def roots_cyclotomic(f, factor=False):
"""Compute roots of cyclotomic polynomials. """
L, U = _inv_totient_estimate(f.degree())
for n in xrange(L, U + 1):
g = cyclotomic_poly(n, f.gen, polys=True)
if f == g:
break
else: # pragma: no cover
raise RuntimeError("failed to find index of a cyclotomic polynomial")
roots = []
if not factor:
for k in xrange(1, n + 1):
if igcd(k, n) == 1:
roots.append(exp(2*k*S.Pi*I/n).expand(complex=True))
else:
g = Poly(f, extension=(-1)**Rational(1, n))
for h, _ in g.factor_list()[1]:
roots.append(-h.TC())
return sorted(roots, key=default_sort_key)
def roots_cyclotomic(f, factor=False):
"""Compute roots of cyclotomic polynomials. """
L, U = _inv_totient_estimate(f.degree())
for n in range(L, U + 1):
g = cyclotomic_poly(n, f.gen, polys=True)
if f == g:
break
else: # pragma: no cover
raise RuntimeError("failed to find index of a cyclotomic polynomial")
roots = []
if not factor:
# get the indices in the right order so the computed
# roots will be sorted
h = n//2
ks = [i for i in range(1, n + 1) if igcd(i, n) == 1]
ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1))
d = 2*I*pi/n
for k in reversed(ks):
roots.append(exp(k*d).expand(complex=True))
else:
g = Poly(f, extension=root(-1, n))
for h, _ in ordered(g.factor_list()[1]):
roots.append(-h.TC())
return roots
def random_poly(x, n, inf, sup, domain=ZZ, polys=False):
"""Return a polynomial of degree ``n`` with coefficients in ``[inf, sup]``. """
poly = Poly(dup_random(n, inf, sup, domain), x, domain=domain)
if not polys:
return poly.as_expr()
else:
return poly
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 _minimal_polynomial_sq(p, n, x):
"""
Returns the minimal polynomial for the ``nth-root`` of a sum of surds
or ``None`` if it fails.
Parameters
==========
p : sum of surds
n : positive integer
x : variable of the returned polynomial
Examples
========
>>> from sympy.polys.numberfields import _minimal_polynomial_sq
>>> from sympy import sqrt
>>> from sympy.abc import x
>>> q = 1 + sqrt(2) + sqrt(3)
>>> _minimal_polynomial_sq(q, 3, x)
x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8
"""
from sympy.simplify.simplify import _is_sum_surds
p = sympify(p)
n = sympify(n)
r = _is_sum_surds(p)
if not n.is_Integer or not n > 0 or not _is_sum_surds(p):
return None
pn = p**Rational(1, n)
# eliminate the square roots
p -= x
while 1:
p1 = _separate_sq(p)
if p1 is p:
p = p1.subs({x:x**n})
break
else:
p = p1
# _separate_sq eliminates field extensions in a minimal way, so that
# if n = 1 then `p = constant*(minimal_polynomial(p))`
# if n > 1 it contains the minimal polynomial as a factor.
if n == 1:
p1 = Poly(p)
if p.coeff(x**p1.degree(x)) < 0:
p = -p
p = p.primitive()[1]
return p
# by construction `p` has root `pn`
# the minimal polynomial is the factor vanishing in x = pn
factors = factor_list(p)[1]
result = _choose_factor(factors, x, pn)
return result
def _eval_sum_hyper(f, i, a):
""" Returns (res, cond). Sums from a to oo. """
from sympy.functions import hyper
from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
from sympy.polys.polytools import Poly, factor
from sympy.core.numbers import Float
if a != 0:
return _eval_sum_hyper(f.subs(i, i + a), i, 0)
if f.subs(i, 0) == 0:
if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0:
return S(0), True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
hs = hypersimp(f, i)
if hs is None:
return None
if isinstance(hs, Float):
from sympy.simplify.simplify import nsimplify
hs = nsimplify(hs)
numer, denom = fraction(factor(hs))
top, topl = numer.as_coeff_mul(i)
bot, botl = denom.as_coeff_mul(i)
ab = [top, bot]
factors = [topl, botl]
params = [[], []]
for k in range(2):
for fac in factors[k]:
mul = 1
if fac.is_Pow:
mul = fac.exp
fac = fac.base
if not mul.is_Integer:
return None
p = Poly(fac, i)
if p.degree() != 1:
return None
m, n = p.all_coeffs()
ab[k] *= m**mul
params[k] += [n/m]*mul
# Add "1" to numerator parameters, to account for implicit n! in
# hypergeometric series.
ap = params[0] + [1]
bq = params[1]
x = ab[0]/ab[1]
h = hyper(ap, bq, x)
return f.subs(i, 0)*hyperexpand(h), h.convergence_statement
def __new__(cls, expr, coeffs=Tuple(), alias=None, **args):
"""Construct a new algebraic number. """
expr = sympify(expr)
if isinstance(expr, (tuple, Tuple)):
minpoly, root = expr
if not minpoly.is_Poly:
minpoly = Poly(minpoly)
elif expr.is_AlgebraicNumber:
minpoly, root = expr.minpoly, expr.root
else:
minpoly, root = minimal_polynomial(
expr, args.get('gen'), polys=True), expr
dom = minpoly.get_domain()
if coeffs != Tuple():
if not isinstance(coeffs, ANP):
rep = DMP.from_sympy_list(sympify(coeffs), 0, dom)
scoeffs = Tuple(*coeffs)
else:
rep = DMP.from_list(coeffs.to_list(), 0, dom)
scoeffs = Tuple(*coeffs.to_list())
if rep.degree() >= minpoly.degree():
rep = rep.rem(minpoly.rep)
sargs = (root, scoeffs)
else:
rep = DMP.from_list([1, 0], 0, dom)
if ask(Q.negative(root)):
rep = -rep
sargs = (root, coeffs)
if alias is not None:
if not isinstance(alias, Symbol):
alias = Symbol(alias)
sargs = sargs + (alias,)
obj = Expr.__new__(cls, *sargs)
obj.rep = rep
obj.root = root
obj.alias = alias
obj.minpoly = minpoly
return obj
def _minpoly_pow(ex, pw, x, dom, mp=None):
"""
Returns ``minpoly(ex**pw, x)``
Parameters
==========
ex : algebraic element
pw : rational number
x : indeterminate of the polynomial
dom: ground domain
mp : minimal polynomial of ``p``
Examples
========
>>> from sympy import sqrt, QQ, Rational
>>> from sympy.polys.numberfields import _minpoly_pow, minpoly
>>> from sympy.abc import x, y
>>> p = sqrt(1 + sqrt(2))
>>> _minpoly_pow(p, 2, x, QQ)
x**2 - 2*x - 1
>>> minpoly(p**2, x)
x**2 - 2*x - 1
>>> _minpoly_pow(y, Rational(1, 3), x, QQ.frac_field(y))
x**3 - y
>>> minpoly(y**Rational(1, 3), x)
x**3 - y
"""
pw = sympify(pw)
if not mp:
mp = _minpoly_compose(ex, x, dom)
if not pw.is_rational:
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
if pw < 0:
if mp == x:
raise ZeroDivisionError('%s is zero' % ex)
mp = _invertx(mp, x)
if pw == -1:
return mp
pw = -pw
ex = 1/ex
y = Dummy(str(x))
mp = mp.subs({x: y})
n, d = pw.as_numer_denom()
res = Poly(resultant(mp, x**d - y**n, gens=[y]), x, domain=dom)
_, factors = res.factor_list()
res = _choose_factor(factors, x, ex**pw, dom)
return res.as_expr()
def horner(f, *gens, **args):
"""
Rewrite a polynomial in Horner form.
Among other applications, evaluation of a polynomial at a point is optimal
when it is applied using the Horner scheme ([1]).
Examples
========
>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e
>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
x*(x*(x*(9*x + 8) + 7) + 6) + 5
>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(a*x + b)))
>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
>>> horner(f, wrt=x)
x*(x*y*(4*y + 2) + y*(2*y + 1))
>>> horner(f, wrt=y)
y*(x*y*(4*x + 2) + x*(2*x + 1))
References
==========
[1] - http://en.wikipedia.org/wiki/Horner_scheme
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as 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 full_coeffs(p,D,s):
'''
Computes coefficients of a polynomial matrix
p : polynomial
D :degree up to which we want to complete the list with zeroes
s: variable of the polynomial matrix
Example: TODO
'''
p=Poly(p,s,domain='QQ') # in order to use the all_coeffs method of polynomial class
c=p.all_coeffs()
if len(c)==D+1:
return c
else:
difference=D+1-len(c)
return difference*[0] +c
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 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 hermite_poly(n, x=None, **args):
"""Generates Hermite polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Hermite polynomial of degree %s" % n)
if x is not None:
x = sympify(x)
else:
x = Dummy('x')
poly = Poly(DMP(dup_hermite(int(n), ZZ), ZZ), x)
if not args.get('polys', False):
return poly.as_basic()
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)
if x is not None:
x = sympify(x)
else:
x = Dummy('x')
poly = Poly(DMP(dup_chebyshevu(int(n), ZZ), ZZ), x)
if not args.get('polys', False):
return poly.as_basic()
else:
return poly
def laguerre_poly(n, x=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 x is not None:
x = sympify(x)
else:
x = Symbol('x', dummy=True)
poly = Poly(DMP(dup_laguerre(int(n), QQ), QQ), x)
if not args.get('polys', False):
return poly.as_basic()
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)
if x is not None:
x = sympify(x)
else:
x = Symbol('x', dummy=True)
poly = Poly(DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ), x)
if not args.get('polys', False):
return poly.as_basic()
else:
return poly
def chebyshevt_poly(n, x=None, **args):
"""Generates Chebyshev polynomial of the first kind of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate 1st kind Chebyshev polynomial of degree %s" % n)
if x is not None:
x = sympify(x)
else:
x = Symbol('x', dummy=True)
poly = Poly(DMP(dup_chebyshevt(int(n), ZZ), ZZ), x)
if not args.get('polys', False):
return poly.as_basic()
else:
return poly
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
if len(factors) == 1:
for root in _try_heuristics(factors[0]):
roots.append(root)
return (roots, False)
add_comment("Use the substitution")
t = Dummy("t1")
add_eq(t, compose_(factors[1:]).as_expr())
g = Poly(factors[0].as_expr().subs(f.gen, t), t)
add_comment("We have")
add_eq(g.as_expr(), 0)
for root in _try_heuristics(g):
roots.append(root)
# f(x) = f1(f2(f3(x))) = 0
# f(x) = f1(t) = 0 --> f2(f3(x)) = t1, t2, ..., tn
i = 1
for factor in factors[1:]:
previous, roots = list(roots), []
h = compose_(factors[i:]).as_expr()
add_comment("Therefore")
for root in previous:
add_eq(h, root)
for root in previous:
if i < len(factors) - 1:
add_comment("Solve the equation")
add_eq(h, root)
add_comment("Use the substitution")
t = Dummy("t" + str(i+1))
add_eq(t, compose_(factors[(i+1):]).as_expr())
g = Poly(factor.as_expr().subs(f.gen, t), t) - Poly(root, t)
else:
g = factor - Poly(root, f.gen)
for root in _try_heuristics(g):
roots.append(root)
i += 1
return (roots, True)
def root_factors(f, *gens, **args):
"""
Returns all factors of a univariate polynomial.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.polys.polyroots import root_factors
>>> root_factors(x**2 - y, x)
[x - sqrt(y), x + sqrt(y)]
"""
args = dict(args)
filter = args.pop('filter', None)
F = Poly(f, *gens, **args)
if not F.is_Poly:
return [f]
if F.is_multivariate:
raise ValueError('multivariate polynomials are not supported')
x = F.gens[0]
zeros = roots(F, filter=filter)
if not zeros:
factors = [F]
else:
factors, N = [], 0
for r, n in ordered(zeros.items()):
factors, N = factors + [Poly(x - r, x)]*n, N + n
if N < F.degree():
G = reduce(lambda p, q: p*q, factors)
factors.append(F.quo(G))
if not isinstance(f, Poly):
factors = [ f.as_expr() for f in factors ]
return factors
def _imag_count_of_factor(f):
"""Return the number of imaginary roots for irreducible
univariate polynomial ``f``.
"""
terms = [(i, j) for (i,), j in f.terms()]
if any(i % 2 for i, j in terms):
return 0
# update signs
even = [(i, I**i*j) for i, j in terms]
even = Poly.from_dict(dict(even), Dummy('x'))
return int(even.count_roots(-oo, oo))
def test_subterm(u, u_diff):
substituted = simplify(integrand / u_diff)
if symbol not in substituted.free_symbols:
# replaced everything already
return False
substituted_ = substituted.subs(u, u_var).cancel()
if symbol not in substituted_.free_symbols:
_, denom = integrand.as_numer_denom()
_, denom2 = substituted_.as_numer_denom()
denom2 = denom2.subs(u_var, u)
constant = denom/denom2
if symbol not in constant.free_symbols:
return (constant, substituted_ / constant)
if u.is_polynomial(symbol):
h = Poly(u, symbol)
if h.degree() == 1 and h.nth(0) != 0:
t = (u_var - h.nth(0)) / h.nth(1)
substituted_ = simplify(integrand.subs(symbol, t))
args = list(get_args(substituted_, u_var))
if symbol not in substituted_.free_symbols and len(args) == 1 and args[0] == u_var:
return (1/h.nth(1), substituted_)
return False
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 random_poly(x, n, inf, sup, domain=ZZ, polys=False):
"""Generates a polynomial of degree ``n`` with coefficients in
``[inf, sup]``.
Parameters
----------
x
`x` is the independent term of polynomial
n : int
`n` decides the order of polynomial
inf
Lower limit of range in which coefficients lie
sup
Upper limit of range in which coefficients lie
domain : optional
Decides what ring the coefficients are supposed
to belong. Default is set to Integers.
polys : bool, optional
``polys=True`` returns an expression, otherwise
(default) returns an expression.
"""
poly = Poly(dup_random(n, inf, sup, domain), x, domain=domain)
return poly if polys else poly.as_expr()
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 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 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 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 test_FiniteExtension_from_sympy():
# Test to_sympy/from_sympy
K = FiniteExtension(Poly(x**3 + 1, x, domain=ZZ))
xf = K(x)
assert K.from_sympy(x) == xf
assert K.to_sympy(xf) == x
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".
"""
from sympy import gcd
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_constantsimp_take_problem():
c = exp(C1) + 2
assert len(Poly(constantsimp(exp(C1) + c + c*x, [C1])).gens) == 2
def test_FiniteExtension_set_domain():
KZ = FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))
KQ = FiniteExtension(Poly(x**2 + 1, x, domain='QQ'))
assert KZ.set_domain(QQ) == KQ
def test_sympify_poly():
p = Poly(x**2 + x + 1, x)
assert _sympify(p) is p
assert sympify(p) is p
def minimal_polynomial(ex, x=None, **args):
"""Computes the minimal polynomial of an algebraic number. """
generator = numbered_symbols('a', cls=Dummy)
mapping, symbols, replace = {}, {}, []
ex = sympify(ex)
if x is not None:
x = sympify(x)
else:
x = S.Pure
def update_mapping(ex, exp, base=None):
a = generator.next()
symbols[ex] = a
if base is not None:
mapping[ex] = a**exp + base
else:
mapping[ex] = exp.as_expr(a)
return a
def bottom_up_scan(ex):
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational and ex.q != 0:
return ex
elif ex.is_Add:
return Add(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Mul:
return Mul(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0 and ex.base.is_Add:
coeff, terms = ex.base.as_coeff_add()
elt, _ = primitive_element(terms, polys=True)
alg = ex.base - coeff
# XXX: turn this into eval()
inverse = invert(elt.gen + coeff, elt).as_expr()
base = inverse.subs(elt.gen, alg).expand()
if ex.exp == -1:
return bottom_up_scan(base)
else:
ex = base**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (ex.base**ex.exp.p).expand(), Rational(
1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex.root not in mapping:
return update_mapping(ex.root, ex.minpoly)
else:
return symbols[ex.root]
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
polys = args.get('polys', False)
if ex.is_AlgebraicNumber:
if not polys:
return ex.minpoly.as_expr(x)
else:
return ex.minpoly.replace(x)
elif ex.is_Rational and ex.q != 0:
result = ex.q * x - ex.p
else:
F = [x - bottom_up_scan(ex)] + mapping.values()
G = groebner(F, symbols.values() + [x], order='lex')
_, factors = factor_list(G[-1])
if len(factors) == 1:
((result, _), ) = factors
else:
for result, _ in factors:
if result.subs(x, ex).evalf(chop=True) == 0:
break
else: # pragma: no cover
raise NotImplementedError(
"multiple candidates for the minimal polynomial of %s" %
ex)
if polys:
return Poly(result, x, field=True)
else:
return result
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))
{-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}
"""
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
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 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 += map(cancel, roots_linear(f))
elif n == 2:
result += map(cancel, roots_quadratic(f))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f)
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()
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:
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.iteritems():
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).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 sorted(zeros, key=default_sort_key)
def test_sympy__polys__polytools__Poly():
from sympy.polys.polytools import Poly
assert _test_args(Poly(2, x, y))
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 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 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_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 make_extension(K):
K = FiniteExtension(Poly(sin(p)**2+cos(p)**2-1, sin(p), domain=K[cos(p)]))
K = FiniteExtension(Poly(sin(t)**2+cos(t)**2-1, sin(t), domain=K[cos(t)]))
return K
def test_CRootOf_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_FiniteExtension_exquo():
# Test exquo
K = FiniteExtension(Poly(x**4 + 1))
xf = K(x)
assert K.exquo(xf**2 - 1, xf - 1) == xf + 1
def lfsr_connection_polynomial(s):
"""
This function computes the lsfr connection polynomial.
INPUT:
``s``: a sequence of elements of even length, with entries in a finite field
OUTPUT:
``C(x)``: the connection polynomial of a minimal LFSR yielding ``s``.
This implements the algorithm in section 3 of J. L. Massey's article [M]_.
References
==========
.. [M] James L. Massey, "Shift-Register Synthesis and BCH Decoding."
IEEE Trans. on Information Theory, vol. 15(1), pp. 122-127, Jan 1969.
Examples
========
>>> from sympy.crypto.crypto import lfsr_sequence, lfsr_connection_polynomial
>>> from sympy.polys.domains import FF
>>> F = FF(2)
>>> fill = [F(1), F(1), F(0), F(1)]
>>> key = [F(1), F(0), F(0), F(1)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**4 + x + 1
>>> fill = [F(1), F(0), F(0), F(1)]
>>> key = [F(1), F(1), F(0), F(1)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**3 + 1
>>> fill = [F(1), F(0), F(1)]
>>> key = [F(1), F(1), F(0)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**3 + x**2 + 1
>>> fill = [F(1), F(0), F(1)]
>>> key = [F(1), F(0), F(1)]
>>> s = lfsr_sequence(key, fill, 20)
>>> lfsr_connection_polynomial(s)
x**3 + x + 1
"""
# Initialization:
p = s[0].mod
F = FF(p)
x = Symbol("x")
C = 1 * x**0
B = 1 * x**0
m = 1
b = 1 * x**0
L = 0
N = 0
while N < len(s):
if L > 0:
dC = Poly(C).degree()
r = min(L + 1, dC + 1)
coeffsC = [C.subs(x, 0)
] + [C.coeff(x**i) for i in range(1, dC + 1)]
d = (s[N].to_int() +
sum([coeffsC[i] * s[N - i].to_int()
for i in range(1, r)])) % p
if L == 0:
d = s[N].to_int() * x**0
if d == 0:
m += 1
N += 1
if d > 0:
if 2 * L > N:
C = (C - d * ((b**(p - 2)) % p) * x**m * B).expand()
m += 1
N += 1
else:
T = C
C = (C - d * ((b**(p - 2)) % p) * x**m * B).expand()
L = N + 1 - L
m = 1
b = d
B = T
N += 1
dC = Poly(C).degree()
coeffsC = [C.subs(x, 0)] + [C.coeff(x**i) for i in range(1, dC + 1)]
return sum([
coeffsC[i] % p * x**i for i in range(dC + 1) if coeffsC[i] is not None
])
def solve_riccati(fx, x, b0, b1, b2, gensol=False):
"""
The main function that gives particular/general
solutions to Riccati ODEs that have atleast 1
rational particular solution.
"""
# Step 1 : Convert to Normal Form
a = -b0*b2 + b1**2/4 - b1.diff(x)/2 + 3*b2.diff(x)**2/(4*b2**2) + b1*b2.diff(x)/(2*b2) - \
b2.diff(x, 2)/(2*b2)
a_t = a.together()
num, den = [Poly(e, x, extension=True) for e in a_t.as_numer_denom()]
num, den = num.cancel(den, include=True)
# Step 2
presol = []
# Step 3 : a(x) is 0
if num == 0:
presol.append(1 / (x + Dummy('C1')))
# Step 4 : a(x) is a non-zero constant
elif x not in num.free_symbols.union(den.free_symbols):
presol.extend([sqrt(a), -sqrt(a)])
# Step 5 : Find poles and valuation at infinity
poles = roots(den, x)
poles, muls = list(poles.keys()), list(poles.values())
val_inf = val_at_inf(num, den, x)
if len(poles):
# Check necessary conditions (outlined in the module docstring)
if not check_necessary_conds(val_inf, muls):
raise ValueError("Rational Solution doesn't exist")
# Step 6
# Construct c-vectors for each singular point
c = construct_c(num, den, x, poles, muls)
# Construct d vectors for each singular point
d = construct_d(num, den, x, val_inf)
# Step 7 : Iterate over all possible combinations and return solutions
# For each possible combination, generate an array of 0's and 1's
# where 0 means pick 1st choice and 1 means pick the second choice.
# NOTE: We could exit from the loop if we find 3 particular solutions,
# but it is not implemented here as -
# a. Finding 3 particular solutions is very rare. Most of the time,
# only 2 particular solutions are found.
# b. In case we exit after finding 3 particular solutions, it might
# happen that 1 or 2 of them are redundant solutions. So, instead of
# spending some more time in computing the particular solutions,
# we will end up computing the general solution from a single
# particular solution which is usually slower than computing the
# general solution from 2 or 3 particular solutions.
c.append(d)
choices = product(*c)
for choice in choices:
m, ybar = compute_m_ybar(x, poles, choice, -val_inf // 2)
numy, deny = [
Poly(e, x, extension=True)
for e in ybar.together().as_numer_denom()
]
# Step 10 : Check if a valid solution exists. If yes, also check
# if m is a non-negative integer
if m.is_nonnegative == True and m.is_integer == True:
# Step 11 : Find polynomial solutions of degree m for the auxiliary equation
psol, coeffs, exists = solve_aux_eq(num, den, numy, deny, x, m)
# Step 12 : If valid polynomial solution exists, append solution.
if exists:
# m == 0 case
if psol == 1 and coeffs == 0:
# p(x) = 1, so p'(x)/p(x) term need not be added
presol.append(ybar)
# m is a positive integer and there are valid coefficients
elif len(coeffs):
# Substitute the valid coefficients to get p(x)
psol = psol.xreplace(coeffs)
# y(x) = ybar(x) + p'(x)/p(x)
presol.append(ybar + psol.diff(x) / psol)
# Remove redundant solutions from the list of existing solutions
remove = set()
for i in range(len(presol)):
for j in range(i + 1, len(presol)):
rem = remove_redundant_sols(presol[i], presol[j], x)
if rem is not None:
remove.add(rem)
sols = [x for x in presol if x not in remove]
# Step 15 : Inverse transform the solutions of the equation in normal form
bp = -b2.diff(x) / (2 * b2**2) - b1 / (2 * b2)
# If general solution is required, compute it from the particular solutions
if gensol:
sols = [get_gen_sol_from_part_sol(sols, a, x)]
# Inverse transform the particular solutions
presol = [
Eq(fx,
riccati_inverse_normal(y, x, b1, b2, bp).cancel(extension=True))
for y in sols
]
return presol
"""
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
>>> from sympy.abc import x
def rational_laurent_series(num, den, x, r, m, n):
r"""
The function computes the Laurent series coefficients
of a rational function.
Parameters
==========
num: A Poly object that is the numerator of `f(x)`.
den: A Poly object that is the denominator of `f(x)`.
x: The variable of expansion of the series.
r: The point of expansion of the series.
m: Multiplicity of r if r is a pole of `f(x)`. Should
be zero otherwise.
n: Order of the term upto which the series is expanded.
Returns
=======
series: A dictionary that has power of the term as key
and coefficient of that term as value.
Below is a basic outline of how the Laurent series of a
rational function `f(x)` about `x_0` is being calculated -
1. Substitute `x + x_0` in place of `x`. If `x_0`
is a pole of `f(x)`, multiply the expression by `x^m`
where `m` is the multiplicity of `x_0`. Denote the
the resulting expression as g(x). We do this substitution
so that we can now find the Laurent series of g(x) about
`x = 0`.
2. We can then assume that the Laurent series of `g(x)`
takes the following form -
.. math:: g(x) = \frac{num(x)}{den(x)} = \sum_{m = 0}^{\infty} a_m x^m
where `a_m` denotes the Laurent series coefficients.
3. Multiply the denominator to the RHS of the equation
and form a recurrence relation for the coefficients `a_m`.
"""
one = Poly(1, x, extension=True)
if r == oo:
# Series at x = oo is equal to first transforming
# the function from x -> 1/x and finding the
# series at x = 0
num, den = inverse_transform_poly(num, den, x)
r = S(0)
if r:
# For an expansion about a non-zero point, a
# transformation from x -> x + r must be made
num = num.transform(Poly(x + r, x, extension=True), one)
den = den.transform(Poly(x + r, x, extension=True), one)
# Remove the pole from the denominator if the series
# expansion is about one of the poles
num, den = (num * x**m).cancel(den, include=True)
# Equate coefficients for the first terms (base case)
maxdegree = 1 + max(num.degree(), den.degree())
syms = symbols(f'a:{maxdegree}', cls=Dummy)
diff = num - den * Poly(syms[::-1], x)
coeff_diffs = diff.all_coeffs()[::-1][:maxdegree]
(coeffs, ) = linsolve(coeff_diffs, syms)
# Use the recursion relation for the rest
recursion = den.all_coeffs()[::-1]
div, rec_rhs = recursion[0], recursion[1:]
series = list(coeffs)
while len(series) < n:
next_coeff = Add(*(c * series[-1 - n]
for n, c in enumerate(rec_rhs))) / div
series.append(-next_coeff)
series = {m - i: val for i, val in enumerate(series)}
return series
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 q(r1, r2, p):
return Poly(((x - r1) * (x - r2)).subs(x, x**p), x)
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):
extension = [ AlgebraicNumber(ext, gen=x) for ext in extension ]
g, coeffs = extension[0].minpoly.replace(x), [1]
for ext in extension[1:]:
s, _, g = sqf_norm(g, x, extension=ext)
coeffs = [ s*c for c in coeffs ] + [1]
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_gosper_normal():
eq = 4 * n + 5, 2 * (4 * n + 1) * (2 * n + 3), n
assert gosper_normal(*eq) == \
(Poly(Rational(1, 4), n), Poly(n + Rational(3, 2)), Poly(n + Rational(1, 4)))
assert gosper_normal(*eq, polys=False) == \
(Rational(1, 4), n + Rational(3, 2), n + Rational(1, 4))
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 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 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
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 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 + 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
"""
_, 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 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)
if f.length == 2 and f.degree() != 1:
# check for foo**n factors in the constant
n = f.degree()
npow_bases = []
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:
other.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, 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.length() == 2:
roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
for r in roots_fun(f):
_update_dict(result, 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, 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 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 test_CRootOf___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))
# issue 8617
raises(PolynomialError, lambda: rootof(exp(x), 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 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.to_sympy(a) for a in k.integral_basis()]
>>> 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