def test_fmpq_mat():
Q = flint.fmpq_mat
Z = flint.fmpz_mat
assert Q(1,2,[3,4]) == Z(1,2,[3,4])
assert Q(1,2,[3,4]) != Z(1,2,[5,4])
assert Q(1,2,[3,4]) != Q(1,2,[5,4])
assert Q(Q(1,2,[3,4])) == Q(1,2,[3,4])
assert Q(Z(1,2,[3,4])) == Q(1,2,[3,4])
assert Q(2,3,[1,2,3,4,5,6]) + Q(2,3,[4,5,6,7,8,9]) == Q(2,3,[5,7,9,11,13,15])
assert Q(2,3,[1,2,3,4,5,6]) - Q(2,3,[4,5,6,7,8,9]) == Q(2,3,[-3,-3,-3,-3,-3,-3])
assert Q(2,3,[1,2,3,4,5,6]) * Q(3,2,[4,5,6,7,8,9]) == Q(2,2,[40,46,94,109])
assert Q(2,3,[1,2,3,4,5,6]) * Z(3,2,[4,5,6,7,8,9]) == Q(2,2,[40,46,94,109])
assert Z(2,3,[1,2,3,4,5,6]) * Q(3,2,[4,5,6,7,8,9]) == Q(2,2,[40,46,94,109])
assert -Q(2,1,[2,5]) == Q(2,1,[-2,-5])
assert +Q(1,1,[3]) == Z(1,1,[3])
assert Q(1,2,[3,4]) * 2 == Q(1,2,[6,8])
assert Q(1,2,[3,4]) * flint.fmpq(1,3) == Q(1,2,[1,flint.fmpq(4,3)])
assert Q(1,2,[3,4]) * flint.fmpq(5,3) == Q(1,2,[5,flint.fmpq(20,3)])
assert 2 * Q(1,2,[3,4]) == Q(1,2,[6,8])
assert flint.fmpq(1,3) * Q(1,2,[3,4]) == Q(1,2,[1,flint.fmpq(4,3)])
assert Q(1,2,[3,4]) / 2 == Q(1,2,[flint.fmpq(3,2),2])
assert Q(1,2,[3,4]) / flint.fmpq(2,3) == Q(1,2,[flint.fmpq(9,2),6])
assert Q(3,2,range(6)).table() == Z(3,2,range(6)).table()
assert Q(3,2,range(6)).entries() == Z(3,2,range(6)).entries()
assert Q(3,2,range(6)).nrows() == 3
assert Q(3,2,range(6)).ncols() == 2
assert Q(2,2,[3,7,4,5]).det() == -13
assert (Q(2,2,[3,7,4,5]) / 5).det() == flint.fmpq(-13,25)
assert raises(lambda: Q(1,2,[1,2]).det(), ValueError)
assert ~~Q(2,2,[1,2,3,4]) == Q(2,2,[1,2,3,4])
assert raises(lambda: ~Q(2,2,[1,1,1,1]), ZeroDivisionError)
assert raises(lambda: ~Q(2,1,[1,1]), ValueError)
def test_nmod():
G = flint.nmod
assert G(0, 2) == G(2, 2) == G(-2, 2)
assert G(1, 2) != G(0, 2)
assert G(0, 2) != G(0, 3)
assert G(3, 5) == G(8, 5)
#assert G(3,5) == 8 # do we want this?
#assert 8 == G(3,5)
assert G(3, 5) != 7
assert 7 != G(3, 5)
assert G(-3, 5) == -G(3, 5) == G(2, 5)
assert G(2, 5) + G(1, 5) == G(3, 5)
assert G(2, 5) + 1 == G(3, 5)
assert 1 + G(2, 5) == G(3, 5)
assert G(2, 5) - G(3, 5) == G(4, 5)
assert G(2, 5) - 3 == G(4, 5)
assert 3 - G(2, 5) == G(1, 5)
assert G(2, 5) * G(3, 5) == G(1, 5)
assert G(2, 5) * 3 == G(1, 5)
assert 3 * G(2, 5) == G(1, 5)
assert G(3, 17) / G(2, 17) == G(10, 17)
assert G(3, 17) / 2 == G(10, 17)
assert 3 / G(2, 17) == G(10, 17)
assert G(3, 17) * flint.fmpq(11, 5) == G(10, 17)
assert G(3, 17) / flint.fmpq(11, 5) == G(6, 17)
assert raises(lambda: G(2, 5) / G(0, 5), ZeroDivisionError)
assert raises(lambda: G(2, 5) / 0, ZeroDivisionError)
assert raises(lambda: G(2, 5) + G(2, 7), ValueError)
assert raises(lambda: G(2, 5) - G(2, 7), ValueError)
assert raises(lambda: G(2, 5) * G(2, 7), ValueError)
assert raises(lambda: G(2, 5) / G(2, 7), ValueError)
assert G(3, 17).modulus() == 17
def test_nmod():
G = flint.nmod
assert G(0,2) == G(2,2) == G(-2,2)
assert G(1,2) != G(0,2)
assert G(0,2) != G(0,3)
assert G(3,5) == G(8,5)
#assert G(3,5) == 8 # do we want this?
#assert 8 == G(3,5)
assert G(3,5) != 7
assert 7 != G(3,5)
assert G(-3,5) == -G(3,5) == G(2,5)
assert G(2,5) + G(1,5) == G(3,5)
assert G(2,5) + 1 == G(3,5)
assert 1 + G(2,5) == G(3,5)
assert G(2,5) - G(3,5) == G(4,5)
assert G(2,5) - 3 == G(4,5)
assert 3 - G(2,5) == G(1,5)
assert G(2,5) * G(3,5) == G(1,5)
assert G(2,5) * 3 == G(1,5)
assert 3 * G(2,5) == G(1,5)
assert G(3,17) / G(2,17) == G(10,17)
assert G(3,17) / 2 == G(10,17)
assert 3 / G(2,17) == G(10,17)
assert G(3,17) * flint.fmpq(11,5) == G(10,17)
assert G(3,17) / flint.fmpq(11,5) == G(6,17)
assert raises(lambda: G(2,5) / G(0,5), ZeroDivisionError)
assert raises(lambda: G(2,5) / 0, ZeroDivisionError)
assert raises(lambda: G(2,5) + G(2,7), ValueError)
assert raises(lambda: G(2,5) - G(2,7), ValueError)
assert raises(lambda: G(2,5) * G(2,7), ValueError)
assert raises(lambda: G(2,5) / G(2,7), ValueError)
assert G(3,17).modulus() == 17
def test_fmpq_mat():
Q = flint.fmpq_mat
Z = flint.fmpz_mat
assert Q(1,2,[3,4]) == Z(1,2,[3,4])
assert Q(1,2,[3,4]) != Z(1,2,[5,4])
assert Q(1,2,[3,4]) != Q(1,2,[5,4])
assert Q(Q(1,2,[3,4])) == Q(1,2,[3,4])
assert Q(Z(1,2,[3,4])) == Q(1,2,[3,4])
assert Q(2,3,[1,2,3,4,5,6]) + Q(2,3,[4,5,6,7,8,9]) == Q(2,3,[5,7,9,11,13,15])
assert Q(2,3,[1,2,3,4,5,6]) - Q(2,3,[4,5,6,7,8,9]) == Q(2,3,[-3,-3,-3,-3,-3,-3])
assert Q(2,3,[1,2,3,4,5,6]) * Q(3,2,[4,5,6,7,8,9]) == Q(2,2,[40,46,94,109])
assert Q(2,3,[1,2,3,4,5,6]) * Z(3,2,[4,5,6,7,8,9]) == Q(2,2,[40,46,94,109])
assert Z(2,3,[1,2,3,4,5,6]) * Q(3,2,[4,5,6,7,8,9]) == Q(2,2,[40,46,94,109])
assert -Q(2,1,[2,5]) == Q(2,1,[-2,-5])
assert +Q(1,1,[3]) == Z(1,1,[3])
assert Q(1,2,[3,4]) * 2 == Q(1,2,[6,8])
assert Q(1,2,[3,4]) * flint.fmpq(1,3) == Q(1,2,[1,flint.fmpq(4,3)])
assert Q(1,2,[3,4]) * flint.fmpq(5,3) == Q(1,2,[5,flint.fmpq(20,3)])
assert 2 * Q(1,2,[3,4]) == Q(1,2,[6,8])
assert flint.fmpq(1,3) * Q(1,2,[3,4]) == Q(1,2,[1,flint.fmpq(4,3)])
assert Q(1,2,[3,4]) / 2 == Q(1,2,[flint.fmpq(3,2),2])
assert Q(1,2,[3,4]) / flint.fmpq(2,3) == Q(1,2,[flint.fmpq(9,2),6])
assert Q(3,2,range(6)).table() == Z(3,2,range(6)).table()
assert Q(3,2,range(6)).entries() == Z(3,2,range(6)).entries()
assert Q(3,2,range(6)).nrows() == 3
assert Q(3,2,range(6)).ncols() == 2
assert Q(2,2,[3,7,4,5]).det() == -13
assert (Q(2,2,[3,7,4,5]) / 5).det() == flint.fmpq(-13,25)
assert raises(lambda: Q(1,2,[1,2]).det(), ValueError)
assert Q(2,2,[1,2,3,4]).inv().inv() == Q(2,2,[1,2,3,4])
assert raises(lambda: Q(2,2,[1,1,1,1]).inv(), ZeroDivisionError)
assert raises(lambda: Q(2,1,[1,1]).inv(), ValueError)
assert raises(lambda: Q([1]), TypeError)
assert raises(lambda: Q([[1],[2,3]]), ValueError)
assert Q([[1,2,3],[4,5,6]]) == Q(2,3,[1,2,3,4,5,6])
def pow(self, n):
if self.is_rational():
c0 = self._minpoly[0]
c1 = self._minpoly[1]
if abs(n) > 2:
check_bits_limit(c0.height_bits() * abs(n))
check_bits_limit(c1.height_bits() * abs(n))
if n >= 0:
return alg(fmpq(c0**n) / (-c1)**n)
else:
return alg((-c1)**(-n) / fmpq(c0**(-n)))
if n == 0:
return alg(1)
if n == 1:
return self
if n < 0:
return (1 / self).pow(-n)
# fast detection of perfect powers
pol, d = self.minpoly().deflation()
if d % n == 0:
# todo: is factoring needed here?
H = list(self.minpoly())
H = H[::n]
H = fmpz_poly(H)
c, factors = H.factor()
prec = 64
orig_prec = ctx.prec
try:
while 1:
ctx.prec = prec
x = self.enclosure(pretty=True)
z = acb(x)**n
maybe_root = [
fac for fac, mult in factors if fac(z).contains(0)
]
if len(maybe_root) == 1:
fac = maybe_root[0]
z2 = alg._validate_root_enclosure(fac, z)
if z2 is not None:
return alg(_minpoly=fac, _enclosure=z2)
prec *= 2
finally:
ctx.prec = orig_prec
if n == 2:
return self * self
v = self.pow(n // 2)
v = v * v
if n % 2:
v *= self
return v
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 guessrational(q):
if isinstance(q, fmpq):
return q
minus, fraction = cfrac(q)
numer = 1
denom = 0
for j in fraction[::-1]:
denom, numer = numer, denom
numer += denom * j
numer *= minus
return fmpq(numer, denom)
def test_misc_bugs(self):
x = (
(1 /
(alg(3).sqrt() + alg(5).sqrt()) + 1).root(3) + 1)**2 + fmpq(1, 3)
x + alg.i()
u = (x + alg.i()).root(3)
P = alg.polynomial_roots([
340282366920938463463374607431768211456,
4454982093016792820459821542203090534400000000,
179577734764660553639051786303255347200000000000000,
-669109415404441101038544285174988800000000000000000000,
17329972089242952969786575138568878173828125
])[0][0].expr()
assert P is not None
def test_examples(self):
assert sum(1 / alg(n) for n in range(1, 101)) == fmpq.harmonic(100)
sqrt = lambda n: alg(n).sqrt()
x = sqrt(2 * sqrt(3) * sqrt(2 * sqrt(10) + 7) + 2 * sqrt(10) + 10)
y = sqrt(2) + sqrt(3) + sqrt(5)
assert x == y
def R(a, n):
return alg(a).root(n)
assert R(2, 1) + R(2, 2) - R(2, 2) - R(2, 1) == 0
assert R(2, 1) + R(2, 2) + R(2, 3) - R(2, 2) - R(2, 1) - R(2, 3) == 0
assert R(2, 4) + R(2, 1) + R(2, 2) + R(2, 3) - R(2, 2) - R(2, 1) - R(
2, 3) - R(2, 4) == 0
assert 7 < sum(R(2, n) for n in [1, 2, 3, 4, 5]) < fmpq(702, 100)
def exp_two_pi_i(x):
x = fmpq(x)
check_degree_limit(10 * x.q)
poly = fmpz_poly.cyclotomic(x.q)
prec = ctx.prec
ctx.prec = 64
try:
while 1:
a = arb.cos_pi_fmpq(2 * x)
b = arb.sin_pi_fmpq(2 * x)
z = acb(a, b)
z2 = alg._validate_root_enclosure(poly, z)
if z2 is not None:
return alg(_minpoly=poly, _enclosure=z2)
ctx.prec *= 2
finally:
ctx.prec = prec
def test_fmpz_mat():
M = flint.fmpz_mat
a = M(2, 3, [1, 2, 3, 4, 5, 6])
b = M(2, 3, [4, 5, 6, 7, 8, 9])
assert a == a
assert a == M(a)
assert a != b
assert a.nrows() == 2
assert a.ncols() == 3
assert a.entries() == [1, 2, 3, 4, 5, 6]
assert a.table() == [[1, 2, 3], [4, 5, 6]]
assert (a + b).entries() == [5, 7, 9, 11, 13, 15]
assert raises(a.det, ValueError)
assert +a == a
assert -a == M(2, 3, [-1, -2, -3, -4, -5, -6])
c = M(2, 2, [1, 2, 3, 4])
assert c.det() == -2
assert raises(lambda: a + c, ValueError)
assert (a * 3).entries() == [3, 6, 9, 12, 15, 18]
assert (3 * a).entries() == [3, 6, 9, 12, 15, 18]
assert (a * long(3)).entries() == [3, 6, 9, 12, 15, 18]
assert (long(3) * a).entries() == [3, 6, 9, 12, 15, 18]
assert (a * flint.fmpz(3)).entries() == [3, 6, 9, 12, 15, 18]
assert (flint.fmpz(3) * a).entries() == [3, 6, 9, 12, 15, 18]
assert M.randrank(5, 7, 3, 10).rank() == 3
A = M.randbits(5, 3, 2)
B = M.randtest(3, 7, 3)
C = M.randtest(7, 2, 4)
assert A * (B * C) == (A * B) * C
assert bool(M(2, 2, [0, 0, 0, 0])) == False
assert bool(M(2, 2, [0, 0, 0, 1])) == True
ctx.pretty = False
assert repr(M(2, 2, [1, 2, 3, 4])) == 'fmpz_mat(2, 2, [1, 2, 3, 4])'
ctx.pretty = True
assert str(M(2, 2, [1, 2, 3, 4])) == '[1, 2]\n[3, 4]'
assert M(1, 2, [3, 4]) * flint.fmpq(1, 3) == flint.fmpq_mat(
1, 2, [1, flint.fmpq(4, 3)])
assert flint.fmpq(1, 3) * M(1, 2, [3, 4]) == flint.fmpq_mat(
1, 2, [1, flint.fmpq(4, 3)])
assert M(1, 2, [3, 4]) / 3 == flint.fmpq_mat(1, 2, [1, flint.fmpq(4, 3)])
assert M(2, 2,
[1, 2, 3, 4
]).inv().det() == flint.fmpq(1) / M(2, 2, [1, 2, 3, 4]).det()
assert M(2, 2, [1, 2, 3, 4]).inv().inv() == M(2, 2, [1, 2, 3, 4])
assert raises(lambda: M.randrank(4, 3, 4, 1), ValueError)
assert raises(lambda: M.randrank(3, 4, 4, 1), ValueError)
assert M(1, 1, [3])**5 == M(1, 1, [3**5])
assert raises(lambda: M(1, 2)**3, ValueError)
assert raises(lambda: M(1, 1)**M(1, 1), TypeError)
assert raises(lambda: 1**M(1, 1), TypeError)
assert raises(lambda: M([1]), TypeError)
assert raises(lambda: M([[1], [2, 3]]), ValueError)
assert M([[1, 2, 3], [4, 5, 6]]) == M(2, 3, [1, 2, 3, 4, 5, 6])
def cos_pi(x, monic=False):
x = fmpq(x)
check_degree_limit(10 * x.q)
x2 = x / 2
poly = fmpz_poly.cos_minpoly(x2.q)
prec = ctx.prec
ctx.prec = 64
try:
while 1:
z = 2 * arb.cos_pi_fmpq(x)
z2 = alg._validate_root_enclosure(poly, z)
if z2 is not None:
v = alg(_minpoly=poly, _enclosure=z2)
if not monic:
v /= 2
return v
ctx.prec *= 2
finally:
ctx.prec = prec
def float_to_fmpq(c):
"""
**Description:**
Converts a float to an fmpq.
**Arguments:**
- `c` *(float)*: The input number.
**Returns:**
*(flint.fmpq)* The rational number that most reasonably approximates the
input.
**Example:**
We convert a few floats to rational numbers.
```python {2}
from cytools.utils import float_to_fmpq
float_to_fmpq(0.1), float_to_fmpq(0.333333333333), float_to_fmpq(2.45)
# (1/10, 1/3, 49/20)
```
"""
f = Fraction(c).limit_denominator()
return fmpq(f.numerator, f.denominator)
def test_fmpz_mat():
M = flint.fmpz_mat
a = M(2,3,[1,2,3,4,5,6])
b = M(2,3,[4,5,6,7,8,9])
assert a == a
assert a == M(a)
assert a != b
assert a.nrows() == 2
assert a.ncols() == 3
assert a.entries() == [1,2,3,4,5,6]
assert a.table() == [[1,2,3],[4,5,6]]
assert (a + b).entries() == [5,7,9,11,13,15]
assert raises(a.det, ValueError)
assert +a == a
assert -a == M(2,3,[-1,-2,-3,-4,-5,-6])
c = M(2,2,[1,2,3,4])
assert c.det() == -2
assert raises(lambda: a + c, ValueError)
assert (a * 3).entries() == [3,6,9,12,15,18]
assert (3 * a).entries() == [3,6,9,12,15,18]
assert (a * 3L).entries() == [3,6,9,12,15,18]
assert (3L * a).entries() == [3,6,9,12,15,18]
assert (a * flint.fmpz(3)).entries() == [3,6,9,12,15,18]
assert (flint.fmpz(3) * a).entries() == [3,6,9,12,15,18]
assert M.randrank(5,7,3,10).rank() == 3
A = M.randbits(5,3,2)
B = M.randtest(3,7,3)
C = M.randtest(7,2,4)
assert A*(B*C) == (A*B)*C
assert bool(M(2,2,[0,0,0,0])) == False
assert bool(M(2,2,[0,0,0,1])) == True
assert repr(M(2,2,[1,2,3,4])) == 'fmpz_mat(2, 2, [1, 2, 3, 4])'
assert str(M(2,2,[1,2,3,4])) == '[1, 2]\n[3, 4]'
assert M(1,2,[3,4]) * flint.fmpq(1,3) == flint.fmpq_mat(1, 2, [1, flint.fmpq(4,3)])
assert flint.fmpq(1,3) * M(1,2,[3,4]) == flint.fmpq_mat(1, 2, [1, flint.fmpq(4,3)])
assert M(1,2,[3,4]) / 3 == flint.fmpq_mat(1, 2, [1, flint.fmpq(4,3)])
assert (~M(2,2,[1,2,3,4])).det() == flint.fmpq(1) / M(2,2,[1,2,3,4]).det()
assert ~~M(2,2,[1,2,3,4]) == M(2,2,[1,2,3,4])
assert raises(lambda: M.randrank(4,3,4,1), ValueError)
assert raises(lambda: M.randrank(3,4,4,1), ValueError)
assert M(1,1,[3]) ** 5 == M(1,1,[3**5])
assert raises(lambda: M(1,2) ** 3, ValueError)
assert raises(lambda: M(1,1) ** M(1,1), TypeError)
assert raises(lambda: 1 ** M(1,1), TypeError)
def reconstruct_lehmer_lower(B, m, ys):
"""
Reconstructs the lower bits :math:`zs` of a system of linear congurential equations in lattice form with
:math:`B \\cdot xs = 0 \\mod m` for solution :math:`xs`, where :math:`xs = ys + zs`.
It works by computing a smaller basis :math:`B` from the basis :math:`L` and solving equations in the smaller basis.
:param B: The system of linear equation in matrix form (evaluating to 0)
:param m: The modulus used for all the equations
:param ys: Partial solutions for the variables (xs = ys + zs)
:return: The remaining operands of the solutions zs
"""
ys = fmpz_mat([[y] for y in ys])
Bys = B * ys
# TODO: There might be a better solution to find the individual ks
# NB: B * (ys + zs) = m * ks for some ks
ks = fmpz_mat([[_round_fmpq(fmpq(By) / m)] for By in Bys])
# We now solve the system of linear equations B zs = m * ks - B ys for zs
Bzs = m * ks - Bys
zs = B.solve(Bzs)
assert all(z.denom() == 1 for z in zs)
return [z.numer() % m for z in zs]
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_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 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
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 fmpq(self):
if not self.is_rational():
raise ValueError("fmpq() requires a rational value")
c0 = self._minpoly[0]
c1 = self._minpoly[1]
return fmpq(c0) / (-c1)
def as_quadratic(self, factor_limit=2**64):
"""
Assuming that *self* has degree 1 or 2, returns (*a*, *b*, *c*)
where *a* and *b* are *fmpq* rational numbers and *c* is a *fmpz*
integer, such that *self* equals `a + b \sqrt{c}`.
The integer *c* will not be a perfect square, but might not be
squarefree since ensuring this property requires integer factorization.
If *|c|* is initially smaller than *factor_limit*, it will be
completely factored into its squarefree part (with the square content
moved to *b*); otherwise, only a partial factorization will be performed.
>>> alg(-23).as_quadratic()
(-23, 0, 0)
>>> (2 + alg.i() / 3).as_quadratic()
(2, 1/3, -1)
>>> x = (alg(5)/2 + alg(24).sqrt())**5
>>> x.as_quadratic()
(353525/32, 36341/8, 6)
>>> x == alg(353525)/32 + alg(36341)/8 * alg(6).sqrt()
True
>>> alg(3 * 29**2 * (10**20+39)**2).sqrt().as_quadratic()
(0, 2900000000000000001131, 3)
>>> x = alg(3 * (10**10+1)**3 * (10**15+1)**2).sqrt()
>>> a, b, c = x.as_quadratic()
>>> (a, b, c)
(0, 39340116598834051, 1938429316132524961593868203)
>>> x == a + b*alg(c).sqrt()
True
>>> a, b, c = x.as_quadratic(factor_limit=10**50)
>>> (a, b, c)
(0, 10000000001000010000000001, 30000000003)
>>> x == a + b*alg(c).sqrt()
True
"""
poly = self._minpoly
if poly.degree() == 1:
return (self.fmpq(), fmpq(), fmpz())
if poly.degree() != 2:
raise ValueError
c, b, a = list(self.minpoly())
D = b**2 - 4 * a * c
Dsqrt = alg(D).sqrt()
x = (-b + Dsqrt) / (2 * a)
if self == x:
bsign = 1
else:
x = (-b - Dsqrt) / (2 * a)
bsign = -1
a, b, c = fmpq(-b) / (2 * a), fmpq(bsign, 2 * a), D
if abs(c) >= 4:
if abs(c) < factor_limit:
fac = c.factor()
else:
fac = c.factor(trial_limit=1000)
# todo: the partial factoring in flint is wonky;
# it should at least do a perfect power test or single-word
# factorisation of the last factor
if 1:
rem, reme = fac[-1]
if rem < factor_limit:
fac = fac[:-1] + [(pp, ee * reme)
for (pp, ee) in rem.factor()]
elif rem.is_perfect_power():
for e in range(64, 1, -1):
p = rem.root(e)
if p**e == rem:
if p < factor_limit:
fac2 = rem.factor()
fac = fac[:-1]
for (p, f) in fac2:
fac.append((p, e * f * reme))
else:
fac = fac[:-1] + [(p, e * reme)]
break
check = 1
for p, e in fac:
check *= p**e
# assert check == abs(c)
square = fmpz(1)
squarefree = fmpz(1)
for p, e in fac:
if e % 2 == 0:
square *= p**(e // 2)
else:
square *= p**(e // 2)
squarefree *= p
if c < 0:
squarefree = -squarefree
b *= square
c = squarefree
# assert self == alg(a) + alg(b) * alg(c).sqrt()
return a, b, c
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 w():
return alg(-1)**fmpq(2, 3)
def enclosure(self, pretty=False):
if self.degree() == 1:
c0 = self._minpoly[0]
c1 = self._minpoly[1]
return arb(c0) / (-c1)
elif pretty:
z = self.enclosure()
re = z.real
im = z.imag
# detect zero real/imag part
if z.real.contains(0):
if self.real_sgn() == 0:
re = arb(0)
if z.imag.contains(0):
if self.imag_sgn() == 0:
im = arb(0)
# detect exact (dyadic) real/imag parts
# fixme: extracting real and imag parts and checking if
# they are exact can be slow at high degree; this is a workaround
# until that operation can be improved
scale = fmpz(2)**max(10, ctx.prec - 20)
if not re.is_exact() and (re * scale).unique_fmpz() is not None:
n = (re * scale).unique_fmpz()
b = self - fmpq(n, scale)
if b.real_sgn() == 0:
re = arb(fmpq(n, scale))
if not im.is_exact() and (im * scale).unique_fmpz() is not None:
n = (im * scale).unique_fmpz()
b = self - fmpq(n, scale) * alg.i()
if b.imag_sgn() == 0:
im = arb(fmpq(n, scale))
if im == 0:
return arb(re)
else:
return acb(re, im)
else:
orig_prec = ctx.prec
x = self._enclosure
if x.rel_accuracy_bits() >= orig_prec - 2:
return x
# Try interval newton refinement
f = self._minpoly
g = self._minpoly.derivative()
try:
ctx.prec = max(x.rel_accuracy_bits(), 32) + 10
for step in range(40):
ctx.prec *= 2
if ctx.prec > 1000000:
raise ValueError("excessive precision")
# print("root-refinement prec", ctx.prec)
# print(x.mid().str(10), x.rad().str(10))
xmid = x.mid()
y = xmid - f(xmid) / g(x)
if y.rel_accuracy_bits() >= 1.1 * orig_prec:
self._enclosure = y
return y
if y.rel_accuracy_bits() < 1.5 * x.rel_accuracy_bits() + 1:
# print("refinement failed -- recomputing roots")
roots = self._minpoly.roots()
near = [r for (r, mult) in roots if acb(r).overlaps(x)]
if len(near) == 1:
y = near[0]
if y.rel_accuracy_bits() >= 1.1 * orig_prec:
self._enclosure = y
return y
x = y
raise ValueError("root refinement did not converge")
finally:
ctx.prec = orig_prec
def sin_pi(x, monic=False):
return alg.cos_pi(fmpq(1, 2) - x, monic=monic)
def exp_pi_i(x):
x = fmpq(x)
return alg.exp_two_pi_i(x / 2)