def test_legendre_poly():
pytest.raises(ValueError, lambda: legendre_poly(-1, x))
assert legendre_poly(1, x, polys=True) == x.as_poly()
assert legendre_poly(0, x) == 1
assert legendre_poly(1, x) == x
assert legendre_poly(2, x) == 3 * x**2 / 2 - Rational(1, 2)
assert legendre_poly(3, x) == 5 * x**3 / 2 - 3 * x / 2
assert legendre_poly(4,
x) == 35 * x**4 / 8 - 30 * x**2 / 8 + Rational(3, 8)
assert legendre_poly(5, x) == 63 * x**5 / 8 - 70 * x**3 / 8 + 15 * x / 8
assert legendre_poly(6, x) == (231 * x**6 / 16 - 315 * x**4 / 16 +
105 * x**2 / 16 - Rational(5, 16))
assert legendre_poly(1, polys=True) == x.as_poly()
def test_RootOf_eval_rational():
p = legendre_poly(4, x, polys=True)
roots = [r.eval_rational(Rational(1, 10)**20) for r in p.real_roots()]
for r in roots:
assert isinstance(r, Rational)
# All we know is that the Rational instance will be at most 1/10^20 from
# the exact root. So if we evaluate to 17 digits, it must be exactly equal
# to:
roots = [str(r.n(17)) for r in roots]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
def test_RootOf_eval_rational():
p = legendre_poly(4, x, polys=True)
roots = [r.eval_rational(Rational(1, 10)**20) for r in p.real_roots()]
for r in roots:
assert isinstance(r, Rational)
# All we know is that the Rational instance will be at most 1/10^20 from
# the exact root. So if we evaluate to 17 digits, it must be exactly equal
# to:
roots = [str(r.evalf(17)) for r in roots]
assert roots == [
'-0.86113631159405258',
'-0.33998104358485626',
'0.33998104358485626',
'0.86113631159405258',
]
pytest.raises(NotImplementedError,
lambda: RootOf(x**3 + x + 3, 1).eval_rational(1e-3))
def test_RootOf_eval_rational():
p = legendre_poly(4, x, polys=True)
roots = [r.eval_rational(Rational(1, 10)**20) for r in p.real_roots()]
for r in roots:
assert isinstance(r, Rational)
# All we know is that the Rational instance will be at most 1/10^20 from
# the exact root. So if we evaluate to 17 digits, it must be exactly equal
# to:
roots = [str(r.evalf(17)) for r in roots]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
pytest.raises(NotImplementedError,
lambda: RootOf(x**3 + x + 3, 1).eval_rational(1e-3))
def test_nroots1():
n = 64
p = legendre_poly(n, x, polys=True)
pytest.raises(mpmath.mp.NoConvergence, lambda: p.nroots(n=3, maxsteps=5))
roots = p.nroots(n=3)
# The order of roots matters. They are ordered from smallest to the
# largest.
assert [str(r) for r in roots] == \
['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961',
'-0.946', '-0.930', '-0.911', '-0.889', '-0.866', '-0.841',
'-0.813', '-0.784', '-0.753', '-0.720', '-0.685', '-0.649',
'-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402',
'-0.357', '-0.311', '-0.265', '-0.217', '-0.170', '-0.121',
'-0.0730', '-0.0243', '0.0243', '0.0730', '0.121', '0.170',
'0.217', '0.265', '0.311', '0.357', '0.402', '0.446', '0.489',
'0.531', '0.572', '0.611', '0.649', '0.685', '0.720', '0.753',
'0.784', '0.813', '0.841', '0.866', '0.889', '0.911', '0.930',
'0.946', '0.961', '0.973', '0.983', '0.991', '0.996', '0.999']
def test_RootOf_evalf():
real = RootOf(x**3 + x + 3, 0).evalf(20)
assert real.epsilon_eq(Float("-1.2134116627622296341"))
re, im = RootOf(x**3 + x + 3, 1).evalf(20).as_real_imag()
assert re.epsilon_eq(+Float("0.60670583138111481707"))
assert im.epsilon_eq(-Float("1.45061224918844152650"))
re, im = RootOf(x**3 + x + 3, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("0.60670583138111481707"))
assert im.epsilon_eq(Float("1.45061224918844152650"))
p = legendre_poly(4, x, polys=True)
roots = [str(r.evalf(17)) for r in p.real_roots()]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
re = RootOf(x**5 - 5 * x + 12, 0).evalf(20)
assert re.epsilon_eq(Float("-1.84208596619025438271"))
re, im = RootOf(x**5 - 5 * x + 12, 1).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("-1.709561043370328882010"))
re, im = RootOf(x**5 - 5 * x + 12, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("+1.709561043370328882010"))
re, im = RootOf(x**5 - 5 * x + 12, 3).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("-0.719798681483861386681"))
re, im = RootOf(x**5 - 5 * x + 12, 4).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("+0.719798681483861386681"))
# issue sympy/sympy#6393
assert str(RootOf(x**5 + 2 * x**4 + x**3 - 68719476736,
0).evalf(3)) == '147.'
eq = (531441 * x**11 + 3857868 * x**10 + 13730229 * x**9 +
32597882 * x**8 + 55077472 * x**7 + 60452000 * x**6 +
32172064 * x**5 - 4383808 * x**4 - 11942912 * x**3 - 1506304 * x**2 +
1453312 * x + 512)
a, b = RootOf(eq, 1).evalf(2).as_real_imag()
c, d = RootOf(eq, 2).evalf(2).as_real_imag()
assert a == c
assert b < d
assert b == -d
# issue sympy/sympy#6451
r = RootOf(legendre_poly(64, x), 7)
assert r.evalf(2) == r.evalf(100).evalf(2)
# issue sympy/sympy#8617
ans = [w[x].evalf(2) for w in solve(x**3 - x - 4)]
assert RootOf(exp(x)**3 - exp(x) - 4, 0).evalf(2) in ans
# issue sympy/sympy#9019
r0 = RootOf(x**2 + 1, 0, radicals=False)
r1 = RootOf(x**2 + 1, 1, radicals=False)
assert r0.evalf(4, chop=True) == -1.0 * I
assert r1.evalf(4, chop=True) == +1.0 * I
# make sure verification is used in case a max/min traps the "root"
assert str(RootOf(4 * x**5 + 16 * x**3 + 12 * x**2 + 7,
0).evalf(3)) == '-0.976'
assert isinstance(RootOf(x**3 + y * x + 1, x, 0).evalf(2), RootOf)
assert RootOf(x**3 + I * x + 2,
0).evalf(7) == (Float('-1.260785326', dps=7) +
I * Float('0.2684419416', dps=7))
r = RootOf(x**2 - 4456178 * x + 60372201703370, 0, radicals=False)
assert r.evalf(2) == Float('2.2282e+6',
dps=2) - I * Float('7.4465e+6', dps=2)
def test_RootOf_evalf():
real = RootOf(x**3 + x + 3, 0).evalf(20)
assert real.epsilon_eq(Float("-1.2134116627622296341"))
re, im = RootOf(x**3 + x + 3, 1).evalf(20).as_real_imag()
assert re.epsilon_eq( Float("0.60670583138111481707"))
assert im.epsilon_eq(-Float("1.45061224918844152650"))
re, im = RootOf(x**3 + x + 3, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("0.60670583138111481707"))
assert im.epsilon_eq(Float("1.45061224918844152650"))
p = legendre_poly(4, x, polys=True)
roots = [str(r.evalf(17)) for r in p.real_roots()]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
re = RootOf(x**5 - 5*x + 12, 0).evalf(20)
assert re.epsilon_eq(Float("-1.84208596619025438271"))
re, im = RootOf(x**5 - 5*x + 12, 1).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("-1.709561043370328882010"))
re, im = RootOf(x**5 - 5*x + 12, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("+1.709561043370328882010"))
re, im = RootOf(x**5 - 5*x + 12, 3).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("-0.719798681483861386681"))
re, im = RootOf(x**5 - 5*x + 12, 4).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("+0.719798681483861386681"))
# issue sympy/sympy#6393
assert str(RootOf(x**5 + 2*x**4 + x**3 - 68719476736, 0).evalf(3)) == '147.'
eq = (531441*x**11 + 3857868*x**10 + 13730229*x**9 + 32597882*x**8 +
55077472*x**7 + 60452000*x**6 + 32172064*x**5 - 4383808*x**4 -
11942912*x**3 - 1506304*x**2 + 1453312*x + 512)
a, b = RootOf(eq, 1).evalf(2).as_real_imag()
c, d = RootOf(eq, 2).evalf(2).as_real_imag()
assert a == c
assert b < d
assert b == -d
# issue sympy/sympy#6451
r = RootOf(legendre_poly(64, x), 7)
assert r.evalf(2) == r.evalf(100).evalf(2)
# issue sympy/sympy#8617
ans = [w[x].evalf(2) for w in solve(x**3 - x - 4)]
assert RootOf(exp(x)**3 - exp(x) - 4, 0).evalf(2) in ans
# issue sympy/sympy#9019
r0 = RootOf(x**2 + 1, 0, radicals=False)
r1 = RootOf(x**2 + 1, 1, radicals=False)
assert r0.evalf(4, chop=True) == -1.0*I
assert r1.evalf(4, chop=True) == +1.0*I
# make sure verification is used in case a max/min traps the "root"
assert str(RootOf(4*x**5 + 16*x**3 + 12*x**2 + 7, 0).evalf(3)) == '-0.976'
assert isinstance(RootOf(x**3 + y*x + 1, x, 0).evalf(2), RootOf)
assert RootOf(x**3 + I*x + 2, 0).evalf(7) == (Float('-1.260785326', dps=7) +
I*Float('0.2684419416', dps=7))
r = RootOf(x**2 - 4456178*x + 60372201703370, 0, radicals=False)
assert r.evalf(2) == Float('2.2282e+6', dps=2) - I*Float('7.4465e+6', dps=2)
