def __new__(cls, f, x, index=None, radicals=False, expand=True):
""" Construct an indexed complex root of a polynomial.
See ``rootof`` for the parameters.
The default value of ``radicals`` is ``False`` to satisfy
``eval(srepr(expr) == expr``.
"""
x = sympify(x)
if index is None and x.is_Integer:
x, index = None, x
else:
index = sympify(index)
if index is not None and index.is_Integer:
index = int(index)
else:
raise ValueError("expected an integer root index, got %s" % index)
poly = PurePoly(f, x, greedy=False, expand=expand)
if not poly.is_univariate:
raise PolynomialError("only univariate polynomials are allowed")
if not poly.gen.is_Symbol:
# PurePoly(sin(x) + 1) == PurePoly(x + 1) but the roots of
# x for each are not the same: issue 8617
raise PolynomialError("generator must be a Symbol")
degree = poly.degree()
if degree <= 0:
raise PolynomialError("can't construct CRootOf object for %s" % f)
if index < -degree or index >= degree:
raise IndexError("root index out of [%d, %d] range, got %d" %
(-degree, degree - 1, index))
elif index < 0:
index += degree
dom = poly.get_domain()
if not dom.is_Exact:
poly = poly.to_exact()
roots = cls._roots_trivial(poly, radicals)
if roots is not None:
return roots[index]
coeff, poly = preprocess_roots(poly)
dom = poly.get_domain()
if not dom.is_ZZ:
raise NotImplementedError("CRootOf is not supported over %s" % dom)
root = cls._indexed_root(poly, index)
return coeff * cls._postprocess_root(root, radicals)
def __new__(cls, f, x, index=None, radicals=True, expand=True):
"""Construct a new ``RootOf`` object for ``k``-th root of ``f``. """
x = sympify(x)
if index is None and x.is_Integer:
x, index = None, x
else:
index = sympify(index)
if index.is_Integer:
index = int(index)
else:
raise ValueError("expected an integer root index, got %d" % index)
poly = PurePoly(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 index < -degree or index >= degree:
raise IndexError("root index out of [%d, %d] range, got %d" %
(-degree, degree - 1, index))
elif index < 0:
index += degree
dom = poly.get_domain()
if not dom.is_Exact:
poly = poly.to_exact()
roots = cls._roots_trivial(poly, radicals)
if roots is not None:
return roots[index]
coeff, poly = preprocess_roots(poly)
dom = poly.get_domain()
if not dom.is_ZZ:
raise NotImplementedError("RootOf is not supported over %s" % dom)
root = cls._indexed_root(poly, index)
return coeff*cls._postprocess_root(root, radicals)
def swinnerton_dyer_poly(n, x=None, **args):
"""Generates n-th Swinnerton-Dyer polynomial in `x`. """
if n <= 0:
raise ValueError(
"can't generate Swinnerton-Dyer polynomial of order %s" % n)
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
p, elts = 2, [[x, -sqrt(2)], [x, sqrt(2)]]
for i in xrange(2, n + 1):
p, _elts = nextprime(p), []
neg_sqrt = -sqrt(p)
pos_sqrt = +sqrt(p)
for elt in elts:
_elts.append(elt + [neg_sqrt])
_elts.append(elt + [pos_sqrt])
elts = _elts
poly = []
for elt in elts:
poly.append(Add(*elt))
if not args.get('polys', False):
return Mul(*poly).expand()
else:
return PurePoly(Mul(*poly), x)
def legendre_poly(n, x=None, polys=False):
"""Generates Legendre polynomial of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError("Cannot generate Legendre polynomial of degree %s" %
n)
poly = DMP(dup_legendre(int(n), QQ), QQ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def chebyshevu_poly(n, x=None, polys=False):
"""Generates Chebyshev polynomial of the second kind of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError(
"can't generate 2nd kind Chebyshev polynomial of degree %s" % n)
poly = DMP(dup_chebyshevu(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def laguerre_poly(n, x=None, alpha=None, polys=False):
"""Generates Laguerre polynomial of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
alpha
Decides minimal domain for the list
of coefficients.
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError("can't generate Laguerre polynomial of degree %s" % n)
if alpha is not None:
K, alpha = construct_domain(
alpha, field=True) # XXX: ground_field=True
else:
K, alpha = QQ, QQ(0)
poly = DMP(dup_laguerre(int(n), alpha, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def gegenbauer_poly(n, a, x=None, polys=False):
"""Generates Gegenbauer polynomial of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
a
Decides minimal domain for the list of
coefficients.
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError(
"can't generate Gegenbauer polynomial of degree %s" % n)
K, a = construct_domain(a, field=True)
poly = DMP(dup_gegenbauer(int(n), a, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def jacobi_poly(n, a, b, x=None, polys=False):
"""Generates Jacobi polynomial of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
a
Lower limit of minimal domain for the list of
coefficients.
b
Upper limit of minimal domain for the list of
coefficients.
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError("can't generate Jacobi polynomial of degree %s" % n)
K, v = construct_domain([a, b], field=True)
poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def swinnerton_dyer_poly(n, x=None, **args):
"""Generates n-th Swinnerton-Dyer polynomial in `x`. """
from .numberfields import minimal_polynomial
if n <= 0:
raise ValueError(
"can't generate Swinnerton-Dyer polynomial of order %s" % n)
if x is not None:
sympify(x)
else:
x = Dummy('x')
if n > 3:
p = 2
a = [sqrt(2)]
for i in range(2, n + 1):
p = nextprime(p)
a.append(sqrt(p))
return minimal_polynomial(Add(*a), x, polys=args.get('polys', False))
if n == 1:
ex = x**2 - 2
elif n == 2:
ex = x**4 - 10 * x**2 + 1
elif n == 3:
ex = x**8 - 40 * x**6 + 352 * x**4 - 960 * x**2 + 576
if not args.get('polys', False):
return ex
else:
return PurePoly(ex, x)
def laguerre_poly(n, x=None, alpha=None, polys=False):
"""Generates Laguerre polynomial of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
alpha
Decides minimal domain for the list
of coefficients.
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError("can't generate Laguerre polynomial of degree %s" % n)
if alpha is not None:
K, alpha = construct_domain(
alpha, field=True) # XXX: ground_field=True
else:
K, alpha = QQ, QQ(0)
poly = DMP(dup_laguerre(int(n), alpha, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def chebyshevu_poly(n, x=None, polys=False):
"""Generates Chebyshev polynomial of the second kind of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError(
"can't generate 2nd kind Chebyshev polynomial of degree %s" % n)
poly = DMP(dup_chebyshevu(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def gegenbauer_poly(n, a, x=None, polys=False):
"""Generates Gegenbauer polynomial of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
a
Decides minimal domain for the list of
coefficients.
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError(
"can't generate Gegenbauer polynomial of degree %s" % n)
K, a = construct_domain(a, field=True)
poly = DMP(dup_gegenbauer(int(n), a, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def jacobi_poly(n, a, b, x=None, polys=False):
"""Generates Jacobi polynomial of degree `n` in `x`.
Parameters
==========
n : int
`n` decides the degree of polynomial
a
Lower limit of minimal domain for the list of
coefficients.
b
Upper limit of minimal domain for the list of
coefficients.
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError("can't generate Jacobi polynomial of degree %s" % n)
K, v = construct_domain([a, b], field=True)
poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def as_poly(self, x=None):
"""Create a Poly instance from ``self``. """
if x is not None:
return Poly.new(self.rep, x)
else:
if self.alias is not None:
return Poly.new(self.rep, self.alias)
else:
return PurePoly.new(self.rep, Dummy('x'))
def test_pickling_polys_polytools():
from sympy.polys.polytools import Poly, PurePoly, GroebnerBasis
x = Symbol('x')
for c in (Poly, Poly(x, x)):
check(c)
for c in (PurePoly, PurePoly(x)):
check(c)
def as_poly(self, x=None):
"""Create a Poly instance from ``self``. """
if x is not None:
return Poly.new(self.rep, x)
else:
if self.alias is not None:
return Poly.new(self.rep, self.alias)
else:
return PurePoly.new(self.rep, Dummy('x'))
def spherical_bessel_fn(n, x=None, polys=False):
"""
Coefficients for the spherical Bessel functions.
Those are only needed in the jn() function.
The coefficients are calculated from:
fn(0, z) = 1/z
fn(1, z) = 1/z**2
fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
Examples
========
>>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
>>> from sympy import Symbol
>>> z = Symbol("z")
>>> fn(1, z)
z**(-2)
>>> fn(2, z)
-1/z + 3/z**3
>>> fn(3, z)
-6/z**2 + 15/z**4
>>> fn(4, z)
1/z - 45/z**3 + 105/z**5
"""
if n < 0:
dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
else:
dup = dup_spherical_bessel_fn(int(n), ZZ)
poly = DMP(dup, ZZ)
if x is not None:
poly = Poly.new(poly, 1/x)
else:
poly = PurePoly.new(poly, 1/Dummy('x'))
return poly if polys else poly.as_expr()
def spherical_bessel_fn(n, x=None, polys=False):
"""
Coefficients for the spherical Bessel functions.
Those are only needed in the jn() function.
The coefficients are calculated from:
fn(0, z) = 1/z
fn(1, z) = 1/z**2
fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)
Parameters
==========
n : int
`n` decides the degree of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
Examples
========
>>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
>>> from sympy import Symbol
>>> z = Symbol("z")
>>> fn(1, z)
z**(-2)
>>> fn(2, z)
-1/z + 3/z**3
>>> fn(3, z)
-6/z**2 + 15/z**4
>>> fn(4, z)
1/z - 45/z**3 + 105/z**5
"""
if n < 0:
dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
else:
dup = dup_spherical_bessel_fn(int(n), ZZ)
poly = DMP(dup, ZZ)
if x is not None:
poly = Poly.new(poly, 1/x)
else:
poly = PurePoly.new(poly, 1/Dummy('x'))
return poly if polys else poly.as_expr()
def __new__(cls, f, x, index=None, radicals=False, expand=True):
""" Construct an indexed complex root of a polynomial.
See ``rootof`` for the parameters.
The default value of ``radicals`` is ``False`` to satisfy
``eval(srepr(expr) == expr``.
"""
x = sympify(x)
if index is None and x.is_Integer:
x, index = None, x
else:
index = sympify(index)
if index is not None and index.is_Integer:
index = int(index)
else:
raise ValueError("expected an integer root index, got %s" % index)
poly = PurePoly(f, x, greedy=False, expand=expand)
if not poly.is_univariate:
raise PolynomialError("only univariate polynomials are allowed")
if not poly.gen.is_Symbol:
# PurePoly(sin(x) + 1) == PurePoly(x + 1) but the roots of
# x for each are not the same: issue 8617
raise PolynomialError("generator must be a Symbol")
degree = poly.degree()
if degree <= 0:
raise PolynomialError("can't construct CRootOf object for %s" % f)
if index < -degree or index >= degree:
raise IndexError("root index out of [%d, %d] range, got %d" %
(-degree, degree - 1, index))
elif index < 0:
index += degree
dom = poly.get_domain()
if not dom.is_Exact:
poly = poly.to_exact()
roots = cls._roots_trivial(poly, radicals)
if roots is not None:
return roots[index]
coeff, poly = preprocess_roots(poly)
dom = poly.get_domain()
if not dom.is_ZZ:
raise NotImplementedError("CRootOf is not supported over %s" % dom)
root = cls._indexed_root(poly, index)
return coeff * cls._postprocess_root(root, radicals)
def _new(cls, poly, index):
"""Construct new ``RootOf`` object from raw data. """
obj = Expr.__new__(cls)
obj.poly = PurePoly(poly)
obj.index = index
try:
_reals_cache[obj.poly] = _reals_cache[poly]
_complexes_cache[obj.poly] = _complexes_cache[poly]
except KeyError:
pass
return obj
def cyclotomic_poly(n, x=None, **args):
"""Generates cyclotomic polynomial of order `n` in `x`. """
if n <= 0:
raise ValueError("can't generate cyclotomic polynomial of order %s" % n)
poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def chebyshevu_poly(n, x=None, **args):
"""Generates Chebyshev polynomial of the second kind of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate 2nd kind Chebyshev polynomial of degree %s" % n)
poly = DMP(dup_chebyshevu(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def legendre_poly(n, x=None, **args):
"""Generates Legendre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Legendre polynomial of degree %s" % n)
poly = DMP(dup_legendre(int(n), QQ), QQ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def legendre_poly(n, x=None, **args):
"""Generates Legendre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Legendre polynomial of degree %s" % n)
poly = DMP(dup_legendre(int(n), QQ), QQ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def chebyshevu_poly(n, x=None, **args):
"""Generates Chebyshev polynomial of the second kind of degree `n` in `x`. """
if n < 0:
raise ValueError(
"can't generate 2nd kind Chebyshev polynomial of degree %s" % n)
poly = DMP(dup_chebyshevu(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def spherical_bessel_fn(n, x=None, **args):
"""
Coefficients for the spherical Bessel functions.
Those are only needed in the jn() function.
The coefficients are calculated from:
fn(0, z) = 1/z
fn(1, z) = 1/z**2
fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)
Examples
========
>>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
>>> from sympy import Symbol
>>> z = Symbol("z")
>>> fn(1, z)
z**(-2)
>>> fn(2, z)
-1/z + 3/z**3
>>> fn(3, z)
-6/z**2 + 15/z**4
>>> fn(4, z)
1/z - 45/z**3 + 105/z**5
"""
from sympy import sympify
if n < 0:
dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
else:
dup = dup_spherical_bessel_fn(int(n), ZZ)
poly = DMP(dup, ZZ)
if x is not None:
poly = Poly.new(poly, 1 / x)
else:
poly = PurePoly.new(poly, 1 / Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def jacobi_poly(n, a, b, x=None, **args):
"""Generates Jacobi polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Jacobi polynomial of degree %s" % n)
K, v = construct_domain([a, b], field=True)
poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def gegenbauer_poly(n, a, x=None, **args):
"""Generates Gegenbauer polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Gegenbauer polynomial of degree %s" % n)
K, a = construct_domain(a, field=True)
poly = DMP(dup_gegenbauer(int(n), a, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def spherical_bessel_fn(n, x=None, **args):
"""
Coefficients for the spherical Bessel functions.
Those are only needed in the jn() function.
The coefficients are calculated from:
fn(0, z) = 1/z
fn(1, z) = 1/z**2
fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)
Examples
========
>>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
>>> from sympy import Symbol
>>> z = Symbol("z")
>>> fn(1, z)
z**(-2)
>>> fn(2, z)
-1/z + 3/z**3
>>> fn(3, z)
-6/z**2 + 15/z**4
>>> fn(4, z)
1/z - 45/z**3 + 105/z**5
"""
from sympy import sympify
if n < 0:
dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
else:
dup = dup_spherical_bessel_fn(int(n), ZZ)
poly = DMP(dup, ZZ)
if x is not None:
poly = Poly.new(poly, 1/x)
else:
poly = PurePoly.new(poly, 1/Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def cyclotomic_poly(n, x=None, **args):
"""Generates cyclotomic polynomial of order `n` in `x`. """
if n <= 0:
raise ValueError("can't generate cyclotomic polynomial of order %s" %
n)
poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def jacobi_poly(n, a, b, x=None, **args):
"""Generates Jacobi polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Jacobi polynomial of degree %s" % n)
K, v = construct_domain([a, b], field=True)
poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def gegenbauer_poly(n, a, x=None, **args):
"""Generates Gegenbauer polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Gegenbauer polynomial of degree %s" %
n)
K, a = construct_domain(a, field=True)
poly = DMP(dup_gegenbauer(int(n), a, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def laguerre_poly(n, x=None, alpha=None, **args):
"""Generates Laguerre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Laguerre polynomial of degree %s" % n)
if alpha is not None:
K, alpha = construct_domain(alpha, field=True) # XXX: ground_field=True
else:
K, alpha = QQ, QQ(0)
poly = DMP(dup_laguerre(int(n), alpha, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def __new__(cls, f, x, index=None, radicals=True, expand=True):
"""Construct a new ``RootOf`` object for ``k``-th root of ``f``. """
x = sympify(x)
if index is None and x.is_Integer:
x, index = None, x
else:
index = sympify(index)
if index.is_Integer:
index = int(index)
else:
raise ValueError("expected an integer root index, got %d" % index)
poly = PurePoly(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 index < -degree or index >= degree:
raise IndexError("root index out of [%d, %d] range, got %d" %
(-degree, degree - 1, index))
elif index < 0:
index += degree
dom = poly.get_domain()
if not dom.is_Exact:
poly = poly.to_exact()
roots = cls._roots_trivial(poly, radicals)
if roots is not None:
return roots[index]
coeff, poly = preprocess_roots(poly)
dom = poly.get_domain()
if not dom.is_ZZ:
raise NotImplementedError("RootOf is not supported over %s" % dom)
root = cls._indexed_root(poly, index)
return coeff * cls._postprocess_root(root, radicals)
def swinnerton_dyer_poly(n, x=None, polys=False):
"""Generates n-th Swinnerton-Dyer polynomial in `x`.
Parameters
----------
n : int
`n` decides the order of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n <= 0:
raise ValueError(
"Cannot generate Swinnerton-Dyer polynomial of order %s" % n)
if x is not None:
sympify(x)
else:
x = Dummy('x')
if n > 3:
from sympy.functions.elementary.miscellaneous import sqrt
from .numberfields import minimal_polynomial
p = 2
a = [sqrt(2)]
for i in range(2, n + 1):
p = nextprime(p)
a.append(sqrt(p))
return minimal_polynomial(Add(*a), x, polys=polys)
if n == 1:
ex = x**2 - 2
elif n == 2:
ex = x**4 - 10 * x**2 + 1
elif n == 3:
ex = x**8 - 40 * x**6 + 352 * x**4 - 960 * x**2 + 576
return PurePoly(ex, x) if polys else ex
def laguerre_poly(n, x=None, alpha=None, **args):
"""Generates Laguerre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Laguerre polynomial of degree %s" % n)
if alpha is not None:
K, alpha = construct_domain(alpha,
field=True) # XXX: ground_field=True
else:
K, alpha = QQ, QQ(0)
poly = DMP(dup_laguerre(int(n), alpha, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def cyclotomic_poly(n, x=None, polys=False):
"""Generates cyclotomic polynomial of order `n` in `x`.
Parameters
----------
n : int
`n` decides the order of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n <= 0:
raise ValueError("can't generate cyclotomic polynomial of order %s" % n)
poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy("x"))
return poly if polys else poly.as_expr()
def legendre_poly(n, x=None, polys=False):
"""Generates Legendre polynomial of degree `n` in `x`.
Parameters
----------
n : int
`n` decides the degree of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n < 0:
raise ValueError("can't generate Legendre polynomial of degree %s" % n)
poly = DMP(dup_legendre(int(n), QQ), QQ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
def cyclotomic_poly(n, x=None, polys=False):
"""Generates cyclotomic polynomial of order `n` in `x`.
Parameters
----------
n : int
`n` decides the order of polynomial
x : optional
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
if n <= 0:
raise ValueError(
"can't generate cyclotomic polynomial of order %s" % n)
poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
return poly if polys else poly.as_expr()
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 _transform(cls, expr, x):
"""Transform an expression to a polynomial. """
poly = PurePoly(expr, x, greedy=False)
return preprocess_roots(poly)
def __setitem__(self, k, v):
if not isinstance(k, PurePoly):
if not (isinstance(k, Expr) and len(k.free_symbols) == 1):
raise ValueError('expecting univariate expression')
k = PurePoly(k, expand=False)
self._dict[k] = v
def test_sympy__polys__polytools__PurePoly():
from sympy.polys.polytools import PurePoly
assert _test_args(PurePoly(2, x, y))
def _pure_factors(poly):
_, factors = poly.factor_list()
return [(PurePoly(f, expand=False), m) for f, m in factors]
def __getitem__(self, k):
if not isinstance(k, PurePoly):
if not (isinstance(k, Expr) and len(k.free_symbols) == 1):
raise KeyError
k = PurePoly(k, expand=False)
return self._dict[k]
