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 test_RootOf_all_roots():
assert Poly(x**5 + x + 1).all_roots() == [
RootOf(x**3 - x**2 + 1, 0),
-Rational(1, 2) - sqrt(3) * I / 2,
-Rational(1, 2) + sqrt(3) * I / 2,
RootOf(x**3 - x**2 + 1, 1),
RootOf(x**3 - x**2 + 1, 2),
]
assert Poly(x**5 + x + 1).all_roots(radicals=False) == [
RootOf(x**3 - x**2 + 1, 0),
RootOf(x**2 + x + 1, 0, radicals=False),
RootOf(x**2 + x + 1, 1, radicals=False),
RootOf(x**3 - x**2 + 1, 1),
RootOf(x**3 - x**2 + 1, 2),
]
r = Poly((x**3 + x + 20) * (x**3 + x + 21)).all_roots()
assert r[0].is_real and r[1].is_real
assert all(not _.is_real for _ in r[2:])
assert r == [
RootOf(x**3 + x + 21, 0),
RootOf(x**3 + x + 20, 0),
RootOf(x**3 + x + 20, 1),
RootOf(x**3 + x + 20, 2),
RootOf(x**3 + x + 21, 1),
RootOf(x**3 + x + 21, 2)
]
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 _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 diofant import sqrt
>>> from diofant.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 diofant.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
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 diofant.functions import hyper
from diofant.simplify import hyperexpand, hypersimp, fraction, simplify
from diofant.polys.polytools import Poly, factor
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 Integer(0), True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
hs = hypersimp(f, i)
if hs is None:
return
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
p = Poly(fac, i)
if p.degree() != 1:
return
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)
e = h
try:
e = hyperexpand(h)
except PolynomialError:
pass
if e is S.NaN and h.convergence_statement:
e = h
return f.subs(i, 0) * e, h.convergence_statement
def _rational_case(cls, poly, func):
"""Handle the rational function case. """
roots = symbols('r:%d' % poly.degree())
var, expr = func.variables[0], func.expr
f = sum(expr.subs(var, r) for r in roots)
p, q = together(f).as_numer_denom()
domain = QQ[roots]
p = p.expand()
q = q.expand()
try:
p = Poly(p, domain=domain, expand=False)
except GeneratorsNeeded:
p, p_coeff = None, (p, )
else:
p_monom, p_coeff = zip(*p.terms())
try:
q = Poly(q, domain=domain, expand=False)
except GeneratorsNeeded:
q, q_coeff = None, (q, )
else:
q_monom, q_coeff = zip(*q.terms())
coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True)
formulas, values = viete(poly, roots), []
for (sym, _), (_, val) in zip(mapping, formulas):
values.append((sym, val))
for i, (coeff, _) in enumerate(coeffs):
coeffs[i] = coeff.subs(values)
n = len(p_coeff)
p_coeff = coeffs[:n]
q_coeff = coeffs[n:]
if p is not None:
p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr()
else:
(p, ) = p_coeff
if q is not None:
q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr()
else:
(q, ) = q_coeff
return factor(p / q)
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 diofant import sqrt, QQ, Rational
>>> from diofant.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 fateman_poly_F_3(n):
"""Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
Y = [Symbol('y_' + str(i)) for i in range(0, n + 1)]
y_0 = Y[0]
u = Add(*[y**(n + 1) for y in Y[1:]])
H = Poly((y_0**(n + 1) + u + 1)**2, *Y)
F = Poly((y_0**(n + 1) - u - 2)**2, *Y)
G = Poly((y_0**(n + 1) + u + 2)**2, *Y)
return H * F, H * G, H
def fateman_poly_F_2(n):
"""Fateman's GCD benchmark: linearly dense quartic inputs """
Y = [Symbol('y_' + str(i)) for i in range(0, n + 1)]
y_0 = Y[0]
u = Add(*[y for y in Y[1:]])
H = Poly((y_0 + u + 1)**2, *Y)
F = Poly((y_0 - u - 2)**2, *Y)
G = Poly((y_0 + u + 2)**2, *Y)
return H * F, H * G, H
def test_RootOf_all_roots():
assert Poly(x**5 + x + 1).all_roots() == [
RootOf(x**3 - x**2 + 1, 0),
-Rational(1, 2) - sqrt(3) * I / 2,
-Rational(1, 2) + sqrt(3) * I / 2,
RootOf(x**3 - x**2 + 1, 1),
RootOf(x**3 - x**2 + 1, 2),
]
assert Poly(x**5 + x + 1).all_roots(radicals=False) == [
RootOf(x**3 - x**2 + 1, 0),
RootOf(x**2 + x + 1, 0, radicals=False),
RootOf(x**2 + x + 1, 1, radicals=False),
RootOf(x**3 - x**2 + 1, 1),
RootOf(x**3 - x**2 + 1, 2),
]
def sdm_to_vector(f, gens, K, n=None):
"""
Convert sdm ``f`` into a list of polynomial expressions.
The generators for the polynomial ring are specified via ``gens``. The rank
of the module is guessed, or passed via ``n``. The ground field is assumed
to be ``K``.
Examples
========
>>> from diofant.abc import x, y, z
>>> from diofant.polys import QQ, lex
>>> f = [((1, 0, 0, 1), QQ(2)), ((0, 2, 0, 0), QQ(1)), ((0, 0, 2, 0), QQ(1))]
>>> sdm_to_vector(f, [x, y, z], QQ)
[x**2 + y**2, 2*z]
"""
dic = sdm_to_dict(f)
dics = {}
for k, v in dic.items():
dics.setdefault(k[0], []).append((k[1:], v))
n = n or len(dics)
res = []
for k in range(n):
if k in dics:
res.append(Poly(dict(dics[k]), gens=gens, domain=K).as_expr())
else:
res.append(S.Zero)
return res
def root_factors(f, *gens, **args):
"""
Returns all factors of a univariate polynomial.
Examples
========
>>> from diofant.abc import x, y
>>> 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 test_RootOf_attributes():
r = RootOf(x**3 + x + 3, 0)
assert r.is_number
assert r.free_symbols == set()
# if the following assertion fails then multivariate polynomials
# are apparently supported and the RootOf.free_symbols routine
# should be changed to return whatever symbols would not be
# the PurePoly dummy symbol
pytest.raises(NotImplementedError,
lambda: RootOf(Poly(x**3 + y * x + 1, x), 0))
def _roots_quartic_euler(p, q, r, a):
"""
Descartes-Euler solution of the quartic equation
Parameters
==========
p, q, r: coefficients of ``x**4 + p*x**2 + q*x + r``
a: shift of the roots
Notes
=====
This is a helper function for ``roots_quartic``.
Look for solutions of the form ::
``x1 = sqrt(R) - sqrt(A + B*sqrt(R))``
``x2 = -sqrt(R) - sqrt(A - B*sqrt(R))``
``x3 = -sqrt(R) + sqrt(A - B*sqrt(R))``
``x4 = sqrt(R) + sqrt(A + B*sqrt(R))``
To satisfy the quartic equation one must have
``p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R``
so that ``R`` must satisfy the Descartes-Euler resolvent equation
``64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0``
If the resolvent does not have a rational solution, return None;
in that case it is likely that the Ferrari method gives a simpler
solution.
Examples
========
>>> from diofant import Rational, Integer
>>> p, q, r = -Rational(64, 5), -Rational(512, 125), -Rational(1024, 3125)
>>> _roots_quartic_euler(p, q, r, Integer(0))[0]
-sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5
"""
# solve the resolvent equation
x = Symbol('x')
eq = 64*x**3 + 32*p*x**2 + (4*p**2 - 16*r)*x - q**2
xsols = list(roots(Poly(eq, x), cubics=False).keys())
xsols = [sol for sol in xsols if sol.is_rational]
if not xsols:
return
R = max(xsols)
c1 = sqrt(R)
B = -q*c1/(4*R)
A = -R - p/2
c2 = sqrt(A + B)
c3 = sqrt(A - B)
return [c1 - c2 - a, -c1 - c3 - a, -c1 + c3 - a, c1 + c2 - a]
def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm. """
if not a.root.is_extended_real or not b.root.is_extended_real:
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
g = b.minpoly.replace(f.gen)
n, m, prev = 100, b.minpoly.degree(), None
for i in range(1, 5):
A = a.root.evalf(n)
B = b.root.evalf(n)
basis = [1, B] + [B**i for i in range(2, m)] + [A]
dps, mp.dps = mp.dps, n
coeffs = pslq(basis, maxcoeff=int(1e10), maxsteps=1000)
mp.dps = dps
if coeffs is None:
break
if coeffs != prev:
prev = coeffs
else:
break
coeffs = [sympify(c) / coeffs[-1] for c in coeffs[:-1]]
while not coeffs[-1]:
coeffs.pop()
coeffs = list(reversed(coeffs))
h = Poly(coeffs, f.gen, domain='QQ')
if f.compose(h).rem(g).is_zero:
d, approx = len(coeffs) - 1, 0
for i, coeff in enumerate(coeffs):
approx += coeff * B**(d - i)
if A * approx < 0:
return [-c for c in coeffs]
else:
return coeffs
elif f.compose(-h).rem(g).is_zero:
return [-c for c in coeffs]
else:
n *= 2
return
def fateman_poly_F_1(n):
"""Fateman's GCD benchmark: trivial GCD """
Y = [Symbol('y_' + str(i)) for i in range(0, n + 1)]
y_0, y_1 = Y[0], Y[1]
u = y_0 + Add(*[y for y in Y[1:]])
v = y_0**2 + Add(*[y**2 for y in Y[1:]])
F = ((u + 1) * (u + 2)).as_poly(*Y)
G = ((v + 1) * (-3 * y_1 * y_0**2 + y_1**2 - 1)).as_poly(*Y)
H = Poly(1, *Y)
return F, G, H
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 symmetric_poly(n, *gens, **args):
"""Generates symmetric polynomial of order `n`. """
gens = _analyze_gens(gens)
if n < 0 or n > len(gens) or not gens:
raise ValueError(
"can't generate symmetric polynomial of order %s for %s" %
(n, gens))
elif not n:
poly = S.One
else:
poly = Add(*[Mul(*s) for s in subsets(gens, int(n))])
if not args.get('polys', False):
return poly
else:
return Poly(poly, *gens)
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 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)
poly = DMP(dup_chebyshevt(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 test_RootOf_conjugate():
p = x**7 + x + 1
assert RootOf(p, 0).conjugate() == RootOf(p, 0)
assert RootOf(p, 1).conjugate() == RootOf(p, 2)
assert RootOf(p, 2).conjugate() == RootOf(p, 1)
assert RootOf(p, 6).conjugate() == RootOf(p, 5)
p2 = p * (x - 123)
assert RootOf(p2, 0).conjugate() == RootOf(p2, 0)
assert RootOf(p2, 1).conjugate() == RootOf(p2, 1)
assert RootOf(p2, 2).conjugate() == RootOf(p2, 3)
assert RootOf(p2, 3).conjugate() == RootOf(p2, 2)
assert RootOf(p2, 7).conjugate() == RootOf(p2, 6)
p3 = Poly(x**7 + x * y + 1, x)
assert RootOf(p3, x, 0).conjugate() == conjugate(RootOf(p3, x, 0),
evaluate=False)
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 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 diofant import Symbol
>>> z = Symbol("z")
>>> spherical_bessel_fn(1, z)
z**(-2)
>>> spherical_bessel_fn(2, z)
-1/z + 3/z**3
>>> spherical_bessel_fn(3, z)
-6/z**2 + 15/z**4
>>> spherical_bessel_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'))
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 test_RootOf_conjugate():
p = x**7 + x + 1
assert RootOf(p, 0).conjugate() == RootOf(p, 0)
assert RootOf(p, 1).conjugate() == RootOf(p, 2)
assert RootOf(p, 2).conjugate() == RootOf(p, 1)
assert RootOf(p, 6).conjugate() == RootOf(p, 5)
p2 = p * (x - 123)
assert RootOf(p2, 0).conjugate() == RootOf(p2, 0)
assert RootOf(p2, 1).conjugate() == RootOf(p2, 1)
assert RootOf(p2, 2).conjugate() == RootOf(p2, 3)
assert RootOf(p2, 3).conjugate() == RootOf(p2, 2)
assert RootOf(p2, 7).conjugate() == RootOf(p2, 6)
p3 = Poly(x**7 + x * y + 1, x)
assert RootOf(p3, x, 0).conjugate() == conjugate(RootOf(p3, x, 0),
evaluate=False)
p4 = x**12 - 4 * x**8 + 2 * x**6 + 4 * x**4 + 4 * x**2 + 1
r4 = RootOf(p4, 4)
r5 = RootOf(p4, 5)
assert r4.conjugate() == r5
assert r4.evalf() == -r5.evalf()
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [Integer(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
def test_sympyissue_15413():
assert Poly(sqrt(2) * x**3 + x,
x).all_roots() == [0, -I * root(2, -4), I * root(2, -4)]
def test_sympyissue_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_sympyissue_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