def test_fmpq_poly():
Q = flint.fmpq_poly
Z = flint.fmpz_poly
assert Q() == Q([]) == Q([0]) == Q([0, 0])
assert Q() != Q([1])
assert Q([1]) == Q([1])
assert bool(Q()) == False
assert bool(Q([1])) == True
assert Q(Q([1, 2])) == Q([1, 2])
assert Q(Z([1, 2])) == Q([1, 2])
assert Q([1, 2]) + 3 == Q([4, 2])
assert 3 + Q([1, 2]) == Q([4, 2])
assert Q([1, 2]) - 3 == Q([-2, 2])
assert 3 - Q([1, 2]) == Q([2, -2])
assert -Q([1, 2]) == Q([-1, -2])
assert Q([flint.fmpq(1, 2), 1]) * 2 == Q([1, 2])
assert Q([1, 2]) == Z([1, 2])
assert Z([1, 2]) == Q([1, 2])
assert Q([1, 2]) != Z([3, 2])
assert Z([1, 2]) != Q([3, 2])
assert Q([1, 2, 3]) * Q([1, 2]) == Q([1, 4, 7, 6])
assert Q([1, 2, 3]) * Z([1, 2]) == Q([1, 4, 7, 6])
assert Q([1, 2, 3]) * 3 == Q([3, 6, 9])
assert 3 * Q([1, 2, 3]) == Q([3, 6, 9])
assert Q([1, 2, 3]) * flint.fmpq(2, 3) == (Q([1, 2, 3]) * 2) / 3
assert flint.fmpq(2, 3) * Q([1, 2, 3]) == (Q([1, 2, 3]) * 2) / 3
assert raises(lambda: Q([1, 2]) / Q([1, 2]), TypeError)
assert Q([1, 2, 3]) / flint.fmpq(2, 3) == Q([1, 2, 3]) * flint.fmpq(3, 2)
assert Q([1, 2, 3])**2 == Q([1, 2, 3]) * Q([1, 2, 3])
assert Q([1, 2, flint.fmpq(1, 2)]).coeffs() == [1, 2, flint.fmpq(1, 2)]
assert Q().coeffs() == []
assert Q().degree() == -1
assert Q([1]).degree() == 0
assert Q([1, 2]).degree() == 1
assert Q().length() == 0
assert Q([1]).length() == 1
assert Q([1, 2]).length() == 2
assert (Q([1, 2, 3]) / 5).numer() == (Q([1, 2, 3]) / 5).p == Z([1, 2, 3])
assert (Q([1, 2, 3]) / 5).denom() == (Q([1, 2, 3]) / 5).q == 5
ctx.pretty = False
assert repr(Q([15, 20, 10]) / 25) == "fmpq_poly([3, 4, 2], 5)"
ctx.pretty = True
assert str(Q([3, 4, 2], 5)) == "2/5*x^2 + 4/5*x + 3/5"
a = Q([2, 2, 3], 4)
assert a[2] == flint.fmpq(3, 4)
a[2] = 4
assert a == Q([1, 1, 8], 2)
p = Q([3, 4, 5], 7)
assert p(2) == flint.fmpq(31, 7)
assert p(flint.fmpq(2, 3)) == flint.fmpq(71, 63)
assert p(Z([1, -1])) == Q([12, -14, 5], 7)
assert p(flint.fmpq_poly([2, 3], 5)) == flint.fmpq_poly([27, 24, 9], 7 * 5)
def composed_op(P1, P2, operation):
d1 = P1.degree()
d2 = P2.degree()
# print("compose", d1, d2)
assert d1 >= 1 and d2 >= 1
cap = d1 * d2 + 1
if operation == operator.truediv:
P2 = list(P2)
if P2[0] == 0:
raise ZeroDivisionError
P2 = fmpq_poly(P2[::-1])
if operation == operator.sub:
P2 = list(P2)
for i in range(1, len(P2), 2):
P2[i] = -P2[i]
P2 = fmpq_poly(P2)
orig_cap = ctx.cap
try:
ctx.cap = cap
P1rev = list(P1)[::-1]
P1drev = list(P1.derivative())[::-1]
P2rev = list(P2)[::-1]
P2drev = list(P2.derivative())[::-1]
NP1 = fmpq_series(P1drev) / fmpq_series(P1rev)
NP2 = fmpq_series(P2drev) / fmpq_series(P2rev)
if operation in (operator.add, operator.sub):
c = fmpq(1)
a1, a2 = [NP1[0]], [NP2[0]]
for j in range(1, cap):
c *= j
a1.append(NP1[j] / c)
a2.append(NP2[j] / c)
NP1E = fmpq_series(a1)
NP2E = fmpq_series(a2)
NP3E = NP1E * NP2E
c = fmpq(-1)
a3 = [fmpq(0)]
for j in range(1, cap):
a3.append(NP3E[j] * c)
c *= j
NP = fmpq_series(a3)
Q = NP
else:
NP = fmpq_series([-NP1[j] * NP2[j] for j in range(1, cap)])
Q = NP.integral()
Q = Q.exp()
Q = fmpq_poly([Q[i] for i in range(cap)][::-1])
Q = Q.p
return Q
finally:
# print("end compose")
ctx.cap = orig_cap
def test_fmpq_poly():
Q = flint.fmpq_poly
Z = flint.fmpz_poly
assert Q() == Q([]) == Q([0]) == Q([0,0])
assert Q() != Q([1])
assert Q([1]) == Q([1])
assert bool(Q()) == False
assert bool(Q([1])) == True
assert Q(Q([1,2])) == Q([1,2])
assert Q(Z([1,2])) == Q([1,2])
assert Q([1,2]) + 3 == Q([4,2])
assert 3 + Q([1,2]) == Q([4,2])
assert Q([1,2]) - 3 == Q([-2,2])
assert 3 - Q([1,2]) == Q([2,-2])
assert -Q([1,2]) == Q([-1,-2])
assert Q([flint.fmpq(1,2),1]) * 2 == Q([1,2])
assert Q([1,2]) == Z([1,2])
assert Z([1,2]) == Q([1,2])
assert Q([1,2]) != Z([3,2])
assert Z([1,2]) != Q([3,2])
assert Q([1,2,3])*Q([1,2]) == Q([1,4,7,6])
assert Q([1,2,3])*Z([1,2]) == Q([1,4,7,6])
assert Q([1,2,3]) * 3 == Q([3,6,9])
assert 3 * Q([1,2,3]) == Q([3,6,9])
assert Q([1,2,3]) * flint.fmpq(2,3) == (Q([1,2,3]) * 2) / 3
assert flint.fmpq(2,3) * Q([1,2,3]) == (Q([1,2,3]) * 2) / 3
assert raises(lambda: Q([1,2]) / Q([1,2]), TypeError)
assert Q([1,2,3]) / flint.fmpq(2,3) == Q([1,2,3]) * flint.fmpq(3,2)
assert Q([1,2,3]) ** 2 == Q([1,2,3]) * Q([1,2,3])
assert Q([1,2,flint.fmpq(1,2)]).coeffs() == [1,2,flint.fmpq(1,2)]
assert Q().coeffs() == []
assert Q().degree() == -1
assert Q([1]).degree() == 0
assert Q([1,2]).degree() == 1
assert Q().length() == 0
assert Q([1]).length() == 1
assert Q([1,2]).length() == 2
assert (Q([1,2,3]) / 5).numer() == (Q([1,2,3]) / 5).p == Z([1,2,3])
assert (Q([1,2,3]) / 5).denom() == (Q([1,2,3]) / 5).q == 5
assert repr(Q([15,20,10]) / 25) == "fmpq_poly([3, 4, 2], 5)"
assert str(Q([3,4,2],5)) == "2/5*x^2 + 4/5*x + 3/5"
a = Q([2,2,3],4)
assert a[2] == flint.fmpq(3,4)
a[2] = 4
assert a == Q([1,1,8],2)
p = Q([3,4,5],7)
assert p(2) == flint.fmpq(31,7)
assert p(flint.fmpq(2,3)) == flint.fmpq(71,63)
assert p(Z([1,-1])) == Q([12,-14,5],7)
assert p(flint.fmpq_poly([2,3],5)) == flint.fmpq_poly([27,24,9],7*5)
def test_fmpz_poly():
Z = flint.fmpz_poly
assert Z() == Z([])
assert Z() == Z([0])
assert Z() == Z([0, flint.fmpz(0), 0])
assert Z() == Z([0, 0, 0])
assert Z() != Z([1])
assert Z([1]) == Z([1])
assert Z([1]) == Z([flint.fmpz(1)])
assert Z(Z([1, 2])) == Z([1, 2])
for ztype in [int, long, flint.fmpz]:
assert Z([1, 2, 3]) + ztype(5) == Z([6, 2, 3])
assert ztype(5) + Z([1, 2, 3]) == Z([6, 2, 3])
assert Z([1, 2, 3]) - ztype(5) == Z([-4, 2, 3])
assert ztype(5) - Z([1, 2, 3]) == Z([4, -2, -3])
assert Z([1, 2, 3]) * ztype(5) == Z([5, 10, 15])
assert ztype(5) * Z([1, 2, 3]) == Z([5, 10, 15])
assert Z([11, 6, 2]) // ztype(5) == Z([2, 1])
assert ztype(5) // Z([-2]) == Z([-3])
assert ztype(5) // Z([1, 2]) == 0
assert Z([11, 6, 2]) % ztype(5) == Z([1, 1, 2])
assert ztype(5) % Z([-2]) == Z([-1])
assert ztype(5) % Z([1, 2]) == 5
assert Z([1, 2, 3])**ztype(0) == 1
assert Z([1, 2, 3])**ztype(1) == Z([1, 2, 3])
assert Z([1, 2, 3])**ztype(2) == Z([1, 4, 10, 12, 9])
assert +Z([1, 2]) == Z([1, 2])
assert -Z([1, 2]) == Z([-1, -2])
assert raises(lambda: Z([1, 2, 3])**-1, (OverflowError, ValueError))
assert raises(lambda: Z([1, 2, 3])**Z([1, 2]), TypeError)
assert raises(lambda: Z([1, 2]) // Z([]), ZeroDivisionError)
assert raises(lambda: Z([]) // Z([]), ZeroDivisionError)
assert raises(lambda: Z([1, 2]) % Z([]), ZeroDivisionError)
assert raises(lambda: divmod(Z([1, 2]), Z([])), ZeroDivisionError)
assert Z([]).degree() == -1
assert Z([]).length() == 0
p = Z([1, 2])
assert p.length() == 2
assert p.degree() == 1
assert p[0] == 1
assert p[1] == 2
assert p[2] == 0
assert p[-1] == 0
assert raises(lambda: p.__setitem__(-1, 1), ValueError)
p[0] = 3
assert p[0] == 3
p[4] = 7
assert p.degree() == 4
assert p[4] == 7
assert p[3] == 0
p[4] = 0
assert p.degree() == 1
assert p.coeffs() == [3, 2]
assert Z([]).coeffs() == []
assert bool(Z([])) == False
assert bool(Z([1])) == True
ctx.pretty = False
assert repr(Z([1, 2])) == "fmpz_poly([1, 2])"
ctx.pretty = True
assert str(Z([1, 2])) == "2*x + 1"
p = Z([3, 4, 5])
assert p(2) == 31
assert p(flint.fmpq(2, 3)) == flint.fmpq(71, 9)
assert p(Z([1, -1])) == Z([12, -14, 5])
assert p(flint.fmpq_poly([2, 3], 5)) == flint.fmpq_poly([27, 24, 9], 5)
assert p(flint.arb("1.1")).overlaps(flint.arb("13.45"))
assert p(flint.acb("1.1", "1.2")).overlaps(flint.acb("6.25", "18.00"))
def test_fmpz_poly():
Z = flint.fmpz_poly
assert Z() == Z([])
assert Z() == Z([0])
assert Z() == Z([0,flint.fmpz(0),0])
assert Z() == Z([0,0L,0])
assert Z() != Z([1])
assert Z([1]) == Z([1L])
assert Z([1]) == Z([flint.fmpz(1)])
assert Z(Z([1,2])) == Z([1,2])
for ztype in [int, long, flint.fmpz]:
assert Z([1,2,3]) + ztype(5) == Z([6,2,3])
assert ztype(5) + Z([1,2,3]) == Z([6,2,3])
assert Z([1,2,3]) - ztype(5) == Z([-4,2,3])
assert ztype(5) - Z([1,2,3]) == Z([4,-2,-3])
assert Z([1,2,3]) * ztype(5) == Z([5,10,15])
assert ztype(5) * Z([1,2,3]) == Z([5,10,15])
assert Z([11,6,2]) // ztype(5) == Z([2,1])
assert ztype(5) // Z([-2]) == Z([-3])
assert ztype(5) // Z([1,2]) == 0
assert Z([11,6,2]) % ztype(5) == Z([1,1,2])
assert ztype(5) % Z([-2]) == Z([-1])
assert ztype(5) % Z([1,2]) == 5
assert Z([1,2,3]) ** ztype(0) == 1
assert Z([1,2,3]) ** ztype(1) == Z([1,2,3])
assert Z([1,2,3]) ** ztype(2) == Z([1,4,10,12,9])
assert +Z([1,2]) == Z([1,2])
assert -Z([1,2]) == Z([-1,-2])
assert raises(lambda: Z([1,2,3]) ** -1, (OverflowError, ValueError))
assert raises(lambda: Z([1,2,3]) ** Z([1,2]), TypeError)
assert raises(lambda: Z([1,2]) // Z([]), ZeroDivisionError)
assert raises(lambda: Z([]) // Z([]), ZeroDivisionError)
assert raises(lambda: Z([1,2]) % Z([]), ZeroDivisionError)
assert raises(lambda: divmod(Z([1,2]), Z([])), ZeroDivisionError)
assert Z([]).degree() == -1
assert Z([]).length() == 0
p = Z([1,2])
assert p.length() == 2
assert p.degree() == 1
assert p[0] == 1
assert p[1] == 2
assert p[2] == 0
assert p[-1] == 0
assert raises(lambda: p.__setitem__(-1, 1), ValueError)
p[0] = 3
assert p[0] == 3
p[4] = 7
assert p.degree() == 4
assert p[4] == 7
assert p[3] == 0
p[4] = 0
assert p.degree() == 1
assert p.coeffs() == [3,2]
assert Z([]).coeffs() == []
assert bool(Z([])) == False
assert bool(Z([1])) == True
assert repr(Z([1,2])) == "fmpz_poly([1, 2])"
assert str(Z([1,2])) == "2*x+1"
p = Z([3,4,5])
assert p(2) == 31
assert p(flint.fmpq(2,3)) == flint.fmpq(71,9)
assert p(Z([1,-1])) == Z([12,-14,5])
assert p(flint.fmpq_poly([2,3],5)) == flint.fmpq_poly([27,24,9],5)
def polynomial_roots(poly, canonical_order=True):
"""
Returns all the roots of the given *fmpz_poly* or *fmpq_poly*,
or list of coefficients that can be converted to an *fmpq_poly*.
(At this time, this function does not support algebraic coefficients.)
The output is a list of pairs (*root*, *m*) where root
is an algebraic number and *m* >= 1 is its multiplicity.
With *canonical_order* set to *True*, the roots are sorted in
a mathematically well-defined way, ensuring a consistent
ordering of algebraic numbers.
"""
poly = fmpq_poly(poly).numer()
# todo: flag to assume irreducible?
c, factors = poly.factor()
roots = []
orig = ctx.prec
maxprec = 64
try:
for fac, mult in factors:
if fac.degree() == 1:
a, b = list(fac)
roots.append((alg(fmpq(-a, b)), mult))
continue
ctx.prec = 64
while 1:
fac_roots = fac.roots()
checked_roots = []
for r, _ in fac_roots:
r = alg._validate_root_enclosure(fac, r)
if r is not None:
checked_roots.append(r)
else:
break
if len(checked_roots) == len(fac_roots):
for i in range(len(fac_roots)):
a = alg(_minpoly=fac, _enclosure=checked_roots[i])
roots.append((a, mult))
break
ctx.prec *= 2
maxprec = max(maxprec, ctx.prec)
if len(roots) > 1 and canonical_order:
from functools import cmp_to_key
real = []
nonreal = []
ctx.prec = 64
for (r, m) in roots:
if r.is_real():
real.append((r, m))
else:
z = r.enclosure()
if z.imag > 0:
nonreal.append((r, m))
else:
assert z.imag < 0
def complex_algebraic_cmp(a, b):
a, _ = a
b, _ = b
ctx.prec = 64
while 1:
ac = a.enclosure()
bc = b.enclosure()
ac = ac.real + ac.imag * arb.pi()
bc = bc.real + bc.imag * arb.pi()
if ac.overlaps(bc):
ctx.prec *= 2
else:
if ac < bc:
return -1
else:
return 1
roots = sorted(real, reverse=True)
nonreal = sorted(nonreal,
key=cmp_to_key(complex_algebraic_cmp),
reverse=True)
for r, m in nonreal:
roots.append((r, m))
roots.append((r.conjugate(), m))
finally:
ctx.prec = orig
return roots
from flint import ctx, arb, acb, fmpz, fmpq, fmpz_poly, fmpq_poly, fmpq_series, fmpz_mat, fmpq_mat
import operator
_fmpz_poly_x = fmpq_poly([0, 1])
# algorithm borrowed from sage
def composed_op(P1, P2, operation):
d1 = P1.degree()
d2 = P2.degree()
# print("compose", d1, d2)
assert d1 >= 1 and d2 >= 1
cap = d1 * d2 + 1
if operation == operator.truediv:
P2 = list(P2)
if P2[0] == 0:
raise ZeroDivisionError
P2 = fmpq_poly(P2[::-1])
if operation == operator.sub:
P2 = list(P2)
for i in range(1, len(P2), 2):
P2[i] = -P2[i]
P2 = fmpq_poly(P2)
orig_cap = ctx.cap
try:
ctx.cap = cap
P1rev = list(P1)[::-1]
P1drev = list(P1.derivative())[::-1]
P2rev = list(P2)[::-1]
P2drev = list(P2.derivative())[::-1]
NP1 = fmpq_series(P1drev) / fmpq_series(P1rev)
