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test_all.py
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import math
import operator
import pickle
import platform
import random
from flint.utils.flint_exceptions import DomainError, IncompatibleContextError
import flint
PYPY = platform.python_implementation() == "PyPy"
ctx = flint.ctx
def raises(f, exception):
try:
f()
except exception:
return True
return False
_default_ctx_string = """\
pretty = True # pretty-print repr() output
unicode = False # use unicode characters in output
prec = 53 # real/complex precision (in bits)
dps = 15 # real/complex precision (in digits)
cap = 10 # power series precision
threads = 1 # max number of threads used internally\
"""
def test_pyflint():
assert flint.__version__ == "0.7.0a5"
ctx = flint.ctx
assert str(ctx) == repr(ctx) == _default_ctx_string
assert ctx.prec == 53
oldprec = ctx.prec
try:
f1 = flint.arb(2).sqrt()
ctx.prec = 10
f2 = flint.arb(2).sqrt()
assert f1.rad() < f2.rad()
assert 1e-17 < f1.rad() < 1e-15
assert 1e-3 < f2.rad() < 1e-2
finally:
ctx.prec = oldprec
assert ctx.dps == 15
olddps = ctx.dps
try:
f1 = flint.arb(2).sqrt()
ctx.dps = 3
f2 = flint.arb(2).sqrt()
assert f1.rad() < f2.rad()
assert 1e-17 < f1.rad() < 1e-15
assert 1e-4 < f2.rad() < 1e-3
finally:
ctx.prec = oldprec
assert ctx.cap == 10
oldcap = ctx.cap
try:
f1 = flint.fmpz_series([1,1])
ctx.cap = 5
f2 = flint.fmpz_series([1,1])
assert f1._equal_repr(flint.fmpz_series([1,1],10))
assert f2._equal_repr(flint.fmpz_series([1,1],5))
finally:
ctx.cap = oldcap
assert raises(lambda: setattr(ctx, "cap", -1), ValueError)
assert raises(lambda: setattr(ctx, "prec", -1), ValueError)
assert raises(lambda: setattr(ctx, "dps", -1), ValueError)
def test_showgood():
from flint import good, showgood
def test_fmpz():
assert flint.fmpz() == flint.fmpz(0)
L = [0, 1, 2, 3, 2**31-1, 2**31, 2**63-1, 2**63, 2**64-1, 2**64]
L += [-x for x in L]
for i in L:
f = flint.fmpz(i)
assert int(f) == i
assert flint.fmpz(f) == f
assert flint.fmpz(str(i)) == f
assert raises(lambda: flint.fmpz(1,2), TypeError)
assert raises(lambda: flint.fmpz("qwe"), ValueError)
assert raises(lambda: flint.fmpz([]), TypeError)
for s in L:
for t in L:
for ltype in (flint.fmpz, int):
for rtype in (flint.fmpz, int):
assert (ltype(s) == rtype(t)) == (s == t)
assert (ltype(s) != rtype(t)) == (s != t)
assert (ltype(s) < rtype(t)) == (s < t)
assert (ltype(s) <= rtype(t)) == (s <= t)
assert (ltype(s) > rtype(t)) == (s > t)
assert (ltype(s) >= rtype(t)) == (s >= t)
assert ltype(s) + rtype(t) == s + t
assert ltype(s) - rtype(t) == s - t
assert ltype(s) * rtype(t) == s * t
assert ltype(s) & rtype(t) == s & t
assert ltype(s) | rtype(t) == s | t
assert ltype(s) ^ rtype(t) == s ^ t
assert ~ltype(s) == ~s
if t == 0:
assert raises(lambda: ltype(s) // rtype(t), ZeroDivisionError)
assert raises(lambda: ltype(s) % rtype(t), ZeroDivisionError)
assert raises(lambda: divmod(ltype(s), rtype(t)), ZeroDivisionError)
else:
assert ltype(s) // rtype(t) == s // t
assert ltype(s) % rtype(t) == s % t
assert divmod(ltype(s), rtype(t)) == divmod(s, t)
if 0 <= t < 10:
assert (ltype(s) ** rtype(t)) == (s ** t)
assert ltype(s) << rtype(t) == s << t
assert ltype(s) >> rtype(t) == s >> t
elif -10 <= t < 0:
assert raises(lambda: ltype(s) << rtype(t), ValueError)
assert raises(lambda: ltype(s) >> rtype(t), ValueError)
assert 2 ** flint.fmpz(2) == 4
assert type(2 ** flint.fmpz(2)) == flint.fmpz
assert raises(lambda: () ** flint.fmpz(1), TypeError)
assert raises(lambda: flint.fmpz(1) ** (), TypeError)
assert raises(lambda: flint.fmpz(1) ** -1, ValueError)
mega = flint.fmpz(2) ** 8000000
assert raises(lambda: mega ** mega, OverflowError)
pow_mod_examples = [
(2, 2, 3, 1),
(2, -1, 5, 3),
(2, 0, 5, 1),
(2, 5, 1000, 32),
]
for a, b, c, ab_mod_c in pow_mod_examples:
assert pow(a, b, c) == ab_mod_c
assert pow(flint.fmpz(a), b, c) == ab_mod_c
assert pow(flint.fmpz(a), flint.fmpz(b), c) == ab_mod_c
assert pow(flint.fmpz(a), b, flint.fmpz(c)) == ab_mod_c
assert pow(flint.fmpz(a), flint.fmpz(b), flint.fmpz(c)) == ab_mod_c
# 3-arg pow cannot be made to work with fmpz on PyPy
# https://github.com/flintlib/python-flint/issues/74
if not PYPY:
assert pow(a, flint.fmpz(b), c) == ab_mod_c
assert pow(a, b, flint.fmpz(c)) == ab_mod_c
assert pow(a, flint.fmpz(b), flint.fmpz(c)) == ab_mod_c
assert raises(lambda: pow(flint.fmpz(2), 2, 0), ValueError)
# XXX: Handle negative modulus like int?
assert raises(lambda: pow(flint.fmpz(2), 2, -1), ValueError)
assert raises(lambda: pow(flint.fmpz(2), "asd", 2), TypeError)
assert raises(lambda: pow(flint.fmpz(2), 2, "asd"), TypeError)
f = flint.fmpz(2)
assert f.numerator == f
assert type(f.numerator) is flint.fmpz
assert f.denominator == 1
assert type(f.denominator) is flint.fmpz
assert int(f) == 2
assert type(int(f)) is int
assert operator.index(f) == 2
assert type(operator.index(f)) is int
assert float(f) == 2.0
assert type(float(f)) is float
assert round(f) == 2
assert type(round(f)) is flint.fmpz
assert round(f, 1) == 2
assert type(round(f, 1)) is flint.fmpz
assert round(f, -1) == 0
assert type(round(f, -1)) is flint.fmpz
assert math.trunc(f) == 2
assert type(math.trunc(f)) is flint.fmpz
assert math.floor(f) == 2
assert type(math.floor(f)) is flint.fmpz
assert math.ceil(f) == 2
assert type(math.ceil(f)) is flint.fmpz
assert flint.fmpz(2) != []
assert +flint.fmpz(0) == 0
assert +flint.fmpz(1) == 1
assert +flint.fmpz(-1) == -1
assert -flint.fmpz(0) == 0
assert -flint.fmpz(1) == -1
assert -flint.fmpz(-1) == 1
assert abs(flint.fmpz(0)) == 0
assert abs(flint.fmpz(1)) == 1
assert abs(flint.fmpz(-1)) == 1
assert bool(flint.fmpz(0)) is False
assert bool(flint.fmpz(1)) is True
assert flint.fmpz(2).bit_length() == 2
assert flint.fmpz(-2).bit_length() == 2
assert flint.fmpz(2).height_bits() == 2
assert flint.fmpz(-2).height_bits() == 2
assert flint.fmpz(2).height_bits(signed=True) == 2
assert flint.fmpz(-2).height_bits(signed=True) == -2
f1 = flint.fmpz(1)
f2 = flint.fmpz(2)
f3 = flint.fmpz(3)
f8 = flint.fmpz(8)
assert f2 << 2 == 8
assert f2 << f2 == 8
assert 2 << f2 == 8
assert raises(lambda: f2 << -1, ValueError)
assert raises(lambda: 2 << -f1, ValueError)
assert f8 >> 2 == f2
assert f8 >> f2 == f2
assert 8 >> f2 == f2
assert raises(lambda: f2 >> -1, ValueError)
assert raises(lambda: 2 >> -f1, ValueError)
assert f2 & 3 == 2
assert f2 & f3 == 2
assert 2 & f3 == 2
assert f2 | 3 == 3
assert f2 | f3 == 3
assert 2 | f3 == 3
assert f2 ^ 3 == 1
assert f2 ^ f3 == 1
assert 2 ^ f3 == 1
assert raises(lambda: f2 << (), TypeError)
assert raises(lambda: () << f2, TypeError)
assert raises(lambda: f2 >> (), TypeError)
assert raises(lambda: () >> f2, TypeError)
assert raises(lambda: f2 & (), TypeError)
assert raises(lambda: () & f2, TypeError)
assert raises(lambda: f2 | (), TypeError)
assert raises(lambda: () | f2, TypeError)
assert raises(lambda: f2 ^ (), TypeError)
assert raises(lambda: () ^ f2, TypeError)
l = [1, 2, 3]
l[flint.fmpz(1)] = -2
assert l == [1, -2, 3]
d = {flint.fmpz(2): 3}
d[flint.fmpz(2)] = -1
assert d == {flint.fmpz(2): -1}
ctx.pretty = False
assert repr(flint.fmpz(0)) == "fmpz(0)"
assert repr(flint.fmpz(-27)) == "fmpz(-27)"
ctx.pretty = True
assert repr(flint.fmpz(0)) == "0"
assert repr(flint.fmpz(-27)) == "-27"
bigstr = '1' * 100
big = flint.fmpz(bigstr)
assert big.str() == bigstr
assert big.str(condense=10) == '1111111111{...80 digits...}1111111111'
def test_fmpz_factor():
assert flint.fmpz(6).gcd(flint.fmpz(9)) == 3
assert flint.fmpz(6).gcd(9) == 3
assert raises(lambda: flint.fmpz(2).gcd('asd'), TypeError)
assert flint.fmpz(6).lcm(flint.fmpz(9)) == 18
assert flint.fmpz(6).lcm(9) == 18
assert raises(lambda: flint.fmpz(2).lcm('asd'), TypeError)
assert flint.fmpz(25).factor() == [(5, 2)]
n = flint.fmpz(10**100 + 1)
assert n.factor() == [
(73, 1), (137, 1), (401, 1), (1201, 1), (1601, 1), (1676321, 1), (5964848081, 1),
(129694419029057750551385771184564274499075700947656757821537291527196801, 1)]
assert n.factor(trial_limit=100) == [
(73, 1), (137, 1), (401, 1),
(2493516234411471571047384039650897753117456334167082044912715710972543643391271845384040149601, 1)]
assert n.factor_smooth() == [
(73, 1), (137, 1), (401, 1), (1201, 1), (1601, 1),
(1296814508839693536173209832765271992846610925502473758289451540212712414540699659186801, 1)]
def test_fmpz_functions():
T, F, VE, OE, DE = True, False, ValueError, OverflowError, DomainError
cases = [
# (f, [f(-1), f(0), f(1), f(2), ... f(10)]),
(lambda n: flint.fmpz(n).is_prime(),
[0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0]),
(lambda n: flint.fmpz(n).is_probable_prime(),
[0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0]),
(lambda n: flint.fmpz(n).is_perfect_power(),
[T, T, T, F, F, T, F, F, F, T, T, F]),
(lambda n: flint.fmpz(n).is_square(),
[F, T, T, F, F, T, F, F, F, F, T, F]),
(lambda n: flint.fmpz(n).partitions_p(),
[0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42]),
(lambda n: flint.fmpz(n).moebius_mu(),
[1, 0, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1]),
(lambda n: flint.fmpz.fac_ui(n),
[OE, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]),
(lambda n: flint.fmpz.primorial_ui(n),
[OE, 1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310]),
(lambda n: flint.fmpz.fib_ui(n),
[OE, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]),
(lambda n: flint.fmpz(n).rising(0),
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]),
(lambda n: flint.fmpz(n).rising(1),
[-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]),
(lambda n: flint.fmpz(n).rising(2),
[0, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110]),
(lambda n: flint.fmpz.bin_uiui(n, 0),
[OE, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]),
(lambda n: flint.fmpz.bin_uiui(n, 1),
[OE, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]),
(lambda n: flint.fmpz.bin_uiui(n, 2),
[OE, 0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45]),
(lambda n: flint.fmpz.bell_number(n),
[OE, 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]),
(lambda n: flint.fmpz.euler_number(n),
[OE, 1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]),
(lambda n: flint.fmpz.stirling_s1(n, 1),
[OE, 0, 1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880]),
(lambda n: flint.fmpz.stirling_s2(n, 2),
[OE, 0, 0, 1, 3, 7, 15, 31, 63, 127, 255, 511]),
(lambda n: flint.fmpz(n).divisor_sigma(2),
[1, 0, 1, 5, 10, 21, 26, 50, 50, 85, 91, 130]),
(lambda n: flint.fmpz(n).euler_phi(),
[0, 0, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4]),
(lambda n: flint.fmpz(n).isqrt(),
[DE, 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3]),
(lambda n: flint.fmpz(n).sqrtrem(),
[DE, (0, 0), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), (2, 3), (2, 4), (3, 0), (3, 1)]),
(lambda n: flint.fmpz(n).sqrtmod(11),
[DE, 0, 1, DE, 5, 2, 4, DE, DE, DE, 3, DE]),
(lambda n: flint.fmpz(n).root(3),
[VE, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2]),
(lambda n: flint.fmpz(n).jacobi(3),
[-1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1]),
(lambda n: flint.fmpz(2).jacobi(n),
[1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1]),
]
is_exception = lambda v: isinstance(v, type) and issubclass(v, Exception)
for func, values in cases:
for n, val in enumerate(values, -1):
if is_exception(val):
assert raises(lambda: func(n), val)
else:
assert func(n) == val
assert raises(lambda: flint.fmpz(1).root(-1), ValueError)
assert raises(lambda: flint.fmpz(1).jacobi('bad'), TypeError)
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])
assert raises(lambda: Z([1,2,[]]), TypeError)
assert raises(lambda: Z({}), TypeError)
# XXX: This should probably be made to work:
assert raises(lambda: Z((1,2,3)), TypeError)
for ztype in [int, 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 divmod(Z([11,6,2]), Z([1,2])) == (Z([2,1]), Z([9,1]))
assert raises(lambda: pow(Z([1,2]), 2, 3), NotImplementedError)
assert +Z([1,2]) == Z([1,2])
assert -Z([1,2]) == Z([-1,-2])
assert raises(lambda: Z([1,2]) + [], TypeError)
assert raises(lambda: Z([1,2]) - [], TypeError)
assert raises(lambda: Z([1,2]) * [], TypeError)
assert raises(lambda: Z([1,2]) / [], TypeError)
assert raises(lambda: Z([1,2]) // [], TypeError)
assert raises(lambda: Z([1,2]) % [], TypeError)
assert raises(lambda: divmod(Z([1,2]), []), TypeError)
assert raises(lambda: [] + Z([1,2]), TypeError)
assert raises(lambda: [] - Z([1,2]), TypeError)
assert raises(lambda: [] * Z([1,2]), TypeError)
assert raises(lambda: [] / Z([1,2]), TypeError)
assert raises(lambda: [] // Z([1,2]), TypeError)
assert raises(lambda: [] % Z([1,2]), TypeError)
assert raises(lambda: divmod([], Z([1,2])), TypeError)
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 raises(lambda: Z([1,2]) < Z([1,2]), TypeError)
assert raises(lambda: Z([1,2]) < [], TypeError)
assert raises(lambda: [] < Z([1,2]), TypeError)
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([])) is False
assert bool(Z([1])) is True
ctx.pretty = False
assert repr(Z([1,2])) == "fmpz_poly([1, 2])"
ctx.pretty = True
assert str(Z([1,2])) == "2*x + 1"
assert str(Z([])) == "0"
assert str(Z([1,0,2])) == "2*x^2 + 1"
assert str(Z([-1,0,2])) == "2*x^2 + (-1)"
assert str(Z([-1,0,1])) == "x^2 + (-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"))
assert raises(lambda: p(None), TypeError)
assert Z([1,2,3]).derivative() == Z([2,6])
assert Z([1,2,-4]).height_bits() == 3
assert Z([1,2,-4]).height_bits(signed=True) == -3
assert Z([1,2,1]).sqrt() == Z([1,1])
assert raises(lambda: Z([1,2,2]).sqrt(), DomainError)
assert Z([1,0,2,0,3]).deflation() == (Z([1,2,3]), 2)
assert Z([]).deflation() == (Z([]), 1)
assert Z([1,1]).deflation() == (Z([1,1]), 1)
[(r,m)] = Z([1,1]).complex_roots()
assert m == 1
assert r.overlaps(-1)
assert Z([]).complex_roots() == []
assert Z([1]).complex_roots() == []
[(r,m)] = Z([1,1]).roots()
assert m == 1
assert r == -1
assert Z([]).roots() == []
assert Z([1]).roots() == []
assert Z([1, 2]).roots() == []
def test_fmpz_poly_factor():
Z = flint.fmpz_poly
assert Z([1,2]).gcd(Z([3,4])) == 1
assert Z([1,2,1]).gcd(Z([1,1])) == Z([1,1])
assert raises(lambda: Z([1,2,1]).gcd([]), TypeError)
assert Z([1,2,1]).factor() == (1, [(Z([1,1]), 2)])
def test_fmpz_poly_functions():
Z = flint.fmpz_poly
assert Z.cyclotomic(4) == Z([1,0,1])
assert Z.cos_minpoly(10) == Z([-1,-1,1])
assert Z.chebyshev_u(4) == Z([1,0,-12,0,16])
assert Z.chebyshev_t(4) == Z([1,0,-8,0,8])
assert Z.swinnerton_dyer(2) == Z([1,0,-10,0,1])
assert Z.swinnerton_dyer(2, use_arb=False) == Z([1,0,-10,0,1])
assert raises(lambda: Z.swinnerton_dyer(21), OverflowError)
assert Z.hilbert_class_poly(-7) == Z([3375,1])
assert raises(lambda: Z.hilbert_class_poly(-2), ValueError)
assert Z([1]).is_cyclotomic() == 0
assert Z([-1,1]).is_cyclotomic() == 1
assert Z([1,1]).is_cyclotomic() == 2
assert Z([1,2]).is_cyclotomic() == 0
assert Z([1,2,1]).is_cyclotomic() == 0
assert Z([2,2,1,1]).is_cyclotomic() == 0
assert Z([2,2,1]).is_cyclotomic() == 0
assert Z([1,2,2]).is_cyclotomic() == 0
assert Z([1,1,1]).is_cyclotomic() == 3
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 (a - b).entries() == [-3,-3,-3,-3,-3,-3]
assert a.transpose() == M(3,2,[1,4,2,5,3,6])
assert raises(lambda: a + 1, TypeError)
assert raises(lambda: 1 + a, TypeError)
assert raises(lambda: a - 1, TypeError)
assert raises(lambda: 1 - a, TypeError)
# XXX: Maybe there should be a ShapeError or something?
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 * 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.nrows(),A.ncols()) == (5,3)
assert (B.nrows(),B.ncols()) == (3,7)
assert (C.nrows(),C.ncols()) == (7,2)
assert A*(B*C) == (A*B)*C
assert raises(lambda: A*C, ValueError)
assert bool(M(2,2,[0,0,0,0])) is False
assert bool(M(2,2,[0,0,0,1])) is 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 raises(lambda: M(1,2,[3,4]) / 3, DomainError)
assert M(1,2,[2,4]) / 2 == M(1,2,[1,2])
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: pow(M([[1]]), 2, 3), NotImplementedError)
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 raises(lambda: M(None), TypeError)
assert raises(lambda: M(2,2,[1,2,3]), ValueError)
assert raises(lambda: M(2,2,2,2), TypeError)
assert M([[1,2,3],[4,5,6]]) == M(2,3,[1,2,3,4,5,6])
assert raises(lambda: M([[1]]) < M([[2]]), TypeError)
assert (M([[1]]) == 1) is False
assert (1 == M([[1]])) is False
assert (M([[1]]) != 1) is True
assert (1 != M([[1]])) is True
D = M([[1,2],[3,4]])
assert (D[0,0],D[0,1],D[1,0],D[1,1]) == (1,2,3,4)
D[0,0] = 3
assert D == M([[3,2],[3,4]])
def set_bad(i,j):
D[i,j] = -1
# XXX: Should be IndexError
raises(lambda: set_bad(2,0), IndexError)
raises(lambda: set_bad(0,2), IndexError)
raises(lambda: D[0,2], IndexError)
raises(lambda: D[0,2], IndexError)
# XXX: Negative indices?
raises(lambda: set_bad(-1,0), IndexError)
raises(lambda: set_bad(0,-1), IndexError)
raises(lambda: D[-1,0], IndexError)
raises(lambda: D[0,-1], IndexError)
assert M.hadamard(2) == M([[1,1],[1,-1]])
assert raises(lambda: M.hadamard(3), ValueError)
assert M.hadamard(2).is_hadamard() is True
assert M([[1,2],[3,4]]).is_hadamard() is False
M1 = M([[1,0],[1,1]])
M2 = M([[1,0],[-1,1]])
x = M([[3],[4]])
b = M([[3],[7]])
assert M1.inv() == M2
assert M2.inv() == M1
assert M1.inv(integer=True) == M2
assert M2.inv(integer=True) == M1
assert M1*x == b
assert M1.solve(b) == x
assert M1.solve(b, integer=True) == x
assert raises(lambda: M1.solve([]), TypeError)
assert raises(lambda: M1.solve(M([[1]])), ValueError)
assert raises(lambda: M([[1,1],[1,1]]).solve(b), ZeroDivisionError)
assert raises(lambda: M([[1,2],[3,4],[5,6]]).solve(b), ValueError)
assert M([[1,0],[1,2]]).solve(b) == flint.fmpq_mat([[3],[2]])
assert raises(lambda: M([[1,0],[1,2]]).solve(b, integer=True), ValueError)
assert raises(lambda: M([[1,2,3],[4,5,6]]).inv(), ValueError)
assert raises(lambda: M([[1,1],[1,1]]).inv(), ZeroDivisionError)
assert raises(lambda: M([[1,0],[1,2]]).inv(integer=True), ValueError)
half = flint.fmpq(1,2)
assert M([[1,0],[1,2]]).inv() == flint.fmpq_mat([[1, 0], [-half, half]])
M3 = M([[1,2,3],[4,5,6],[7,8,9]])
M3_copy = M(M3)
M3r = M([[-3,0,3],[0,-3,-6],[0,0,0]])
assert M3.rref() == (M3r, -3, 2)
assert M3 != M3r
assert M3.rref(inplace=True) == (M3r, -3, 2)
assert M3 == M3r
M3 = M3_copy
M3n = M([[3,0,0],[-6,0,0],[3,0,0]])
assert M3.nullspace() == (M3n, 1)
assert M3 * M3.nullspace()[0] == M(3,3,[0]*9)
# XXX: lll core dumps on a singular matrix
M4 = M([[1,2,3],[4,5,6],[7,8,10]])
L4 = M([[0,0,1],[-1,1,0],[2,1,0]])
T4 = M([[1,-2,1],[0,5,-3],[-2,1,0]])
assert L4 == T4 * M4
assert M4.lll() == L4
assert M4.lll(transform=True) == (L4, T4)
# XXX: rep="gram" consumes all memory in the system and core dumps
#for rep in "zbasis", "gram":
rep = "zbasis"
for gram in "approx", "exact":
assert M4.lll(rep=rep, gram=gram) == L4
assert M4.lll(rep=rep, gram=gram, transform=True) == (L4, T4)
assert raises(lambda: M4.lll(rep="bad"), ValueError)
assert raises(lambda: M4.lll(gram="bad"), ValueError)
M5 = M([[1,2,3],[4,5,6]])
H5 = M([[1,2,3],[0,3,6]])
T5 = M([[1,0],[4,-1]])
assert H5 == T5 * M5
assert M5.hnf() == H5
assert M5.hnf(transform=True) == (H5, T5)
assert M5.is_hnf() is False
assert H5.is_hnf() is True
S5 = M([[1,0,0],[0,3,0]])
assert M5.snf() == S5
assert M5.is_snf() is False
assert S5.is_snf() is True
M6 = M([[2,0,0],[0,2,1],[0,0,2]])
assert M6.charpoly() == flint.fmpz_poly([-8,12,-6,1])
assert M6.minpoly() == flint.fmpz_poly([4,-4,1])
assert list(M6) == [2,0,0,0,2,1,0,0,2]
def test_fmpz_series():
Zp = flint.fmpz_poly
Z = flint.fmpz_series
ctx = flint.ctx
assert ctx.cap == 10
s1 = Z([1,2])
s2 = Z([1,2])
s3 = Z([1,1])
s4 = Z([1,2],11)
p1 = Zp([1,2])
assert s1._equal_repr(s1) is True
assert s1._equal_repr(s2) is True
assert s1._equal_repr(s3) is False
assert s1._equal_repr(s4) is False
assert s1._equal_repr(p1) is False
assert s1._equal_repr(flint.fmpz(1)) is False
assert s1._equal_repr(1) is False
assert Z([])._equal_repr(Z([0])) is True
assert Z([1])._equal_repr(Z([0])) is False
# XXX: this gives a core dump:
# s = Z([1,2])
# s[10**10] = 1
assert Z(Z([1]))._equal_repr(Z([1]))
assert Z(Zp([1]))._equal_repr(Z([1]))
assert Z(1)._equal_repr(Z([1]))
assert raises(lambda: Z([1]) < Z([1]), TypeError)
assert len(Z([1,2])) == 2
assert Z([1,2]).length() == 2
s5 = Z([1,2])
assert s5[1] == 2
assert s5[2] == 0
assert s5[-1] == 0
# XXX: This goes beyond cap. Should it give an error?
assert s5[100] == 0
s5[2] = -1
assert s5[2] == -1
assert s5._equal_repr(Z([1,2,-1]))
def set_bad():
s5[-1] = 3
assert raises(set_bad, ValueError)
assert Z([1,2,0,4]).str() == "1 + 2*x + 4*x^3 + O(x^10)"
assert Z([1,2,0,4]).repr() == "fmpz_series([1, 2, 0, 4], prec=10)"
assert Z([],0).str() == "O(x^0)"
assert Z([],-1).str() == "(invalid power series)"
assert (+Z([1,2]))._equal_repr(Z([1,2]))
assert (-Z([1,2]))._equal_repr(Z([-1,-2]))
assert (Z([1,2]) + Z([3,4,5]))._equal_repr(Z([4,6,5]))
assert (Z([1,2]) + 1)._equal_repr(Z([2,2]))
assert (Z([1,2]) + Zp([3,4,5]))._equal_repr(Z([4,6,5]))
assert (1 + Z([1,2]))._equal_repr(Z([2,2]))
assert (Zp([1,2]) + Z([3,4,5]))._equal_repr(Z([4,6,5]))
assert raises(lambda: Z([1,2]) + [], TypeError)
assert raises(lambda: [] + Z([1,2]), TypeError)
assert (Z([1,2]) - Z([3,5]))._equal_repr(Z([-2,-3]))
assert (Z([1,2]) - 1)._equal_repr(Z([0,2]))
assert (1 - Z([1,2]))._equal_repr(Z([0,-2]))
assert raises(lambda: [] - Z([1,2]), TypeError)
assert raises(lambda: Z([1,2]) - [], TypeError)
assert (Z([1,2]) * Z([1,2]))._equal_repr(Z([1,4,4]))
assert (2 * Z([1,2]))._equal_repr(Z([2,4]))
assert (Z([1,2]) * 2)._equal_repr(Z([2,4]))
assert raises(lambda: [] * Z([1,2]), TypeError)
assert raises(lambda: Z([1,2]) * [], TypeError)
assert Z([1,2]).valuation() == 0
assert Z([0,2]).valuation() == 1
assert Z([0,0]).valuation() == -1
assert (Z([1,1]) / Z([1,-1]))._equal_repr(Z([1,2,2,2,2,2,2,2,2,2]))
assert raises(lambda: Z([1,1]) / Z([]), ZeroDivisionError)
assert (Z([]) / Z([1,-1]))._equal_repr(Z([]))
# quotient would not be a power series
assert raises(lambda: Z([1,1]) / Z([0,1]), ValueError)
# leading term in denominator is not a unit
assert raises(lambda: Z([1,1]) / Z([2,1]), ValueError)
assert (Z([0,1,1]) / Z([0,1,-1]))._equal_repr(Z([1,2,2,2,2,2,2,2,2],9))
# XXX: This gives leading term not a unit:
# assert Z([2,4]) / 2 == Z([1,2])
assert (Z([1,1]) / -1)._equal_repr(Z([-1,-1]))
assert raises(lambda: Z([1,1]) / [], TypeError)
assert raises(lambda: [] / Z([1,1]), TypeError)
assert (Z([1,2]) ** 2)._equal_repr(Z([1,4,4]))
assert raises(lambda: pow(Z([1,2]), 3, 5), NotImplementedError)
assert (Z([1,2])(Z([0,1,2])))._equal_repr(Z([1,2,4]))
assert raises(lambda: Z([1,2])(Z([1,2])), ValueError)
assert raises(lambda: Z([1,2])([]), TypeError)
coeffs = [0, 1, -2, 8, -40, 224, -1344, 8448, -54912, 366080]
assert Z([0,1,2]).reversion()._equal_repr(Z(coeffs))
# power series reversion must have valuation 1
assert raises(lambda: Z([1,1]).reversion(), ValueError)
# leading term is not a unit
assert raises(lambda: Z([0,2,1]).reversion(), ValueError)
def test_fmpq():
Q = flint.fmpq
assert Q() == Q(0)
assert Q(0) != Q(1)
assert Q(0) == 0
assert 0 == Q(0)
assert Q(2) != 1
assert 1 != Q(2)
assert Q(1) != ()
assert Q(1,2) != 1
assert Q(2,3) == Q(flint.fmpz(2),3)
assert Q(-2,-4) == Q(1,2)
assert Q("1") == Q(1)
assert Q("1/2") == Q(1,2)
assert raises(lambda: Q("1.0"), ValueError)
assert raises(lambda: Q("1.5"), ValueError)
assert raises(lambda: Q("1/2/3"), ValueError)
assert raises(lambda: Q([]), TypeError)
assert raises(lambda: Q(1, []), TypeError)
assert raises(lambda: Q([], 1), TypeError)
assert bool(Q(0)) is False
assert bool(Q(1)) is True
assert Q(1,3) + Q(2,3) == 1
assert Q(1,3) - Q(2,3) == Q(-1,3)
assert Q(1,3) * Q(2,3) == Q(2,9)
assert Q(1,3) + 2 == Q(7,3)
assert 2 + Q(1,3) == Q(7,3)
assert Q(1,3) - 2 == Q(-5,3)
assert 2 - Q(1,3) == Q(5,3)
assert Q(1,3) * 2 == Q(2,3)
assert 2 * Q(1,3) == Q(2,3)
assert Q(2,3) / Q(4,5) == Q(5,6)
assert Q(2,3) / 5 == Q(2,15)
assert Q(2,3) / flint.fmpz(5) == Q(2,15)
assert 5 / Q(2,3) == Q(15,2)
assert flint.fmpz(5) / Q(2,3) == Q(15,2)
assert Q(2,3)/5 == Q(2,15)
assert Q(1,2) ** 2 == Q(1,4)
assert Q(1,2) ** -2 == Q(4)
assert raises(lambda: Q(0) ** -1, ZeroDivisionError)
assert raises(lambda: Q(1,2) ** Q(1,2), TypeError)
assert raises(lambda: Q(1,2) ** [], TypeError)
assert raises(lambda: [] ** Q(1,2), TypeError)
# XXX: This should NotImplementedError or something.
assert raises(lambda: pow(Q(1,2),2,3), AssertionError)
megaz = flint.fmpz(2) ** 8000000
megaq = Q(megaz)
assert raises(lambda: megaq ** megaz, OverflowError)
assert raises(lambda: Q(1,2) + [], TypeError)
assert raises(lambda: Q(1,2) - [], TypeError)
assert raises(lambda: Q(1,2) * [], TypeError)
assert raises(lambda: Q(1,2) / [], TypeError)
assert raises(lambda: [] + Q(1,2), TypeError)
assert raises(lambda: [] - Q(1,2), TypeError)
assert raises(lambda: [] * Q(1,2), TypeError)
assert raises(lambda: [] / Q(1,2), TypeError)
assert (Q(1,2) == 1) is False
assert (Q(1,2) != 1) is True
assert (Q(1,2) < 1) is True
assert (Q(1,2) <= 1) is True
assert (Q(1,2) > 1) is False
assert (Q(1,2) >= 1) is False
assert (Q(1,2) == Q(3,4)) is False
assert (Q(1,2) != Q(3,4)) is True
assert (Q(1,2) < Q(3,4)) is True
assert (Q(1,2) <= Q(3,4)) is True
assert (Q(1,2) > Q(3,4)) is False
assert (Q(1,2) >= Q(3,4)) is False
assert (Q(1,2) == Q(1,2)) is True
assert (Q(1,2) != Q(1,2)) is False
assert (Q(1,2) < Q(1,2)) is False
assert (Q(1,2) <= Q(1,2)) is True
assert (Q(1,2) > Q(1,2)) is False
assert (Q(1,2) >= Q(1,2)) is True
assert raises(lambda: Q(1,2) > [], TypeError)
assert raises(lambda: [] < Q(1,2), TypeError)
ctx.pretty = False
assert repr(Q(-2,3)) == "fmpq(-2,3)"
assert repr(Q(3)) == "fmpq(3)"
ctx.pretty = True
assert str(Q(-2,3)) == "-2/3"
assert str(Q(3)) == "3"
assert Q(2,3).p == Q(2,3).numer() == Q(2,3).numerator == 2
assert Q(2,3).q == Q(2,3).denom() == Q(2,3).denominator == 3
assert +Q(5,7) == Q(5,7)
assert -Q(5,7) == Q(-5,7)
assert -Q(-5,7) == Q(5,7)
assert abs(Q(5,7)) == Q(5,7)
assert abs(-Q(5,7)) == Q(5,7)
assert raises(lambda: Q(1,0), ZeroDivisionError)
assert raises(lambda: Q(1,2) / Q(0), ZeroDivisionError)
assert raises(lambda: Q(1,2) / 0, ZeroDivisionError)
assert Q(2,3).gcd(Q(4,9)) == Q(2,9)
assert Q(2,3).gcd(5) == Q(1,3)
assert raises(lambda: Q(2,3).gcd([]), TypeError)
assert Q(5,3).floor() == flint.fmpz(1)
assert Q(-5,3).floor() == flint.fmpz(-2)
assert Q(5,3).ceil() == flint.fmpz(2)
assert Q(-5,3).ceil() == flint.fmpz(-1)
assert type(int(Q(5,3))) is int
assert type(math.floor(Q(5,3))) is flint.fmpz
assert type(math.ceil(Q(5,3))) is flint.fmpz
assert type(math.trunc(Q(5,3))) is flint.fmpz
assert type(round(Q(5,3))) is flint.fmpz
assert type(round(Q(5,3))) is flint.fmpz
assert type(round(Q(5,3), 0)) is flint.fmpq
assert type(round(Q(5,3), 1)) is flint.fmpq
assert int(Q(5,3)) == 1
assert math.floor(Q(5,3)) == flint.fmpz(1)
assert math.ceil(Q(5,3)) == flint.fmpz(2)
assert math.trunc(Q(5,3)) == flint.fmpz(1)
assert round(Q(5,3)) == flint.fmpz(2)
assert int(Q(-5,3)) == flint.fmpz(-1)
assert math.floor(Q(-5,3)) == flint.fmpz(-2)
assert math.ceil(Q(-5,3)) == flint.fmpz(-1)
assert math.trunc(Q(-5,3)) == flint.fmpz(-1)
assert round(Q(-5,3)) == -2
assert round(Q(100,3), 2) == Q(3333,100)
assert round(Q(100,3), 0) == Q(33,1)
assert round(Q(100,3), -1) == Q(30,1)
assert round(Q(100,3), -2) == Q(0)
d = {}
d[Q(1,2)] = 3
d[Q(1,2)] = 4
assert d == {Q(1,2):4}
assert Q(-5,3).height_bits() == 3
assert Q(-5,3).height_bits(signed=True) == -3
cases = [
(lambda q: q.next(),
[Q(0), Q(1), Q(-1), Q(1,2), Q(-1,2), Q(2), Q(-2), Q(1,3), Q(-1,3), Q(3)]),
(lambda q: q.next(signed=False),
[Q(0), Q(1), Q(1,2), Q(2), Q(1,3), Q(3), Q(2,3), Q(3,2), Q(1,4), Q(4)]),
(lambda q: q.next(minimal=False),
[Q(0), Q(1), Q(-1), Q(1,2), Q(-1,2), Q(2), Q(-2), Q(1,3), Q(-1,3), Q(3,2)]),
(lambda q: q.next(signed=False, minimal=False),
[Q(0), Q(1), Q(1,2), Q(2), Q(1,3), Q(3,2), Q(2,3), Q(3), Q(1,4), Q(4,3)]),
]
for func, values in cases:
for val1, val2 in zip(values[:-1], values[1:]):
assert func(val1) == val2
raises(lambda: Q(-1).next(signed=False), ValueError)
OE = OverflowError
cases = [
(flint.fmpq.bernoulli,
[OE, Q(1), Q(-1,2), Q(1,6), Q(0), Q(-1,30)]),
(lambda n: flint.fmpq.bernoulli(n, cache=True),
[OE, Q(1), Q(-1,2), Q(1,6), Q(0), Q(-1,30)]),
(flint.fmpq.harmonic,
[OE, Q(0), Q(1), Q(3,2), Q(11, 6), Q(25, 12)]),
(lambda n: flint.fmpq.dedekind_sum(n, 3),
[-Q(1,18), 0, Q(1,18), -Q(1,18), 0, Q(1,18), -Q(1,18)]),
]
is_exception = lambda v: isinstance(v, type) and issubclass(v, Exception)
for func, values in cases:
for n, val in enumerate(values, -1):
if is_exception(val):
assert raises(lambda: func(n), val)
else:
assert func(n) == val
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 Q([1], 2) == Q([flint.fmpq(1,2)])
assert raises(lambda: Q([1,[]]), TypeError)
assert raises(lambda: Q({}), TypeError)
assert raises(lambda: Q([1], []), TypeError)
assert raises(lambda: Q([1], 0), ZeroDivisionError)
assert bool(Q()) is False
assert bool(Q([1])) is 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 raises(lambda: Q([1]) + [], TypeError)
assert raises(lambda: [] + Q([1]), TypeError)
assert raises(lambda: Q([1]) - [], TypeError)
assert raises(lambda: [] - Q([1]), TypeError)
assert raises(lambda: Q([1]) * [], TypeError)
assert raises(lambda: [] * Q([1]), TypeError)
assert Q([1,2,1]) // Q([1,1]) == Q([1,1])
assert Q([1,2,1]) % Q([1,1]) == 0
assert divmod(Q([1,2,1]), Q([1,1])) == (Q([1,1]), 0)
assert raises(lambda: Q([1,2,1]) // [], TypeError)
assert raises(lambda: [] // Q([1,2,1]), TypeError)
assert raises(lambda: Q([1,2,1]) % [], TypeError)
assert raises(lambda: [] % Q([1,2,1]), TypeError)
assert raises(lambda: divmod(Q([1,2,1]), []), TypeError)
assert raises(lambda: divmod([], Q([1,2,1])), TypeError)
assert raises(lambda: Q([1,2,1]) / 0, ZeroDivisionError)
assert raises(lambda: Q([1,2,1]) // 0, ZeroDivisionError)
assert raises(lambda: Q([1,2,1]) % 0, ZeroDivisionError)
assert raises(lambda: divmod(Q([1,2,1]), 0), ZeroDivisionError)
assert +Q([1,2]) == Q([1,2])
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]) != []
assert raises(lambda: Q([1,2]) < Q([1,2]), TypeError)
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 Q([1,2]) / Q([1,2]) == Q([1])
assert raises(lambda: Q([1,2]) / Q([2,2]), DomainError)
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 raises(lambda: pow(Q([1,2]), 3, 5), NotImplementedError)
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])) == "fmpq_poly([15, 20, 10])"
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)
assert a[-1] == flint.fmpq(0)