,
# but at least it tests the exp cancellation case
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-2*x*t0, t0),
Poly(I*k*t1, t1)]})
DE.decrement_level()
assert special_denom(Poly(1, t0), Poly(I*k, t0), Poly(1, t0), Poly(t0, t0),
Poly(1, t0), DE) == \
(Poly(1, t0), Poly(I*k, t0), Poly(t0, t0), Poly(1, t0))
def test_integrate_hyperexponential_polynomial():
# Without proper cancellation within integrate_hyperexponential_polynomial(),
# this will take a long time to complete, and will return a complicated
# expression
p = Poly(
(-28 * x**11 * t0 - 6 * x**8 * t0 + 6 * x**9 * t0 - 15 * x**8 * t0**2 +
15 * x**7 * t0**2 + 84 * x**10 * t0**2 - 140 * x**9 * t0**3 -
20 * x**6 * t0**3 + 20 * x**7 * t0**3 - 15 * x**6 * t0**4 +
15 * x**5 * t0**4 + 140 * x**8 * t0**4 - 84 * x**7 * t0**5 -
6 * x**4 * t0**5 + 6 * x**5 * t0**5 + x**3 * t0**6 - x**4 * t0**6 +
28 * x**6 * t0**6 - 4 * x**5 * t0**7 + x**9 - x**10 + 4 * x**12) /
(-8 * x**11 * t0 + 28 * x**10 * t0**2 - 56 * x**9 * t0**3 +
70 * x**8 * t0**4 - 56 * x**7 * t0**5 + 28 * x**6 * t0**6 -
8 * x**5 * t0**7 + x**4 * t0**8 + x**12) * t1**2 +
(-28 * x**11 * t0 - 12 * x**8 * t0 + 12 * x**9 * t0 -
30 * x**8 * t0**2 + 30 * x**7 * t0**2 + 84 * x**10 * t0**2 -
140 * x**9 * t0**3 - 40 * x**6 * t0**3 + 40 * x**7 * t0**3 -
30 * x**6 * t0**4 + 30 * x**5 * t0**4 + 140 * x**8 * t0**4 -
84 * x**7 * t0**5 - 12 * x**4 * t0**5 + 12 * x**5 * t0**5 -
2 * x**4 * t0**6 + 2 * x**3 * t0**6 + 28 * x**6 * t0**6 -
4 * x**5 * t0**7 + 2 * x**9 - 2 * x**10 + 4 * x**12) /
(-8 * x**11 * t0 + 28 * x**10 * t0**2 - 56 * x**9 * t0**3 +
70 * x**8 * t0**4 - 56 * x**7 * t0**5 + 28 * x**6 * t0**6 -
8 * x**5 * t0**7 + x**4 * t0**8 + x**12) * t1 +
(-2 * x**2 * t0 + 2 * x**3 * t0 + x * t0**2 - x**2 * t0**2 + x**3 -
x**4) / (-4 * x**5 * t0 + 6 * x**4 * t0**2 - 4 * x**3 * t0**3 +
x**2 * t0**4 + x**6),
t1,
z,
expand=False)
DE = DifferentialExtension(
extension={'D': [Poly(1, x),
Poly(1 / x, t0),
Poly(2 * x * t1, t1)]})
assert integrate_hyperexponential_polynomial(p, DE, z) == (Poly(
(x - t0) * t1**2 + (-2 * t0 + 2 * x) * t1,
t1), Poly(-2 * x * t0 + x**2 + t0**2, t1), True)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0)]})
assert integrate_hyperexponential_polynomial(Poly(0, t0),
DE, z) == (Poly(0, t0),
Poly(1, t0), True)
def test_ratint_logpart():
assert ratint_logpart(x, x**2 - 9, x, t) == \
[(Poly(x**2 - 9, x), Poly(-2*t + 1, t))]
assert ratint_logpart(x**2, x**3 - 5, x, t) == \
[(Poly(x**3 - 5, x), Poly(-3*t + 1, t))]
#!/usr/bin/python3
from itertools import product
import sys
from sympy import Poly
sys.path.append('../../')
from crypto.math import x, coeffs2poly, reduce_gf28
f = Poly(x**5 + 1, x, modulus=2)
# Iterate over all 8 term (7th degree) polynomials
for coeffs in product([0, 1], repeat=8):
poly = coeffs2poly(coeffs)
result = reduce_gf28(f * poly)
# If the multiplication gave us the multiplicative identity
# we found the multiplicative inverse
if result == Poly(1, x, modulus=2):
print(poly, 'is the multiplicative inverse of', f)
def test_AlgebraicNumber():
minpoly, root = x**2 - 2, sqrt(2)
a = AlgebraicNumber(root, gen=x)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_aliased == False
assert a.coeffs() == [S(1), S(0)]
assert a.native_coeffs() == [QQ(1), QQ(0)]
a = AlgebraicNumber(root, gen=x, alias='y')
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_aliased == True
a = AlgebraicNumber(root, gen=x, alias=Symbol('y'))
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_aliased == True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [S(8) / 3]).rep == DMP([QQ(8, 3)], QQ)
assert AlgebraicNumber(sqrt(2), [7, 3]).rep == DMP([QQ(7), QQ(3)], QQ)
assert AlgebraicNumber(sqrt(2), [S(7) / 9, S(3) / 2]).rep == DMP(
[QQ(7, 9), QQ(3, 2)], QQ)
assert AlgebraicNumber(sqrt(2), [1, 2, 3]).rep == DMP([QQ(2), QQ(5)], QQ)
a = AlgebraicNumber(AlgebraicNumber(root, gen=x), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_aliased == False
assert a.coeffs() == [S(1), S(2)]
assert a.native_coeffs() == [QQ(1), QQ(2)]
a = AlgebraicNumber((minpoly, root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_aliased == False
a = AlgebraicNumber((Poly(minpoly), root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_aliased == False
assert AlgebraicNumber(sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
assert AlgebraicNumber(-sqrt(3)).rep == DMP([-QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(2))
assert a == b
c = AlgebraicNumber(sqrt(2), gen=x)
d = AlgebraicNumber(sqrt(2), gen=x)
assert a == b
assert a == c
a = AlgebraicNumber(sqrt(2), [1, 2])
b = AlgebraicNumber(sqrt(2), [1, 3])
assert a != b and a != sqrt(2) + 3
assert (a == x) == False and (a != x) == True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x)
assert b.as_poly() == Poly(y)
assert a.as_expr() == sqrt(2)
assert a.as_expr(x) == x
assert b.as_expr() == sqrt(2)
assert b.as_expr(x) == x
a = AlgebraicNumber(sqrt(2), [2, 3])
b = AlgebraicNumber(sqrt(2), [2, 3], alias=y)
p = a.as_poly()
assert p == Poly(2 * p.gen + 3)
assert a.as_poly(x) == Poly(2 * x + 3)
assert b.as_poly() == Poly(2 * y + 3)
assert a.as_expr() == 2 * sqrt(2) + 3
assert a.as_expr(x) == 2 * x + 3
assert b.as_expr() == 2 * sqrt(2) + 3
assert b.as_expr(x) == 2 * x + 3
a = AlgebraicNumber(sqrt(2))
b = to_number_field(sqrt(2))
assert a.args == b.args == (sqrt(2), Tuple())
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(1, 2, 3))