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, domain='ZZ(x)'),
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 roots_quintic(f):
"""
Calulate exact roots of a solvable quintic
"""
result = []
coeff_5, coeff_4, p, q, r, s = f.all_coeffs()
# Eqn must be of the form x^5 + px^3 + qx^2 + rx + s
if coeff_4:
return result
if coeff_5 != 1:
l = [p / coeff_5, q / coeff_5, r / coeff_5, s / coeff_5]
if not all(coeff.is_Rational for coeff in l):
return result
f = Poly(f / coeff_5)
quintic = PolyQuintic(f)
# Eqn standardized. Algo for solving starts here
if not f.is_irreducible:
return result
f20 = quintic.f20
# Check if f20 has linear factors over domain Z
if f20.is_irreducible:
return result
# Now, we know that f is solvable
for _factor in f20.factor_list()[1]:
if _factor[0].is_linear:
theta = _factor[0].root(0)
break
d = discriminant(f)
delta = sqrt(d)
# zeta = a fifth root of unity
zeta1, zeta2, zeta3, zeta4 = quintic.zeta
T = quintic.T(theta, d)
tol = S(1e-10)
alpha = T[1] + T[2] * delta
alpha_bar = T[1] - T[2] * delta
beta = T[3] + T[4] * delta
beta_bar = T[3] - T[4] * delta
disc = alpha**2 - 4 * beta
disc_bar = alpha_bar**2 - 4 * beta_bar
l0 = quintic.l0(theta)
l1 = _quintic_simplify((-alpha + sqrt(disc)) / S(2))
l4 = _quintic_simplify((-alpha - sqrt(disc)) / S(2))
l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / S(2))
l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / S(2))
order = quintic.order(theta, d)
test = (order * delta.n()) - ((l1.n() - l4.n()) * (l2.n() - l3.n()))
# Comparing floats
# Problems importing on top
from sympy.utilities.randtest import comp
if not comp(test, 0, tol):
l2, l3 = l3, l2
# Now we have correct order of l's
R1 = l0 + l1 * zeta1 + l2 * zeta2 + l3 * zeta3 + l4 * zeta4
R2 = l0 + l3 * zeta1 + l1 * zeta2 + l4 * zeta3 + l2 * zeta4
R3 = l0 + l2 * zeta1 + l4 * zeta2 + l1 * zeta3 + l3 * zeta4
R4 = l0 + l4 * zeta1 + l3 * zeta2 + l2 * zeta3 + l1 * zeta4
Res = [None, [None] * 5, [None] * 5, [None] * 5, [None] * 5]
Res_n = [None, [None] * 5, [None] * 5, [None] * 5, [None] * 5]
sol = Symbol('sol')
# Simplifying improves performace a lot for exact expressions
R1 = _quintic_simplify(R1)
R2 = _quintic_simplify(R2)
R3 = _quintic_simplify(R3)
R4 = _quintic_simplify(R4)
# Solve imported here. Causing problems if imported as 'solve'
# and hence the changed name
from sympy.solvers.solvers import solve as _solve
a, b = symbols('a b', cls=Dummy)
_sol = _solve(sol**5 - a - I * b, sol)
for i in range(5):
_sol[i] = factor(_sol[i])
R1 = R1.as_real_imag()
R2 = R2.as_real_imag()
R3 = R3.as_real_imag()
R4 = R4.as_real_imag()
for i, root in enumerate(_sol):
Res[1][i] = _quintic_simplify(root.subs({a: R1[0], b: R1[1]}))
Res[2][i] = _quintic_simplify(root.subs({a: R2[0], b: R2[1]}))
Res[3][i] = _quintic_simplify(root.subs({a: R3[0], b: R3[1]}))
Res[4][i] = _quintic_simplify(root.subs({a: R4[0], b: R4[1]}))
for i in range(1, 5):
for j in range(5):
Res_n[i][j] = Res[i][j].n()
Res[i][j] = _quintic_simplify(Res[i][j])
r1 = Res[1][0]
r1_n = Res_n[1][0]
for i in range(5):
if comp(im(r1_n * Res_n[4][i]), 0, tol):
r4 = Res[4][i]
break
u, v = quintic.uv(theta, d)
sqrt5 = math.sqrt(5)
# Now we have various Res values. Each will be a list of five
# values. We have to pick one r value from those five for each Res
u, v = quintic.uv(theta, d)
testplus = (u + v * delta * sqrt(5)).n()
testminus = (u - v * delta * sqrt(5)).n()
# Evaluated numbers suffixed with _n
# We will use evaluated numbers for calculation. Much faster.
r4_n = r4.n()
r2 = r3 = None
for i in range(5):
r2temp_n = Res_n[2][i]
for j in range(5):
# Again storing away the exact number and using
# evaluated numbers in computations
r3temp_n = Res_n[3][j]
if (comp(r1_n * r2temp_n**2 + r4_n * r3temp_n**2 - testplus, 0,
tol) and comp(
r3temp_n * r1_n**2 + r2temp_n * r4_n**2 - testminus,
0, tol)):
r2 = Res[2][i]
r3 = Res[3][j]
break
if r2:
break
# Now, we have r's so we can get roots
x1 = (r1 + r2 + r3 + r4) / 5
x2 = (r1 * zeta4 + r2 * zeta3 + r3 * zeta2 + r4 * zeta1) / 5
x3 = (r1 * zeta3 + r2 * zeta1 + r3 * zeta4 + r4 * zeta2) / 5
x4 = (r1 * zeta2 + r2 * zeta4 + r3 * zeta1 + r4 * zeta3) / 5
x5 = (r1 * zeta1 + r2 * zeta2 + r3 * zeta3 + r4 * zeta4) / 5
result = [x1, x2, x3, x4, x5]
# Now check if solutions are distinct
saw = set()
for r in result:
r = r.n(2)
if r in saw:
# Roots were identical. Abort, return []
# and fall back to usual solve
return []
saw.add(r)
return result
def test_sympy__polys__polytools__Poly():
from sympy.polys.polytools import Poly
assert _test_args(Poly(2, x, y))
def test_RootOf_real_roots():
assert Poly(x**5 + x + 1).real_roots() == [RootOf(x**3 - x**2 + 1, 0)]
assert Poly(x**5 + x +
1).real_roots(radicals=False) == [RootOf(x**3 - x**2 + 1, 0)]
def test_issue_8316():
f = Poly(7 * x**8 - 9)
assert len(f.all_roots()) == 8
f = Poly(7 * x**8 - 10)
assert len(f.all_roots()) == 8
def test__imag_count():
from sympy.polys.rootoftools import _imag_count_of_factor
def imag_count(p):
return sum(
[_imag_count_of_factor(f) * m for f, m in p.factor_list()[1]])
assert imag_count(Poly(x**6 + 10 * x**2 + 1)) == 2
assert imag_count(Poly(x**2)) == 0
assert imag_count(Poly([1] * 3 + [-1], x)) == 0
assert imag_count(Poly(x**3 + 1)) == 0
assert imag_count(Poly(x**2 + 1)) == 2
assert imag_count(Poly(x**2 - 1)) == 0
assert imag_count(Poly(x**4 - 1)) == 2
assert imag_count(Poly(x**4 + 1)) == 0
assert imag_count(Poly([1, 2, 3], x)) == 0
assert imag_count(Poly(x**3 + x + 1)) == 0
assert imag_count(Poly(x**4 + x + 1)) == 0
def q(r1, r2, p):
return Poly(((x - r1) * (x - r2)).subs(x, x**p), x)
assert imag_count(q(-1, -2, 2)) == 4
assert imag_count(q(-1, 2, 2)) == 2
assert imag_count(q(1, 2, 2)) == 0
assert imag_count(q(1, 2, 4)) == 4
assert imag_count(q(-1, 2, 4)) == 2
assert imag_count(q(-1, -2, 4)) == 0
def primitive_element(extension, x=None, **args):
"""Construct a common number field for all extensions. """
if not extension:
raise ValueError("can't compute primitive element for empty extension")
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not args.get('ex', False):
gen, coeffs = extension[0], [1]
# XXX when minimal_polynomial is extended to work
# with AlgebraicNumbers this test can be removed
if isinstance(gen, AlgebraicNumber):
g = gen.minpoly.replace(x)
else:
g = minimal_polynomial(gen, x, polys=True)
for ext in extension[1:]:
_, factors = factor_list(g, extension=ext)
g = _choose_factor(factors, x, gen)
s, _, g = g.sqf_norm()
gen += s * ext
coeffs.append(s)
if not args.get('polys', False):
return g.as_expr(), coeffs
else:
return cls(g), coeffs
generator = numbered_symbols('y', cls=Dummy)
F, Y = [], []
for ext in extension:
y = next(generator)
if ext.is_Poly:
if ext.is_univariate:
f = ext.as_expr(y)
else:
raise ValueError("expected minimal polynomial, got %s" % ext)
else:
f = minpoly(ext, y)
F.append(f)
Y.append(y)
coeffs_generator = args.get('coeffs', _coeffs_generator)
for coeffs in coeffs_generator(len(Y)):
f = x - sum([c * y for c, y in zip(coeffs, Y)])
G = groebner(F + [f], Y + [x], order='lex', field=True)
H, g = G[:-1], cls(G[-1], x, domain='QQ')
for i, (h, y) in enumerate(zip(H, Y)):
try:
H[i] = Poly(y - h, x,
domain='QQ').all_coeffs() # XXX: composite=False
except CoercionFailed: # pragma: no cover
break # G is not a triangular set
else:
break
else: # pragma: no cover
raise RuntimeError("run out of coefficient configurations")
_, g = g.clear_denoms()
if not args.get('polys', False):
return g.as_expr(), coeffs, H
else:
return g, coeffs, H
def test_smith_normal_deprecated():
from sympy.polys.solvers import RawMatrix as Matrix
with warns_deprecated_sympy():
m = Matrix([[12, 6, 4,8],[3,9,6,12],[2,16,14,28],[20,10,10,20]])
setattr(m, 'ring', ZZ)
with warns_deprecated_sympy():
smf = Matrix([[1, 0, 0, 0], [0, 10, 0, 0], [0, 0, -30, 0], [0, 0, 0, 0]])
assert smith_normal_form(m) == smf
x = Symbol('x')
with warns_deprecated_sympy():
m = Matrix([[Poly(x-1), Poly(1, x),Poly(-1,x)],
[0, Poly(x), Poly(-1,x)],
[Poly(0,x),Poly(-1,x),Poly(x)]])
setattr(m, 'ring', QQ[x])
invs = (Poly(1, x, domain='QQ'), Poly(x - 1, domain='QQ'), Poly(x**2 - 1, domain='QQ'))
assert invariant_factors(m) == invs
with warns_deprecated_sympy():
m = Matrix([[2, 4]])
setattr(m, 'ring', ZZ)
with warns_deprecated_sympy():
smf = Matrix([[2, 0]])
assert smith_normal_form(m) == smf
def _eval_scalar_mul(self, other):
mat = [
Poly(a.as_expr() * other, *a.gens) if isinstance(a, Poly) else a *
other for a in self._mat
]
return self.__class__(self.rows, self.cols, mat, copy=False)
def _transform(cls, expr, x):
"""Transform an expression to a polynomial. """
poly = Poly(expr, x, greedy=False)
return _rootof_preprocess(poly)
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
return exc.expr
form, gen = S.Zero, F.gen
if F.is_univariate:
for coeff in F.all_coeffs():
form = form * gen + coeff
else:
F, gens = Poly(F, gen), gens[1:]
for coeff in F.all_coeffs():
form = form * gen + horner(coeff, *gens, **args)
return form
def interpolate(data, x):
"""
Construct an interpolating polynomial for the data points.
Examples
========
>>> from sympy.polys.polyfuncs import interpolate
def __new__(cls, f, x=None, indices=None, radicals=True, expand=True):
"""Construct a new ``RootOf`` object for ``k``-th root of ``f``. """
if indices is None and (not isinstance(x, Basic) or x.is_Integer):
x, indices = None, x
poly = Poly(f, x, greedy=False, expand=expand)
if not poly.is_univariate:
raise PolynomialError("only univariate polynomials are allowed")
degree = poly.degree()
if degree <= 0:
raise PolynomialError("can't construct RootOf object for %s" % f)
if indices is not None and indices is not True:
if hasattr(indices, '__iter__'):
indices, iterable = list(indices), True
else:
indices, iterable = [indices], False
indices = map(int, indices)
for i, index in enumerate(indices):
if index < -degree or index >= degree:
raise IndexError(
"root index out of [%d, %d] range, got %d" %
(-degree, degree - 1, index))
elif index < 0:
indices[i] += degree
else:
iterable = True
if indices is True:
indices = range(degree)
dom = poly.get_domain()
if not dom.is_Exact:
poly = poly.to_exact()
roots = roots_trivial(poly, radicals)
if roots is not None:
if indices is not None:
result = [roots[index] for index in indices]
else:
result = [root for root in roots if root.is_real]
else:
coeff, poly = _rootof_preprocess(poly)
dom = poly.get_domain()
if not dom.is_ZZ:
raise NotImplementedError("RootOf is not supported over %s" %
dom)
result = []
for data in _rootof_data(poly, indices):
poly, index, pointer, conjugate = data
roots = roots_trivial(poly, radicals)
if roots is not None:
result.append(coeff * roots[index])
else:
result.append(coeff *
cls._new(poly, index, pointer, conjugate))
if not iterable:
return result[0]
else:
return result
def reduce_poly(self, f, gen=None):
r"""
Reduce a univariate :py:class:`~.Poly` *f*, or an :py:class:`~.Expr`
expressing the same, modulo this :py:class:`~.PrimeIdeal`.
Explanation
===========
If our second generator $\alpha$ is zero, then we simply reduce the
coefficients of *f* mod the rational prime $p$ lying under this ideal.
Otherwise we first reduce *f* mod $\alpha$ (as a polynomial in the same
variable as *f*), and then mod $p$.
Examples
========
>>> from sympy import QQ, cyclotomic_poly, symbols
>>> zeta = symbols('zeta')
>>> Phi = cyclotomic_poly(7, zeta)
>>> k = QQ.algebraic_field((Phi, zeta))
>>> P = k.primes_above(11)
>>> frp = P[0]
>>> B = k.integral_basis(fmt='sympy')
>>> print([frp.reduce_poly(b, zeta) for b in B])
[1, zeta, zeta**2, -5*zeta**2 - 4*zeta + 1, -zeta**2 - zeta - 5,
4*zeta**2 - zeta - 1]
Parameters
==========
f : :py:class:`~.Poly`, :py:class:`~.Expr`
The univariate polynomial to be reduced.
gen : :py:class:`~.Symbol`, None, optional (default=None)
Symbol to use as the variable in the polynomials. If *f* is a
:py:class:`~.Poly` or a non-constant :py:class:`~.Expr`, this
replaces its variable. If *f* is a constant :py:class:`~.Expr`,
then *gen* must be supplied.
Returns
=======
:py:class:`~.Poly`, :py:class:`~.Expr`
Type is same as that of given *f*. If returning a
:py:class:`~.Poly`, its domain will be the finite field
$\mathbb{F}_p$.
Raises
======
GeneratorsNeeded
If *f* is a constant :py:class:`~.Expr` and *gen* is ``None``.
NotImplementedError
If *f* is other than :py:class:`~.Poly` or :py:class:`~.Expr`,
or is not univariate.
"""
if isinstance(f, Expr):
try:
g = Poly(f)
except GeneratorsNeeded as e:
if gen is None:
raise e from None
g = Poly(f, gen)
return self.reduce_poly(g).as_expr()
if isinstance(f, Poly) and f.is_univariate:
a = self.alpha.poly(f.gen)
if a != 0:
f = f.rem(a)
return f.set_modulus(self.p)
raise NotImplementedError
def round_two(T, radicals=None):
r"""
Zassenhaus's "Round 2" algorithm.
Explanation
===========
Carry out Zassenhaus's "Round 2" algorithm on a monic irreducible
polynomial *T* over :ref:`ZZ`. This computes an integral basis and the
discriminant for the field $K = \mathbb{Q}[x]/(T(x))$.
Alternatively, you may pass an :py:class:`~.AlgebraicField` instance, in
place of the polynomial *T*, in which case the algorithm is applied to the
minimal polynomial for the field's primitive element.
Ordinarily this function need not be called directly, as one can instead
access the :py:meth:`~.AlgebraicField.maximal_order`,
:py:meth:`~.AlgebraicField.integral_basis`, and
:py:meth:`~.AlgebraicField.discriminant` methods of an
:py:class:`~.AlgebraicField`.
Examples
========
Working through an AlgebraicField:
>>> from sympy import Poly, QQ
>>> from sympy.abc import x
>>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
>>> K = QQ.alg_field_from_poly(T, "theta")
>>> print(K.maximal_order())
Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2
>>> print(K.discriminant())
-503
>>> print(K.integral_basis(fmt='sympy'))
[1, theta, theta/2 + theta**2/2]
Calling directly:
>>> from sympy import Poly
>>> from sympy.abc import x
>>> from sympy.polys.numberfields.basis import round_two
>>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
>>> print(round_two(T))
(Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2, -503)
The nilradicals mod $p$ that are sometimes computed during the Round Two
algorithm may be useful in further calculations. Pass a dictionary under
`radicals` to receive these:
>>> T = Poly(x**3 + 3*x**2 + 5)
>>> rad = {}
>>> ZK, dK = round_two(T, radicals=rad)
>>> print(rad)
{3: Submodule[[-1, 1, 0], [-1, 0, 1]]}
Parameters
==========
T : :py:class:`~.Poly`, :py:class:`~.AlgebraicField`
Either (1) the irreducible monic polynomial over :ref:`ZZ` defining the
number field, or (2) an :py:class:`~.AlgebraicField` representing the
number field itself.
radicals : dict, optional
This is a way for any $p$-radicals (if computed) to be returned by
reference. If desired, pass an empty dictionary. If the algorithm
reaches the point where it computes the nilradical mod $p$ of the ring
of integers $Z_K$, then an $\mathbb{F}_p$-basis for this ideal will be
stored in this dictionary under the key ``p``. This can be useful for
other algorithms, such as prime decomposition.
Returns
=======
Pair ``(ZK, dK)``, where:
``ZK`` is a :py:class:`~sympy.polys.numberfields.modules.Submodule`
representing the maximal order.
``dK`` is the discriminant of the field $K = \mathbb{Q}[x]/(T(x))$.
See Also
========
.AlgebraicField.maximal_order
.AlgebraicField.integral_basis
.AlgebraicField.discriminant
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
"""
K = None
if isinstance(T, AlgebraicField):
K, T = T, T.ext.minpoly_of_element()
if T.domain == QQ:
try:
T = Poly(T, domain=ZZ)
except CoercionFailed:
pass # Let the error be raised by the next clause.
if (not T.is_univariate or not T.is_irreducible or not T.is_monic
or not T.domain == ZZ):
raise ValueError(
'Round 2 requires a monic irreducible univariate polynomial over ZZ.'
)
n = T.degree()
D = T.discriminant()
D_modulus = ZZ.from_sympy(abs(D))
# D must be 0 or 1 mod 4 (see Cohen Sec 4.4), which ensures we can write
# it in the form D = D_0 * F**2, where D_0 is 1 or a fundamental discriminant.
_, F = extract_fundamental_discriminant(D)
Ztheta = PowerBasis(K or T)
H = Ztheta.whole_submodule()
nilrad = None
while F:
# Next prime:
p, e = F.popitem()
U_bar, m = _apply_Dedekind_criterion(T, p)
if m == 0:
continue
# For a given prime p, the first enlargement of the order spanned by
# the current basis can be done in a simple way:
U = Ztheta.element_from_poly(Poly(U_bar, domain=ZZ))
# TODO:
# Theory says only first m columns of the U//p*H term below are needed.
# Could be slightly more efficient to use only those. Maybe `Submodule`
# class should support a slice operator?
H = H.add(U // p * H, hnf_modulus=D_modulus)
if e <= m:
continue
# A second, and possibly more, enlargements for p will be needed.
# These enlargements require a more involved procedure.
q = p
while q < n:
q *= p
H1, nilrad = _second_enlargement(H, p, q)
while H1 != H:
H = H1
H1, nilrad = _second_enlargement(H, p, q)
# Note: We do not store all nilradicals mod p, only the very last. This is
# because, unless computed against the entire integral basis, it might not
# be accurate. (In other words, if H was not already equal to ZK when we
# passed it to `_second_enlargement`, then we can't trust the nilradical
# so computed.) Example: if T(x) = x ** 3 + 15 * x ** 2 - 9 * x + 13, then
# F is divisible by 2, 3, and 7, and the nilradical mod 2 as computed above
# will not be accurate for the full, maximal order ZK.
if nilrad is not None and isinstance(radicals, dict):
radicals[p] = nilrad
ZK = H
# Pre-set expensive boolean properties which we already know to be true:
ZK._starts_with_unity = True
ZK._is_sq_maxrank_HNF = True
dK = (D * ZK.matrix.det()**2) // ZK.denom**(2 * n)
return ZK, dK
def polytope_integrate(poly, expr=None, *, clockwise=False, max_degree=None):
"""Integrates polynomials over 2/3-Polytopes.
Explanation
===========
This function accepts the polytope in ``poly`` and the function in ``expr``
(uni/bi/trivariate polynomials are implemented) and returns
the exact integral of ``expr`` over ``poly``.
Parameters
==========
poly : The input Polygon.
expr : The input polynomial.
clockwise : Binary value to sort input points of 2-Polytope clockwise.(Optional)
max_degree : The maximum degree of any monomial of the input polynomial.(Optional)
Examples
========
>>> from sympy.abc import x, y
>>> from sympy import Point, Polygon
>>> from sympy.integrals.intpoly import polytope_integrate
>>> polygon = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
>>> polys = [1, x, y, x*y, x**2*y, x*y**2]
>>> expr = x*y
>>> polytope_integrate(polygon, expr)
1/4
>>> polytope_integrate(polygon, polys, max_degree=3)
{1: 1, x: 1/2, y: 1/2, x*y: 1/4, x*y**2: 1/6, x**2*y: 1/6}
"""
if clockwise:
if isinstance(poly, Polygon):
poly = Polygon(*point_sort(poly.vertices), evaluate=False)
else:
raise TypeError("clockwise=True works for only 2-Polytope"
"V-representation input")
if isinstance(poly, Polygon):
# For Vertex Representation(2D case)
hp_params = hyperplane_parameters(poly)
facets = poly.sides
elif len(poly[0]) == 2:
# For Hyperplane Representation(2D case)
plen = len(poly)
if len(poly[0][0]) == 2:
intersections = [
intersection(poly[(i - 1) % plen], poly[i], "plane2D")
for i in range(0, plen)
]
hp_params = poly
lints = len(intersections)
facets = [
Segment2D(intersections[i], intersections[(i + 1) % lints])
for i in range(0, lints)
]
else:
raise NotImplementedError("Integration for H-representation 3D"
"case not implemented yet.")
else:
# For Vertex Representation(3D case)
vertices = poly[0]
facets = poly[1:]
hp_params = hyperplane_parameters(facets, vertices)
if max_degree is None:
if expr is None:
raise TypeError(
'Input expression must be a valid SymPy expression')
return main_integrate3d(expr, facets, vertices, hp_params)
if max_degree is not None:
result = {}
if expr is not None:
f_expr = []
for e in expr:
_ = decompose(e)
if len(_) == 1 and not _.popitem()[0]:
f_expr.append(e)
elif Poly(e).total_degree() <= max_degree:
f_expr.append(e)
expr = f_expr
if not isinstance(expr, list) and expr is not None:
raise TypeError('Input polynomials must be list of expressions')
if len(hp_params[0][0]) == 3:
result_dict = main_integrate3d(0, facets, vertices, hp_params,
max_degree)
else:
result_dict = main_integrate(0, facets, hp_params, max_degree)
if expr is None:
return result_dict
for poly in expr:
poly = _sympify(poly)
if poly not in result:
if poly.is_zero:
result[S.Zero] = S.Zero
continue
integral_value = S.Zero
monoms = decompose(poly, separate=True)
for monom in monoms:
monom = nsimplify(monom)
coeff, m = strip(monom)
integral_value += result_dict[m] * coeff
result[poly] = integral_value
return result
if expr is None:
raise TypeError('Input expression must be a valid SymPy expression')
return main_integrate(expr, facets, hp_params)
def test_shape():
c0, c1, c2 = symbols('c0:3')
x = Symbol('x')
assert CompanionMatrix(Poly([1, c0], x)).shape == (1, 1)
assert CompanionMatrix(Poly([1, c1, c0], x)).shape == (2, 2)
assert CompanionMatrix(Poly([1, c2, c1, c0], x)).shape == (3, 3)
def bivariate_type(f, x, y, **kwargs):
"""Given an expression, f, 3 tests will be done to see what type
of composite bivariate it might be, options for u(x, y) are::
x*y
x+y
x*y+x
x*y+y
If it matches one of these types, ``u(x, y)``, ``P(u)`` and dummy
variable ``u`` will be returned. Solving ``P(u)`` for ``u`` and
equating the solutions to ``u(x, y)`` and then solving for ``x`` or
``y`` is equivalent to solving the original expression for ``x`` or
``y``. If ``x`` and ``y`` represent two functions in the same
variable, e.g. ``x = g(t)`` and ``y = h(t)``, then if ``u(x, y) - p``
can be solved for ``t`` then these represent the solutions to
``P(u) = 0`` when ``p`` are the solutions of ``P(u) = 0``.
Only positive values of ``u`` are considered.
Examples
========
>>> from sympy.solvers.solvers import solve
>>> from sympy.solvers.bivariate import bivariate_type
>>> from sympy.abc import x, y
>>> eq = (x**2 - 3).subs(x, x + y)
>>> bivariate_type(eq, x, y)
(x + y, _u**2 - 3, _u)
>>> uxy, pu, u = _
>>> usol = solve(pu, u); usol
[sqrt(3)]
>>> [solve(uxy - s) for s in solve(pu, u)]
[[{x: -y + sqrt(3)}]]
>>> all(eq.subs(s).equals(0) for sol in _ for s in sol)
True
"""
u = Dummy('u', positive=True)
if kwargs.pop('first', True):
p = Poly(f, x, y)
f = p.as_expr()
_x = Dummy()
_y = Dummy()
rv = bivariate_type(Poly(f.subs({x: _x, y: _y}), _x, _y), _x, _y, first=False)
if rv:
reps = {_x: x, _y: y}
return rv[0].xreplace(reps), rv[1].xreplace(reps), rv[2]
return
p = f
f = p.as_expr()
# f(x*y)
args = Add.make_args(p.as_expr())
new = []
for a in args:
a = _mexpand(a.subs(x, u/y))
free = a.free_symbols
if x in free or y in free:
break
new.append(a)
else:
return x*y, Add(*new), u
def ok(f, v, c):
new = _mexpand(f.subs(v, c))
free = new.free_symbols
return None if (x in free or y in free) else new
# f(a*x + b*y)
new = []
d = p.degree(x)
if p.degree(y) == d:
a = root(p.coeff_monomial(x**d), d)
b = root(p.coeff_monomial(y**d), d)
new = ok(f, x, (u - b*y)/a)
if new is not None:
return a*x + b*y, new, u
# f(a*x*y + b*y)
new = []
d = p.degree(x)
if p.degree(y) == d:
for itry in range(2):
a = root(p.coeff_monomial(x**d*y**d), d)
b = root(p.coeff_monomial(y**d), d)
new = ok(f, x, (u - b*y)/a/y)
if new is not None:
return a*x*y + b*y, new, u
x, y = y, x
def pole_zero_numerical_data(system):
"""
Returns the numerical data of poles and zeros of the system.
It is internally used by ``pole_zero_plot`` to get the data
for plotting poles and zeros. Users can use this data to further
analyse the dynamics of the system or plot using a different
backend/plotting-module.
Parameters
==========
system : SISOLinearTimeInvariant
The system for which the pole-zero data is to be computed.
Returns
=======
tuple : (zeros, poles)
zeros = Zeros of the system. NumPy array of complex numbers.
poles = Poles of the system. NumPy array of complex numbers.
Raises
======
NotImplementedError
When a SISO LTI system is not passed.
When time delay terms are present in the system.
ValueError
When more than one free symbol is present in the system.
The only variable in the transfer function should be
the variable of the Laplace transform.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import pole_zero_numerical_data
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> pole_zero_numerical_data(tf1) # doctest: +SKIP
([-0.+1.j 0.-1.j], [-2. +0.j -0.5+0.8660254j -0.5-0.8660254j -1. +0.j ])
See Also
========
pole_zero_plot
"""
_check_system(system)
system = system.doit() # Get the equivalent TransferFunction object.
num_poly = Poly(system.num, system.var).all_coeffs()
den_poly = Poly(system.den, system.var).all_coeffs()
num_poly = np.array(num_poly, dtype=np.complex128)
den_poly = np.array(den_poly, dtype=np.complex128)
zeros = np.roots(num_poly)
poles = np.roots(den_poly)
return zeros, poles
def q(r1, r2, p):
return Poly(((x - r1) * (x - r2)).subs(x, x**p), x)
def roots_quartic(f):
r"""
Returns a list of roots of a quartic polynomial.
There are many references for solving quartic expressions available [1-5].
This reviewer has found that many of them require one to select from among
2 or more possible sets of solutions and that some solutions work when one
is searching for real roots but don't work when searching for complex roots
(though this is not always stated clearly). The following routine has been
tested and found to be correct for 0, 2 or 4 complex roots.
The quasisymmetric case solution [6] looks for quartics that have the form
`x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`.
Although there is a general solution, simpler results can be obtained for
certain values of the coefficients. In all cases, 4 roots are returned:
1) `f = c + a*(a**2/8 - b/2) == 0`
2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0`
3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then
a) `p == 0`
b) `p != 0`
**Examples**
>>> from sympy import Poly, symbols, I
>>> from sympy.polys.polyroots import roots_quartic
>>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20'))
>>> # 4 complex roots: 1+-I*sqrt(3), 2+-I
>>> sorted(str(tmp.evalf(n=2)) for tmp in r)
['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + I', '2.0 - 1.0*I']
**References**
1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html
2. http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html
4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
5. http://www.albmath.org/files/Math_5713.pdf
6. http://www.statemaster.com/encyclopedia/Quartic-equation
"""
_, a, b, c, d = f.monic().all_coeffs()
if not d:
return [S.Zero] + roots([1, a, b, c], multiple=True)
elif (c / a)**2 == d:
x, m = f.gen, c / a
g = Poly(x**2 + a * x + b - 2 * m, x)
z1, z2 = roots_quadratic(g)
h1 = Poly(x**2 - z1 * x + m, x)
h2 = Poly(x**2 - z2 * x + m, x)
r1 = roots_quadratic(h1)
r2 = roots_quadratic(h2)
return r1 + r2
else:
a2 = a**2
e = b - 3 * a2 / 8
f = c + a * (a2 / 8 - b / 2)
g = d - a * (a * (3 * a2 / 256 - b / 16) + c / 4)
aon4 = a / 4
ans = []
if f is S.Zero:
y1, y2 = [tmp**S.Half for tmp in roots([1, e, g], multiple=True)]
return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
if g is S.Zero:
y = [S.Zero] + roots([1, 0, e, f], multiple=True)
return [tmp - aon4 for tmp in y]
else:
p = -e**2 / 12 - g
q = -e**3 / 108 + e * g / 3 - f**2 / 8
TH = Rational(1, 3)
if p is S.Zero:
y = -5 * e / 6 - q**TH
else:
# with p !=0 then u below is not 0
root = sqrt(q**2 / 4 + p**3 / 27)
r = -q / 2 + root # or -q/2 - root
u = r**TH # primary root of solve(x**3-r, x)
y = -5 * e / 6 + u - p / u / 3
w = sqrt(e + 2 * y)
arg1 = 3 * e + 2 * y
arg2 = 2 * f / w
for s in [-1, 1]:
root = sqrt(-(arg1 + s * arg2))
for t in [-1, 1]:
ans.append((s * w - t * root) / 2 - aon4)
return ans
def test_RootOf___new__():
assert RootOf(x, 0) == 0
assert RootOf(x, -1) == 0
assert RootOf(x, S.Zero) == 0
assert RootOf(x - 1, 0) == 1
assert RootOf(x - 1, -1) == 1
assert RootOf(x + 1, 0) == -1
assert RootOf(x + 1, -1) == -1
assert RootOf(x**2 + 2 * x + 3, 0) == -1 - I * sqrt(2)
assert RootOf(x**2 + 2 * x + 3, 1) == -1 + I * sqrt(2)
assert RootOf(x**2 + 2 * x + 3, -1) == -1 + I * sqrt(2)
assert RootOf(x**2 + 2 * x + 3, -2) == -1 - I * sqrt(2)
r = RootOf(x**2 + 2 * x + 3, 0, radicals=False)
assert isinstance(r, RootOf) is True
r = RootOf(x**2 + 2 * x + 3, 1, radicals=False)
assert isinstance(r, RootOf) is True
r = RootOf(x**2 + 2 * x + 3, -1, radicals=False)
assert isinstance(r, RootOf) is True
r = RootOf(x**2 + 2 * x + 3, -2, radicals=False)
assert isinstance(r, RootOf) is True
assert RootOf((x - 1) * (x + 1), 0, radicals=False) == -1
assert RootOf((x - 1) * (x + 1), 1, radicals=False) == 1
assert RootOf((x - 1) * (x + 1), -1, radicals=False) == 1
assert RootOf((x - 1) * (x + 1), -2, radicals=False) == -1
assert RootOf((x - 1) * (x + 1), 0, radicals=True) == -1
assert RootOf((x - 1) * (x + 1), 1, radicals=True) == 1
assert RootOf((x - 1) * (x + 1), -1, radicals=True) == 1
assert RootOf((x - 1) * (x + 1), -2, radicals=True) == -1
assert RootOf((x - 1) * (x**3 + x + 3), 0) == RootOf(x**3 + x + 3, 0)
assert RootOf((x - 1) * (x**3 + x + 3), 1) == 1
assert RootOf((x - 1) * (x**3 + x + 3), 2) == RootOf(x**3 + x + 3, 1)
assert RootOf((x - 1) * (x**3 + x + 3), 3) == RootOf(x**3 + x + 3, 2)
assert RootOf((x - 1) * (x**3 + x + 3), -1) == RootOf(x**3 + x + 3, 2)
assert RootOf((x - 1) * (x**3 + x + 3), -2) == RootOf(x**3 + x + 3, 1)
assert RootOf((x - 1) * (x**3 + x + 3), -3) == 1
assert RootOf((x - 1) * (x**3 + x + 3), -4) == RootOf(x**3 + x + 3, 0)
assert RootOf(x**4 + 3 * x**3, 0) == -3
assert RootOf(x**4 + 3 * x**3, 1) == 0
assert RootOf(x**4 + 3 * x**3, 2) == 0
assert RootOf(x**4 + 3 * x**3, 3) == 0
raises(GeneratorsNeeded, lambda: RootOf(0, 0))
raises(GeneratorsNeeded, lambda: RootOf(1, 0))
raises(PolynomialError, lambda: RootOf(Poly(0, x), 0))
raises(PolynomialError, lambda: RootOf(Poly(1, x), 0))
raises(PolynomialError, lambda: RootOf(x - y, 0))
raises(NotImplementedError, lambda: RootOf(x**3 - x + sqrt(2), 0))
raises(NotImplementedError, lambda: RootOf(x**3 - x + I, 0))
raises(IndexError, lambda: RootOf(x**2 - 1, -4))
raises(IndexError, lambda: RootOf(x**2 - 1, -3))
raises(IndexError, lambda: RootOf(x**2 - 1, 2))
raises(IndexError, lambda: RootOf(x**2 - 1, 3))
raises(ValueError, lambda: RootOf(x**2 - 1, x))
assert RootOf(Poly(x - y, x), 0) == y
assert RootOf(Poly(x**2 - y, x), 0) == -sqrt(y)
assert RootOf(Poly(x**2 - y, x), 1) == sqrt(y)
assert RootOf(Poly(x**3 - y, x), 0) == y**Rational(1, 3)
assert RootOf(y * x**3 + y * x + 2 * y, x, 0) == -1
raises(NotImplementedError, lambda: RootOf(x**3 + x + 2 * y, x, 0))
assert RootOf(x**3 + x + 1, 0).is_commutative is True
def roots(f, *gens, **flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set `cubics=False` or `quartics=False` respectively.
To get roots from a specific domain set the `filter` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting `filter='C'`).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a tuple containing all
those roots set the `multiple` flag to True.
**Examples**
>>> from sympy import Poly, roots
>>> from sympy.abc import x, y
>>> roots(x**2 - 1, x)
{1: 1, -1: 1}
>>> p = Poly(x**2-1, x)
>>> roots(p)
{1: 1, -1: 1}
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x))
{y**(1/2): 1, -y**(1/2): 1}
>>> roots(x**2 - y, x)
{y**(1/2): 1, -y**(1/2): 1}
>>> roots([1, 0, -1])
{1: 1, -1: 1}
"""
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
quartics = flags.pop('quartics', True)
multiple = flags.pop('multiple', False)
filter = flags.pop('filter', None)
predicate = flags.pop('predicate', None)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
f = Poly(f, *gens, **flags)
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
if auto and f.get_domain().has_Ring:
f = f.to_field()
x = f.gen
def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for root in _try_heuristics(factors[0]):
roots.append(root)
for factor in factors[1:]:
previous, roots = list(roots), []
for root in previous:
g = factor - Poly(root, x)
for root in _try_heuristics(g):
roots.append(root)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S(0)] * f.degree()
if f.length() == 2:
if f.degree() == 1:
return map(cancel, roots_linear(f))
else:
return roots_binomial(f)
result = []
for i in [-1, 1]:
if not f.eval(i):
f = f.exquo(Poly(x - i, x))
result.append(i)
break
n = f.degree()
if n == 1:
result += map(cancel, roots_linear(f))
elif n == 2:
result += map(cancel, roots_quadratic(f))
elif n == 3 and cubics:
result += roots_cubic(f)
elif n == 4 and quartics:
result += roots_quartic(f)
return result
if f.is_monomial == 1:
if f.is_ground:
if multiple:
return []
else:
return {}
else:
result = {S(0): f.degree()}
else:
(k, ), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S(0): k}
result = {}
if f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
elif f.length() == 2:
for r in roots_binomial(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and factors[0][1] == 1:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for factor, k in factors:
for r in _try_heuristics(Poly(factor, x, field=True)):
_update_dict(result, r, k)
result.update(zeros)
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).iterkeys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).iterkeys():
if not predicate(zero):
del result[zero]
if not multiple:
return result
else:
zeros = []
for zero, k in result.iteritems():
zeros.extend([zero] * k)
return zeros
def test_issue_7876():
l1 = Poly(x**6 - x + 1, x).all_roots()
l2 = [RootOf(x**6 - x + 1, i) for i in range(6)]
assert frozenset(l1) == frozenset(l2)
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 roots_quartic(f):
r"""
Returns a list of roots of a quartic polynomial.
There are many references for solving quartic expressions available [1-5].
This reviewer has found that many of them require one to select from among
2 or more possible sets of solutions and that some solutions work when one
is searching for real roots but don't work when searching for complex roots
(though this is not always stated clearly). The following routine has been
tested and found to be correct for 0, 2 or 4 complex roots.
The quasisymmetric case solution [6] looks for quartics that have the form
`x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`.
Although no general solution that is always applicable for all
coefficients is known to this reviewer, certain conditions are tested
to determine the simplest 4 expressions that can be returned:
1) `f = c + a*(a**2/8 - b/2) == 0`
2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0`
3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then
a) `p == 0`
b) `p != 0`
Examples
========
>>> from sympy import Poly, symbols, I
>>> from sympy.polys.polyroots import roots_quartic
>>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20'))
>>> # 4 complex roots: 1+-I*sqrt(3), 2+-I
>>> sorted(str(tmp.evalf(n=2)) for tmp in r)
['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I']
References
==========
1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html
2. http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html
4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
5. http://www.albmath.org/files/Math_5713.pdf
6. http://www.statemaster.com/encyclopedia/Quartic-equation
7. eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf
"""
_, a, b, c, d = f.monic().all_coeffs()
if not d:
return [S.Zero] + roots([1, a, b, c], multiple=True)
elif (c / a)**2 == d:
x, m = f.gen, c / a
g = Poly(x**2 + a * x + b - 2 * m, x)
z1, z2 = roots_quadratic(g)
h1 = Poly(x**2 - z1 * x + m, x)
h2 = Poly(x**2 - z2 * x + m, x)
r1 = roots_quadratic(h1)
r2 = roots_quadratic(h2)
return r1 + r2
else:
a2 = a**2
e = b - 3 * a2 / 8
f = c + a * (a2 / 8 - b / 2)
g = d - a * (a * (3 * a2 / 256 - b / 16) + c / 4)
aon4 = a / 4
if f is S.Zero:
y1, y2 = [sqrt(tmp) for tmp in roots([1, e, g], multiple=True)]
return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
if g is S.Zero:
y = [S.Zero] + roots([1, 0, e, f], multiple=True)
return [tmp - aon4 for tmp in y]
else:
# Descartes-Euler method, see [7]
sols = _roots_quartic_euler(e, f, g, aon4)
if sols:
return sols
# Ferrari method, see [1, 2]
a2 = a**2
e = b - 3 * a2 / 8
f = c + a * (a2 / 8 - b / 2)
g = d - a * (a * (3 * a2 / 256 - b / 16) + c / 4)
p = -e**2 / 12 - g
q = -e**3 / 108 + e * g / 3 - f**2 / 8
TH = Rational(1, 3)
def _ans(y):
w = sqrt(e + 2 * y)
arg1 = 3 * e + 2 * y
arg2 = 2 * f / w
ans = []
for s in [-1, 1]:
root = sqrt(-(arg1 + s * arg2))
for t in [-1, 1]:
ans.append((s * w - t * root) / 2 - aon4)
return ans
# p == 0 case
y1 = -5 * e / 6 - q**TH
if p.is_zero:
return _ans(y1)
# if p != 0 then u below is not 0
root = sqrt(q**2 / 4 + p**3 / 27)
r = -q / 2 + root # or -q/2 - root
u = r**TH # primary root of solve(x**3 - r, x)
y2 = -5 * e / 6 + u - p / u / 3
if p.is_nonzero:
return _ans(y2)
# sort it out once they know the values of the coefficients
return [
Piecewise((a1, Eq(p, 0)), (a2, True))
for a1, a2 in zip(_ans(y1), _ans(y2))
]
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_number
assert a.is_aliased is False
assert a.coeffs() == [S.One, S.Zero]
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_number
assert a.is_aliased is 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_number
assert a.is_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), ()).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), (0, 0)).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [Rational(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), [Rational(7, 9), Rational(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_number
assert a.is_aliased is False
assert a.coeffs() == [S.One, 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_number
assert a.is_aliased is 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_number
assert a.is_aliased is 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)
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) is False and (a != x) is True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x, domain='QQ')
assert b.as_poly() == Poly(y, domain='QQ')
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, domain='QQ')
assert b.as_poly() == Poly(2*y + 3, domain='QQ')
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(1, 0))
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(1, 0), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(1, 2, 3))
a = AlgebraicNumber(sqrt(2), [1, 2], "alpha")
b = AlgebraicNumber(a)
c = AlgebraicNumber(a, alias="gamma")
assert a == b
assert c.alias.name == "gamma"
a = AlgebraicNumber(sqrt(2) + sqrt(3), [S(1)/2, 0, S(-9)/2, 0])
b = AlgebraicNumber(a, [1, 0, 0])
assert b.root == a.root
assert a.to_root() == sqrt(2)
assert b.to_root() == 2
a = AlgebraicNumber(2)
assert a.is_primitive_element is True
def roots(f, *gens, **flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set ``cubics=False`` or ``quartics=False`` respectively. If cubic
roots are real but are expressed in terms of complex numbers
(casus irreducibilis [1]) the ``trig`` flag can be set to True to
have the solutions returned in terms of cosine and inverse cosine
functions.
To get roots from a specific domain set the ``filter`` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting ``filter='C'``).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a list containing all
those roots set the ``multiple`` flag to True; the list will
have identical roots appearing next to each other in the result.
(For a given Poly, the all_roots method will give the roots in
sorted numerical order.)
Examples
========
>>> from sympy import Poly, roots
>>> from sympy.abc import x, y
>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}
>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}
>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}
>>> roots([1, 0, -1])
{-1: 1, 1: 1}
References
==========
1. http://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
trig = flags.pop('trig', False)
quartics = flags.pop('quartics', True)
quintics = flags.pop('quintics', False)
multiple = flags.pop('multiple', False)
filter = flags.pop('filter', None)
predicate = flags.pop('predicate', None)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
f = Poly(f, *gens, **flags)
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for root in _try_heuristics(factors[0]):
roots.append(root)
for factor in factors[1:]:
previous, roots = list(roots), []
for root in previous:
g = factor - Poly(root, f.gen)
for root in _try_heuristics(g):
roots.append(root)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S(0)] * f.degree()
if f.length() == 2:
if f.degree() == 1:
return list(map(cancel, roots_linear(f)))
else:
return roots_binomial(f)
result = []
for i in [-1, 1]:
if not f.eval(i):
f = f.quo(Poly(f.gen - i, f.gen))
result.append(i)
break
n = f.degree()
if n == 1:
result += list(map(cancel, roots_linear(f)))
elif n == 2:
result += list(map(cancel, roots_quadratic(f)))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f, trig=trig)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
(k, ), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S(0): k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().has_Ring:
f = f.to_field()
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
if not f.get_domain().is_Exact:
for r in f.nroots():
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
elif f.length() == 2:
for r in roots_binomial(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for factor, k in factors:
for r in _try_heuristics(Poly(factor, f.gen, field=True)):
_update_dict(result, r, k)
if coeff is not S.One:
_result, result, = result, {}
for root, k in _result.items():
result[coeff * root] = k
result.update(zeros)
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).keys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).keys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k * rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero] * result[zero])
return zeros
def roots(f,
*gens,
auto=True,
cubics=True,
trig=False,
quartics=True,
quintics=False,
multiple=False,
filter=None,
predicate=None,
**flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set ``cubics=False`` or ``quartics=False`` respectively. If cubic
roots are real but are expressed in terms of complex numbers
(casus irreducibilis [1]) the ``trig`` flag can be set to True to
have the solutions returned in terms of cosine and inverse cosine
functions.
To get roots from a specific domain set the ``filter`` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting ``filter='C'``).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a list containing all
those roots set the ``multiple`` flag to True; the list will
have identical roots appearing next to each other in the result.
(For a given Poly, the all_roots method will give the roots in
sorted numerical order.)
Examples
========
>>> from sympy import Poly, roots
>>> from sympy.abc import x, y
>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}
>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}
>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}
>>> roots([1, 0, -1])
{-1: 1, 1: 1}
References
==========
.. [1] https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
F = Poly(f, *gens, **flags)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
raise PolynomialError("generator must be a Symbol")
else:
f = F
if f.length == 2 and f.degree() != 1:
# check for foo**n factors in the constant
n = f.degree()
npow_bases = []
others = []
expr = f.as_expr()
con = expr.as_independent(*gens)[0]
for p in Mul.make_args(con):
if p.is_Pow and not p.exp % n:
npow_bases.append(p.base**(p.exp / n))
else:
others.append(p)
if npow_bases:
b = Mul(*npow_bases)
B = Dummy()
d = roots(
Poly(expr - con + B**n * Mul(*others), *gens,
**flags), *gens, **flags)
rv = {}
for k, v in d.items():
rv[k.subs(B, b)] = v
return rv
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, zeros, currentroot, k):
if currentroot == S.Zero:
if S.Zero in zeros:
zeros[S.Zero] += k
else:
zeros[S.Zero] = k
if currentroot in result:
result[currentroot] += k
else:
result[currentroot] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for currentroot in _try_heuristics(factors[0]):
roots.append(currentroot)
for currentfactor in factors[1:]:
previous, roots = list(roots), []
for currentroot in previous:
g = currentfactor - Poly(currentroot, f.gen)
for currentroot in _try_heuristics(g):
roots.append(currentroot)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S.Zero] * f.degree()
if f.length() == 2:
if f.degree() == 1:
return list(map(cancel, roots_linear(f)))
else:
return roots_binomial(f)
result = []
for i in [-1, 1]:
if not f.eval(i):
f = f.quo(Poly(f.gen - i, f.gen))
result.append(i)
break
n = f.degree()
if n == 1:
result += list(map(cancel, roots_linear(f)))
elif n == 2:
result += list(map(cancel, roots_quadratic(f)))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f, trig=trig)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
# Convert the generators to symbols
dumgens = symbols('x:%d' % len(f.gens), cls=Dummy)
f = f.per(f.rep, dumgens)
(k, ), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S.Zero: k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().is_Ring:
f = f.to_field()
# Use EX instead of ZZ_I or QQ_I
if f.get_domain().is_QQ_I:
f = f.per(f.rep.convert(EX))
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
dom = f.get_domain()
if not dom.is_Exact and dom.is_Numerical:
for r in f.nroots():
_update_dict(result, zeros, r, 1)
elif f.degree() == 1:
_update_dict(result, zeros, roots_linear(f)[0], 1)
elif f.length() == 2:
roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
for r in roots_fun(f):
_update_dict(result, zeros, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, zeros, r, 1)
else:
if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for currentroot in _try_decompose(f):
_update_dict(result, zeros, currentroot, 1)
else:
for r in _try_heuristics(f):
_update_dict(result, zeros, r, 1)
else:
for currentroot in _try_decompose(f):
_update_dict(result, zeros, currentroot, 1)
else:
for currentfactor, k in factors:
for r in _try_heuristics(
Poly(currentfactor, f.gen, field=True)):
_update_dict(result, zeros, r, k)
if coeff is not S.One:
_result, result, = result, {}
for currentroot, k in _result.items():
result[coeff * currentroot] = k
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: all(a.is_real for a in r.as_numer_denom()),
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).keys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).keys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k * rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1
# adding zero roots after non-trivial roots have been translated
result.update(zeros)
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero] * result[zero])
return zeros
def _minpoly_op_algebraic_element(op, ex1, ex2, x, dom, mp1=None, mp2=None):
"""
return the minimal polynomial for ``op(ex1, ex2)``
Parameters
==========
op : operation ``Add`` or ``Mul``
ex1, ex2 : expressions for the algebraic elements
x : indeterminate of the polynomials
dom: ground domain
mp1, mp2 : minimal polynomials for ``ex1`` and ``ex2`` or None
Examples
========
>>> from sympy import sqrt, Add, Mul, QQ
>>> from sympy.polys.numberfields import _minpoly_op_algebraic_element
>>> from sympy.abc import x, y
>>> p1 = sqrt(sqrt(2) + 1)
>>> p2 = sqrt(sqrt(2) - 1)
>>> _minpoly_op_algebraic_element(Mul, p1, p2, x, QQ)
x - 1
>>> q1 = sqrt(y)
>>> q2 = 1 / y
>>> _minpoly_op_algebraic_element(Add, q1, q2, x, QQ.frac_field(y))
x**2*y**2 - 2*x*y - y**3 + 1
References
==========
[1] http://en.wikipedia.org/wiki/Resultant
[2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638
"Degrees of sums in a separable field extension".
"""
y = Dummy(str(x))
if mp1 is None:
mp1 = _minpoly_compose(ex1, x, dom)
if mp2 is None:
mp2 = _minpoly_compose(ex2, y, dom)
else:
mp2 = mp2.subs({x: y})
if op is Add:
# mp1a = mp1.subs({x: x - y})
if dom == QQ:
R, X = ring('X', QQ)
p1 = R(dict_from_expr(mp1)[0])
p2 = R(dict_from_expr(mp2)[0])
else:
(p1, p2), _ = parallel_poly_from_expr((mp1, x - y), x, y)
r = p1.compose(p2)
mp1a = r.as_expr()
elif op is Mul:
mp1a = _muly(mp1, x, y)
else:
raise NotImplementedError('option not available')
if op is Mul or dom != QQ:
r = resultant(mp1a, mp2, gens=[y, x])
else:
r = rs_compose_add(p1, p2)
r = expr_from_dict(r.as_expr_dict(), x)
deg1 = degree(mp1, x)
deg2 = degree(mp2, y)
if op is Mul and deg1 == 1 or deg2 == 1:
# if deg1 = 1, then mp1 = x - a; mp1a = x - y - a;
# r = mp2(x - a), so that `r` is irreducible
return r
r = Poly(r, x, domain=dom)
_, factors = r.factor_list()
res = _choose_factor(factors, x, op(ex1, ex2), dom)
return res.as_expr()
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])
