def __new__(cls, name, index, system, pretty_str, latex_str):
name = str(name)
pretty_str = str(pretty_str)
latex_str = str(latex_str)
# Verify arguments
if index not in range(0, 3):
raise ValueError("index must be 0, 1 or 2")
if not isinstance(system, CoordSysCartesian):
raise TypeError("system should be a CoordSysCartesian")
# Initialize an object
obj = super(BaseVector, cls).__new__(cls, Symbol(name), Integer(index),
system, Symbol(pretty_str),
Symbol(latex_str))
# Assign important attributes
obj._base_instance = obj
obj._components = {obj: Integer(1)}
obj._measure_number = Integer(1)
obj._name = name
obj._pretty_form = pretty_str
obj._latex_form = latex_str
obj._system = system
assumptions = {}
assumptions['commutative'] = True
obj._assumptions = StdFactKB(assumptions)
# This attr is used for re-expression to one of the systems
# involved in the definition of the Vector. Applies to
# VectorMul and VectorAdd too.
obj._sys = system
return obj
def test_Singletons():
protocols = [0, 1, 2, 3]
copiers = [copy.copy, copy.deepcopy]
copiers += [lambda x: pickle.loads(pickle.dumps(x, p)) for p in protocols]
for obj in (Integer(-1), Integer(0), Integer(1), Rational(1, 2), pi, E, I,
oo, -oo, zoo, nan, GoldenRatio, EulerGamma, Catalan,
S.EmptySet, S.IdentityFunction):
for func in copiers:
assert func(obj) is obj
def _eval_number(cls, arg):
if arg.is_Number:
if arg.is_Rational:
return -Integer(-arg.p // arg.q)
elif arg.is_Float:
return Integer(int(arg.ceiling()))
else:
return arg
if arg.is_NumberSymbol:
return arg.approximation_interval(Integer)[1]
def test_Number():
assert precedence(Integer(0)) == PRECEDENCE["Atom"]
assert precedence(Integer(1)) == PRECEDENCE["Atom"]
assert precedence(Integer(-1)) == PRECEDENCE["Add"]
assert precedence(Integer(10)) == PRECEDENCE["Atom"]
assert precedence(Rational(5, 2)) == PRECEDENCE["Mul"]
assert precedence(Rational(-5, 2)) == PRECEDENCE["Add"]
assert precedence(Float(5)) == PRECEDENCE["Atom"]
assert precedence(Float(-5)) == PRECEDENCE["Add"]
assert precedence(oo) == PRECEDENCE["Atom"]
assert precedence(-oo) == PRECEDENCE["Add"]
def test_continued_fraction():
pytest.raises(ValueError, lambda: cf_p(1, 0, 0))
pytest.raises(ValueError, lambda: cf_p(1, 1, -1))
assert cf_p(4, 3, 0) == [1, 3]
assert cf_p(0, 3, 5) == [0, 1, [2, 1, 12, 1, 2, 2]]
assert cf_p(1, 1, 0) == [1]
assert cf_p(3, 4, 0) == [0, 1, 3]
assert cf_p(4, 5, 0) == [0, 1, 4]
assert cf_p(5, 6, 0) == [0, 1, 5]
assert cf_p(11, 13, 0) == [0, 1, 5, 2]
assert cf_p(16, 19, 0) == [0, 1, 5, 3]
assert cf_p(27, 32, 0) == [0, 1, 5, 2, 2]
assert cf_p(1, 2, 5) == [[1]]
assert cf_p(0, 1, 2) == [1, [2]]
assert cf_p(6, 7, 49) == [1, 1, 6]
assert cf_p(3796, 1387, 0) == [2, 1, 2, 1, 4]
assert cf_p(3245, 10000) == [0, 3, 12, 4, 13]
assert cf_p(1932, 2568) == [0, 1, 3, 26, 2]
assert cf_p(6589, 2569) == [2, 1, 1, 3, 2, 1, 3, 1, 23]
assert list(itertools.islice(cf_i(Phi), 7)) == [1, 1, 1, 1, 1, 1, 1]
assert list(itertools.islice(cf_i(pi), 7)) == [3, 7, 15, 1, 292, 1, 1]
assert list(cf_i(Rational(17, 12))) == [1, 2, 2, 2]
assert list(cf_i(Rational(-17, 12))) == [-2, 1, 1, 2, 2]
assert list(cf_c([1, 6, 1, 8])) == [
Integer(1),
Rational(7, 6),
Rational(8, 7),
Rational(71, 62)
]
assert list(cf_c([2])) == [Integer(2)]
assert list(cf_c([1, 1, 1, 1, 1, 1, 1])) == [
1,
Integer(2),
Rational(3, 2),
Rational(5, 3),
Rational(8, 5),
Rational(13, 8),
Rational(21, 13)
]
assert list(cf_c([1, 6, Rational(
-1, 2), 4])) == [1, Rational(7, 6),
Rational(5, 4),
Rational(3, 2)]
assert cf_r([1, 6, 1, 8]) == Rational(71, 62)
assert cf_r([3]) == Integer(3)
assert cf_r([-1, 5, 1, 4]) == Rational(-24, 29)
assert (cf_r([0, 1, 1, 7, [24, 8]]) - (sqrt(3) + 2) / 7).expand() == 0
assert cf_r([1, 5, 9]) == Rational(55, 46)
assert (cf_r([[1]]) - (sqrt(5) + 1) / 2).expand() == 0
def eval(cls, x, k=None):
if k is S.Zero:
return cls(x)
elif k is None:
k = S.Zero
if k is S.Zero:
if x is S.Zero:
return S.Zero
if x is S.Exp1:
return S.One
if x == -1/S.Exp1:
return S.NegativeOne
if x == -log(2)/2:
return -log(2)
if x is S.Infinity:
return S.Infinity
if k.is_nonzero:
if x is S.Zero:
return S.NegativeInfinity
if k is S.NegativeOne:
if x == -S.Pi/2:
return -S.ImaginaryUnit*S.Pi/2
elif x == -1/S.Exp1:
return S.NegativeOne
elif x == -2*exp(-2):
return -Integer(2)
def __iter__(self):
yield S.Zero
i = Integer(1)
while True:
yield i
yield -i
i = i + 1
def __new__(cls, name, index, system, pretty_str, latex_str):
name = str(name)
pretty_str = str(pretty_str)
latex_str = str(latex_str)
from diofant.vector.coordsysrect import CoordSysCartesian
obj = super(BaseScalar, cls).__new__(cls, name)
if not isinstance(system, CoordSysCartesian):
raise TypeError("system should be a CoordSysCartesian")
if index not in range(0, 3):
raise ValueError("Invalid index specified.")
# The _id is used for equating purposes, and for hashing
obj._id = (index, system)
obj._name = name
obj._pretty_form = pretty_str
obj._latex_form = latex_str
obj._system = system
# Change the args for the object
obj._args = tuple([
Symbol(name),
Integer(index), system,
Symbol(pretty_str),
Symbol(latex_str)
])
return obj
def eval(cls, n):
n = sympify(n)
if n.is_Number:
if n is S.Zero:
return S.One
elif n is S.Infinity:
return S.Infinity
elif n.is_Integer:
if n.is_negative:
return S.ComplexInfinity
else:
n, result = n.p, 1
if n < 20:
for i in range(2, n + 1):
result *= i
else:
N, bits = n, 0
while N != 0:
if N & 1 == 1:
bits += 1
N = N >> 1
result = cls._recursive(n)*2**(n - bits)
return Integer(result)
def continued_fraction_convergents(cf):
"""Return an iterator over the convergents of a continued fraction.
The parameter should be an iterable returning successive
partial quotients of the continued fraction, such as might be
returned by continued_fraction_iterator. In computing the
convergents, the continued fraction need not be strictly in
canonical form (all integers, all but the first positive).
Rational and negative elements may be present in the expansion.
Examples
========
>>> from diofant.core import Rational, pi
>>> from diofant import S
>>> from diofant.ntheory.continued_fraction import \
continued_fraction_convergents, continued_fraction_iterator
>>> list(continued_fraction_convergents([0, 2, 1, 2]))
[0, 1/2, 1/3, 3/8]
>>> list(continued_fraction_convergents([1, Rational(1, 2), -7, Rational(1, 4)]))
[1, 3, 19/5, 7]
>>> it = continued_fraction_convergents(continued_fraction_iterator(pi))
>>> for n in range(7):
... print(next(it))
3
22/7
333/106
355/113
103993/33102
104348/33215
208341/66317
See Also
========
continued_fraction_iterator
"""
p_2, q_2 = Integer(0), Integer(1)
p_1, q_1 = Integer(1), Integer(0)
for a in cf:
p, q = a * p_1 + p_2, a * q_1 + q_2
p_2, q_2 = p_1, q_1
p_1, q_1 = p, q
yield p / q
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 _eval_nseries(self, x, n, logx):
if len(self.args) == 1:
from diofant import O, Add, Integer, factorial
x = self.args[0]
o = O(x**n, x)
l = S.Zero
if n > 0:
l += Add(*[Integer(-k)**(k - 1)*x**k/factorial(k)
for k in range(1, n)])
return l + o
return super(LambertW, self)._eval_nseries(x, n=n, logx=logx)
def test_visual_factorint():
assert factorint(1, visual=1) == 1
forty2 = factorint(42, visual=True)
assert type(forty2) == Mul
assert str(forty2) == '2**1*3**1*7**1'
assert factorint(1, visual=True) is Integer(1)
no = {'evaluate': False}
assert factorint(42**2, visual=True) == Mul(Pow(2, 2, **no),
Pow(3, 2, **no),
Pow(7, 2, **no), **no)
assert -1 in factorint(-42, visual=True).args
def bkey(b, e=None):
'''Return (b**s, c.q), c.p where e -> c*s. If e is not given then
it will be taken by using as_base_exp() on the input b.
e.g.
x**3/2 -> (x, 2), 3
x**y -> (x**y, 1), 1
x**(2*y/3) -> (x**y, 3), 2
exp(x/2) -> (exp(a), 2), 1
'''
if e is not None: # coming from c_powers or from below
if e.is_Integer:
return (b, S.One), e
elif e.is_Rational:
return (b, Integer(e.q)), Integer(e.p)
else:
c, m = e.as_coeff_Mul(rational=True)
if c is not S.One and b.is_positive:
return (b**m, Integer(c.q)), Integer(c.p)
else:
return (b**e, S.One), S.One
else:
return bkey(*b.as_base_exp())
def test_scalar_potential_difference():
point1 = C.origin.locate_new('P1', 1 * i + 2 * j + 3 * k)
point2 = C.origin.locate_new('P2', 4 * i + 5 * j + 6 * k)
genericpointC = C.origin.locate_new('RP', x * i + y * j + z * k)
genericpointP = P.origin.locate_new('PP',
P.x * P.i + P.y * P.j + P.z * P.k)
assert scalar_potential_difference(Integer(0), C, point1, point2) == 0
assert (scalar_potential_difference(scalar_field, C, C.origin,
genericpointC) == scalar_field)
assert (scalar_potential_difference(grad_field, C, C.origin,
genericpointC) == scalar_field)
assert scalar_potential_difference(grad_field, C, point1, point2) == 948
assert (scalar_potential_difference(y * z * i + x * z * j + x * y * k, C,
point1,
genericpointC) == x * y * z - 6)
potential_diff_P = (2 * P.z * (P.x * sin(q) + P.y * cos(q)) *
(P.x * cos(q) - P.y * sin(q))**2)
assert (scalar_potential_difference(
grad_field, P, P.origin, genericpointP).simplify() == potential_diff_P)
pytest.raises(
TypeError,
lambda: scalar_potential_difference(Integer(0), 1, point1, point2))
def _eval_aseries(self, n, args0, x, logx):
from diofant import Order
if args0[0] != oo:
return super(loggamma, self)._eval_aseries(n, args0, x, logx)
z = self.args[0]
m = min(n, ceiling((n + Integer(1)) / 2))
r = log(z) * (z - Rational(1, 2)) - z + log(2 * pi) / 2
l = [
bernoulli(2 * k) / (2 * k * (2 * k - 1) * z**(2 * k - 1))
for k in range(1, m)
]
o = None
if m == 0:
o = Order(1, x)
else:
o = Order(1 / z**(2 * m - 1), x)
# It is very inefficient to first add the order and then do the nseries
return (r + Add(*l))._eval_nseries(x, n, logx) + o
def _real_to_rational(expr, tolerance=None):
"""
Replace all reals in expr with rationals.
>>> from diofant import nsimplify
>>> from diofant.abc import x
>>> nsimplify(.76 + .1*x**.5, rational=True)
sqrt(x)/10 + 19/25
"""
p = expr
reps = {}
reduce_num = None
if tolerance is not None and tolerance < 1:
reduce_num = ceiling(1 / tolerance)
for float in p.atoms(Float):
key = float
if reduce_num is not None:
r = Rational(float).limit_denominator(reduce_num)
elif (tolerance is not None and tolerance >= 1
and float.is_Integer is False):
r = Rational(tolerance *
round(float / tolerance)).limit_denominator(
int(tolerance))
else:
r = nsimplify(float, rational=False)
# e.g. log(3).n() -> log(3) instead of a Rational
if float and not r:
r = Rational(float)
elif not r.is_Rational:
if float < 0:
float = -float
d = Pow(10, int((mpmath.log(float) / mpmath.log(10))))
r = -Rational(str(float / d)) * d
elif float > 0:
d = Pow(10, int((mpmath.log(float) / mpmath.log(10))))
r = Rational(str(float / d)) * d
else:
r = Integer(0)
reps[key] = r
return p.subs(reps, simultaneous=True)
def _eval(cls, n, k):
# n.is_Number and k.is_Integer and k != 1 and n != k
if k.is_Integer:
if n.is_Integer and n >= 0:
n, k = int(n), int(k)
if k > n:
return S.Zero
elif k > n // 2:
k = n - k
M, result = int(_sqrt(n)), 1
for prime in sieve.primerange(2, n + 1):
if prime > n - k:
result *= prime
elif prime > n // 2:
continue
elif prime > M:
if n % prime < k % prime:
result *= prime
else:
N, K = n, k
exp = a = 0
while N > 0:
a = int((N % prime) < (K % prime + a))
N, K = N // prime, K // prime
exp = a + exp
if exp > 0:
result *= prime**exp
return Integer(result)
else:
d = result = n - k + 1
for i in range(2, k + 1):
d += 1
result *= d
result /= i
return result
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 __new__(cls, *args):
from diofant.functions.elementary.integers import ceiling
# expand range
slc = slice(*args)
start, stop, step = slc.start or 0, slc.stop, slc.step or 1
try:
start, stop, step = [
w if w in [S.NegativeInfinity, S.Infinity] else Integer(
as_int(w)) for w in (start, stop, step)
]
except ValueError:
raise ValueError("Inputs to Range must be Integer Valued\n" +
"Use ImageSets of Ranges for other cases")
if not step.is_finite:
raise ValueError("Infinite step is not allowed")
if start == stop:
return S.EmptySet
n = ceiling((stop - start) / step)
if n <= 0:
return S.EmptySet
# normalize args: regardless of how they are entered they will show
# canonically as Range(inf, sup, step) with step > 0
if n.is_finite:
start, stop = sorted((start, start + (n - 1) * step))
else:
start, stop = sorted((start, stop - step))
step = abs(step)
if (start, stop) == (S.NegativeInfinity, S.Infinity):
raise ValueError("Both the start and end value of "
"Range cannot be unbounded")
else:
return Basic.__new__(cls, start, stop + step, step)
def eval(cls, arg):
from diofant import im
if arg.is_integer:
return arg
if arg.func is cls:
return arg
if arg.is_imaginary or (S.ImaginaryUnit * arg).is_extended_real:
i = im(arg)
if not i.has(S.ImaginaryUnit):
return cls(i) * S.ImaginaryUnit
return cls(arg, evaluate=False)
v = cls._eval_number(arg)
if v is not None:
return v
# Integral, numerical, symbolic part
ipart = npart = spart = S.Zero
# Extract integral (or complex integral) terms
terms = Add.make_args(arg)
for t in terms:
if t.is_integer or (t.is_imaginary and im(t).is_integer):
ipart += t
elif t.has(Dummy, Symbol):
spart += t
else:
npart += t
if not (npart or spart):
return ipart
# Evaluate npart numerically if independent of spart
if npart and (not spart or npart.is_extended_real and
(spart.is_imaginary or
(S.ImaginaryUnit * spart).is_extended_real)
or npart.is_imaginary and spart.is_extended_real):
try:
from diofant.core.evalf import DEFAULT_MAXPREC as TARGET
prec = 10
while True:
r, i = cls(npart,
evaluate=False).evalf(prec).as_real_imag()
if 2**prec > max(abs(int(r)), abs(int(i))) + 10:
break
else:
if prec >= TARGET:
raise PrecisionExhausted
prec += 10
ipart += Integer(r) + Integer(i) * S.ImaginaryUnit
npart = S.Zero
except (PrecisionExhausted, NotImplementedError):
pass
spart += npart
if not spart:
return ipart
elif spart.is_imaginary or (S.ImaginaryUnit * spart).is_extended_real:
return ipart + cls(im(spart), evaluate=False) * S.ImaginaryUnit
else:
return ipart + cls(spart, evaluate=False)
def test_del_operator():
# Tests for curl
assert (delop
^ Vector.zero == (Derivative(0, C.y) - Derivative(0, C.z)) * C.i +
(-Derivative(0, C.x) + Derivative(0, C.z)) * C.j +
(Derivative(0, C.x) - Derivative(0, C.y)) * C.k)
assert ((delop ^ Vector.zero).doit() == Vector.zero == curl(
Vector.zero, C))
assert delop.cross(Vector.zero) == delop ^ Vector.zero
assert (delop ^ i).doit() == Vector.zero
assert delop.cross(2 * y**2 * j, doit=True) == Vector.zero
assert delop.cross(2 * y**2 * j) == delop ^ 2 * y**2 * j
v = x * y * z * (i + j + k)
assert ((delop ^ v).doit() == (-x * y + x * z) * i + (x * y - y * z) * j +
(-x * z + y * z) * k == curl(v, C))
assert delop ^ v == delop.cross(v)
assert (delop.cross(
2 * x**2 *
j) == (Derivative(0, C.y) - Derivative(2 * C.x**2, C.z)) * C.i +
(-Derivative(0, C.x) + Derivative(0, C.z)) * C.j +
(-Derivative(0, C.y) + Derivative(2 * C.x**2, C.x)) * C.k)
assert (delop.cross(2 * x**2 * j, doit=True) == 4 * x * k == curl(
2 * x**2 * j, C))
# Tests for divergence
assert delop & Vector.zero == Integer(0) == divergence(Vector.zero, C)
assert (delop & Vector.zero).doit() == Integer(0)
assert delop.dot(Vector.zero) == delop & Vector.zero
assert (delop & i).doit() == Integer(0)
assert (delop & x**2 * i).doit() == 2 * x == divergence(x**2 * i, C)
assert (delop.dot(v, doit=True) == x * y + y * z + z * x == divergence(
v, C))
assert delop & v == delop.dot(v)
assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \
- 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z)
v = x * i + y * j + z * k
assert (delop & v == Derivative(C.x, C.x) + Derivative(C.y, C.y) +
Derivative(C.z, C.z))
assert delop.dot(v, doit=True) == 3 == divergence(v, C)
assert delop & v == delop.dot(v)
assert simplify((delop & v).doit()) == 3
# Tests for gradient
assert (delop.gradient(0, doit=True) == Vector.zero == gradient(0, C))
assert delop.gradient(0) == delop(0)
assert (delop(Integer(0))).doit() == Vector.zero
assert (delop(x) == (Derivative(C.x, C.x)) * C.i +
(Derivative(C.x, C.y)) * C.j + (Derivative(C.x, C.z)) * C.k)
assert (delop(x)).doit() == i == gradient(x, C)
assert (delop(x * y * z) == (Derivative(C.x * C.y * C.z, C.x)) * C.i +
(Derivative(C.x * C.y * C.z, C.y)) * C.j +
(Derivative(C.x * C.y * C.z, C.z)) * C.k)
assert (delop.gradient(x * y * z, doit=True) ==
y * z * i + z * x * j + x * y * k == gradient(x * y * z, C))
assert delop(x * y * z) == delop.gradient(x * y * z)
assert (delop(2 * x**2)).doit() == 4 * x * i
assert ((delop(a * sin(y) / x)).doit() == -a * sin(y) / x**2 * i +
a * cos(y) / x * j)
# Tests for directional derivative
assert (Vector.zero & delop)(a) == Integer(0)
assert ((Vector.zero & delop)(a)).doit() == Integer(0)
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
assert ((v & delop)(Integer(0))).doit() == Integer(0)
assert ((i & delop)(x)).doit() == 1
assert ((j & delop)(y)).doit() == 1
assert ((k & delop)(z)).doit() == 1
assert ((i & delop)(x * y * z)).doit() == y * z
assert ((v & delop)(x)).doit() == x
assert ((v & delop)(x * y * z)).doit() == 3 * x * y * z
assert (v & delop)(x + y + z) == C.x + C.y + C.z
assert ((v & delop)(x + y + z)).doit() == x + y + z
assert ((v & delop)(v)).doit() == v
assert ((i & delop)(v)).doit() == i
assert ((j & delop)(v)).doit() == j
assert ((k & delop)(v)).doit() == k
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
def continued_fraction_reduce(cf):
"""Reduce a continued fraction to a rational or quadratic irrational.
Compute the rational or quadratic irrational number from its
terminating or periodic continued fraction expansion. The
continued fraction expansion (cf) should be supplied as a
terminating iterator supplying the terms of the expansion. For
terminating continued fractions, this is equivalent to
``list(continued_fraction_convergents(cf))[-1]``, only a little more
efficient. If the expansion has a repeating part, a list of the
repeating terms should be returned as the last element from the
iterator. This is the format returned by
continued_fraction_periodic.
For quadratic irrationals, returns the largest solution found,
which is generally the one sought, if the fraction is in canonical
form (all terms positive except possibly the first).
Examples
========
>>> from diofant.ntheory.continued_fraction import continued_fraction_reduce
>>> continued_fraction_reduce([1, 2, 3, 4, 5])
225/157
>>> continued_fraction_reduce([-2, 1, 9, 7, 1, 2])
-256/233
>>> continued_fraction_reduce([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8]).n(10)
2.718281835
>>> continued_fraction_reduce([1, 4, 2, [3, 1]])
(sqrt(21) + 287)/238
>>> continued_fraction_reduce([[1]])
1/2 + sqrt(5)/2
>>> from diofant.ntheory.continued_fraction import continued_fraction_periodic
>>> continued_fraction_reduce(continued_fraction_periodic(8, 5, 13))
(sqrt(13) + 8)/5
See Also
========
continued_fraction_periodic
"""
from diofant.core.symbol import Dummy
from diofant.solvers import solve
period = []
x = Dummy('x')
def untillist(cf):
for nxt in cf:
if isinstance(nxt, list):
period.extend(nxt)
yield x
break
yield nxt
a = Integer(0)
for a in continued_fraction_convergents(untillist(cf)):
pass
if period:
y = Dummy('y')
solns = solve(continued_fraction_reduce(period + [y]) - y, y)
solns.sort()
pure = solns[-1]
return a.subs(x, pure).radsimp()
else:
return a
def roots_cubic(f, trig=False):
"""Returns a list of roots of a cubic polynomial.
References
==========
.. [1] https://en.wikipedia.org/wiki/Cubic_function, General
formula for roots, (accessed November 17, 2014).
"""
if trig:
a, b, c, d = f.all_coeffs()
p = (3*a*c - b**2)/3/a**2
q = (2*b**3 - 9*a*b*c + 27*a**2*d)/(27*a**3)
D = 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2
if (D > 0) is S.true:
rv = []
for k in range(3):
rv.append(2*sqrt(-p/3)*cos(acos(3*q/2/p*sqrt(-3/p))/3 - k*2*pi/3))
return [i - b/3/a for i in rv]
_, a, b, c = f.monic().all_coeffs()
if c is S.Zero:
x1, x2 = roots([1, a, b], multiple=True)
return [x1, S.Zero, x2]
p = b - a**2/3
q = c - a*b/3 + 2*a**3/27
pon3 = p/3
aon3 = a/3
u1 = None
if p is S.Zero:
if q is S.Zero:
return [-aon3]*3
if q.is_extended_real:
if q.is_positive:
u1 = -root(q, 3)
elif q.is_negative:
u1 = root(-q, 3)
elif q is S.Zero:
y1, y2 = roots([1, 0, p], multiple=True)
return [tmp - aon3 for tmp in [y1, S.Zero, y2]]
elif q.is_extended_real and q.is_negative:
u1 = -root(-q/2 + sqrt(q**2/4 + pon3**3), 3)
coeff = I*sqrt(3)/2
if u1 is None:
u1 = Integer(1)
u2 = -S.Half + coeff
u3 = -S.Half - coeff
a, b, c, d = Integer(1), a, b, c
D0 = b**2 - 3*a*c
D1 = 2*b**3 - 9*a*b*c + 27*a**2*d
C = root((D1 + sqrt(D1**2 - 4*D0**3))/2, 3)
return [-(b + uk*C + D0/C/uk)/3/a for uk in [u1, u2, u3]]
u2 = u1*(-S.Half + coeff)
u3 = u1*(-S.Half - coeff)
if p is S.Zero:
return [u1 - aon3, u2 - aon3, u3 - aon3]
soln = [
-u1 + pon3/u1 - aon3,
-u2 + pon3/u2 - aon3,
-u3 + pon3/u3 - aon3
]
return soln
def roots_quintic(f):
"""
Calulate exact roots of a solvable quintic
"""
result = []
coeff_5, coeff_4, p, q, r, s = f.all_coeffs()
# Eqn must be of the form x^5 + px^3 + qx^2 + rx + s
if coeff_4:
return result
if coeff_5 != 1:
l = [p/coeff_5, q/coeff_5, r/coeff_5, s/coeff_5]
if not all(coeff.is_Rational for coeff in l):
return result
f = Poly(f/coeff_5)
quintic = PolyQuintic(f)
# Eqn standardized. Algo for solving starts here
if not f.is_irreducible:
return result
f20 = quintic.f20
# Check if f20 has linear factors over domain Z
if f20.is_irreducible:
return result
# Now, we know that f is solvable
for _factor in f20.factor_list()[1]:
if _factor[0].is_linear:
theta = _factor[0].root(0)
break
d = discriminant(f)
delta = sqrt(d)
# zeta = a fifth root of unity
zeta1, zeta2, zeta3, zeta4 = quintic.zeta
T = quintic.T(theta, d)
tol = Float(1e-10)
alpha = T[1] + T[2]*delta
alpha_bar = T[1] - T[2]*delta
beta = T[3] + T[4]*delta
beta_bar = T[3] - T[4]*delta
disc = alpha**2 - 4*beta
disc_bar = alpha_bar**2 - 4*beta_bar
l0 = quintic.l0(theta)
l1 = _quintic_simplify((-alpha + sqrt(disc)) / Integer(2))
l4 = _quintic_simplify((-alpha - sqrt(disc)) / Integer(2))
l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / Integer(2))
l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / Integer(2))
order = quintic.order(theta, d)
test = (order*delta.n()) - ( (l1.n() - l4.n())*(l2.n() - l3.n()) )
# Comparing floats
if not comp(test, 0, tol):
l2, l3 = l3, l2
# Now we have correct order of l's
R1 = l0 + l1*zeta1 + l2*zeta2 + l3*zeta3 + l4*zeta4
R2 = l0 + l3*zeta1 + l1*zeta2 + l4*zeta3 + l2*zeta4
R3 = l0 + l2*zeta1 + l4*zeta2 + l1*zeta3 + l3*zeta4
R4 = l0 + l4*zeta1 + l3*zeta2 + l2*zeta3 + l1*zeta4
Res = [None, [None]*5, [None]*5, [None]*5, [None]*5]
Res_n = [None, [None]*5, [None]*5, [None]*5, [None]*5]
sol = Symbol('sol')
# Simplifying improves performace a lot for exact expressions
R1 = _quintic_simplify(R1)
R2 = _quintic_simplify(R2)
R3 = _quintic_simplify(R3)
R4 = _quintic_simplify(R4)
# Solve imported here. Causing problems if imported as 'solve'
# and hence the changed name
from diofant.solvers.solvers import solve as _solve
a, b = symbols('a b', cls=Dummy)
_sol = _solve( sol**5 - a - I*b, sol)
for i in range(5):
_sol[i] = factor(_sol[i])
R1 = R1.as_real_imag()
R2 = R2.as_real_imag()
R3 = R3.as_real_imag()
R4 = R4.as_real_imag()
for i, root in enumerate(_sol):
Res[1][i] = _quintic_simplify(root.subs({ a: R1[0], b: R1[1] }))
Res[2][i] = _quintic_simplify(root.subs({ a: R2[0], b: R2[1] }))
Res[3][i] = _quintic_simplify(root.subs({ a: R3[0], b: R3[1] }))
Res[4][i] = _quintic_simplify(root.subs({ a: R4[0], b: R4[1] }))
for i in range(1, 5):
for j in range(5):
Res_n[i][j] = Res[i][j].n()
Res[i][j] = _quintic_simplify(Res[i][j])
r1 = Res[1][0]
r1_n = Res_n[1][0]
for i in range(5):
if comp(im(r1_n*Res_n[4][i]), 0, tol):
r4 = Res[4][i]
break
u, v = quintic.uv(theta, d)
sqrt5 = math.sqrt(5)
# Now we have various Res values. Each will be a list of five
# values. We have to pick one r value from those five for each Res
u, v = quintic.uv(theta, d)
testplus = (u + v*delta*sqrt(5)).n()
testminus = (u - v*delta*sqrt(5)).n()
# Evaluated numbers suffixed with _n
# We will use evaluated numbers for calculation. Much faster.
r4_n = r4.n()
r2 = r3 = None
for i in range(5):
r2temp_n = Res_n[2][i]
for j in range(5):
# Again storing away the exact number and using
# evaluated numbers in computations
r3temp_n = Res_n[3][j]
if (comp(r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus, 0, tol) and
comp(r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus, 0, tol)):
r2 = Res[2][i]
r3 = Res[3][j]
break
if r2:
break
# Now, we have r's so we can get roots
x1 = (r1 + r2 + r3 + r4)/5
x2 = (r1*zeta4 + r2*zeta3 + r3*zeta2 + r4*zeta1)/5
x3 = (r1*zeta3 + r2*zeta1 + r3*zeta4 + r4*zeta2)/5
x4 = (r1*zeta2 + r2*zeta4 + r3*zeta1 + r4*zeta3)/5
x5 = (r1*zeta1 + r2*zeta2 + r3*zeta3 + r4*zeta4)/5
result = [x1, x2, x3, x4, x5]
# Now check if solutions are distinct
saw = set()
for r in result:
r = r.n(2)
if r in saw:
# Roots were identical. Abort, return []
# and fall back to usual solve
return []
saw.add(r)
return result
def roots(f, *gens, **flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set ``cubics=False`` or ``quartics=False`` respectively. If cubic
roots are real but are expressed in terms of complex numbers
(casus irreducibilis [1]) the ``trig`` flag can be set to True to
have the solutions returned in terms of cosine and inverse cosine
functions.
To get roots from a specific domain set the ``filter`` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting ``filter='C'``).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a list containing all
those roots set the ``multiple`` flag to True; the list will
have identical roots appearing next to each other in the result.
(For a given Poly, the all_roots method will give the roots in
sorted numerical order.)
Examples
========
>>> from diofant import Poly, roots, sqrt
>>> from diofant.abc import x, y
>>> roots(x**2 - 1, x) == {-1: 1, 1: 1}
True
>>> p = Poly(x**2-1, x)
>>> roots(p) == {-1: 1, 1: 1}
True
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x)) == {-sqrt(y): 1, sqrt(y): 1}
True
>>> roots(x**2 - y, x) == {-sqrt(y): 1, sqrt(y): 1}
True
>>> roots([1, 0, -1]) == {-1: 1, 1: 1}
True
References
==========
.. [1] http://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
"""
from diofant.polys.polytools import to_rational_coeffs
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
trig = flags.pop('trig', False)
quartics = flags.pop('quartics', True)
quintics = flags.pop('quintics', False)
multiple = flags.pop('multiple', False)
filter = flags.pop('filter', None)
predicate = flags.pop('predicate', None)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
f = Poly(f, *gens, **flags)
if f.length == 2 and f.degree() != 1:
# check for foo**n factors in the constant
n = f.degree()
npow_bases = []
expr = f.as_expr()
con = expr.as_independent(*gens)[0]
for p in Mul.make_args(con):
if p.is_Pow and not p.exp % n:
npow_bases.append(p.base**(p.exp/n))
else:
other.append(p)
if npow_bases:
b = Mul(*npow_bases)
B = Dummy()
d = roots(Poly(expr - con + B**n*Mul(*others), *gens,
**flags), *gens, **flags)
rv = {}
for k, v in d.items():
rv[k.subs(B, b)] = v
return rv
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k
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 _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
(k,), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {Integer(0): k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().has_Ring:
f = f.to_field()
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
if not f.get_domain().is_Exact:
for r in f.nroots():
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.length() == 2:
roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
for r in roots_fun(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
else:
if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for factor, k in factors:
for r in _try_heuristics(Poly(factor, f.gen, field=True)):
_update_dict(result, r, k)
if coeff is not S.One:
_result, result, = result, {}
for root, k in _result.items():
result[coeff*root] = k
result.update(zeros)
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_extended_real,
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).keys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).keys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k*rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero]*result[zero])
return zeros
def test_core_numbers():
for c in (Integer(2), Rational(2, 3), Float("1.2")):
check(c)
def test_sympyissue_6981():
S = set(divisors(4)).union(set(divisors(Integer(2))))
assert S == {1, 2, 4}
def dot(self, other):
"""
Returns the dot product of this Vector, either with another
Vector, or a Dyadic, or a Del operator.
If 'other' is a Vector, returns the dot product scalar (Diofant
expression).
If 'other' is a Dyadic, the dot product is returned as a Vector.
If 'other' is an instance of Del, returns the directional
derivate operator as a Python function. If this function is
applied to a scalar expression, it returns the directional
derivative of the scalar field wrt this Vector.
Parameters
==========
other: Vector/Dyadic/Del
The Vector or Dyadic we are dotting with, or a Del operator .
Examples
========
>>> from diofant.vector import CoordSysCartesian
>>> C = CoordSysCartesian('C')
>>> C.i.dot(C.j)
0
>>> C.i & C.i
1
>>> v = 3*C.i + 4*C.j + 5*C.k
>>> v.dot(C.k)
5
>>> (C.i & C.delop)(C.x*C.y*C.z)
C.y*C.z
>>> d = C.i.outer(C.i)
>>> C.i.dot(d)
C.i
"""
from diofant.vector.functions import express
# Check special cases
if isinstance(other, Dyadic):
if isinstance(self, VectorZero):
return Vector.zero
outvec = Vector.zero
for k, v in other.components.items():
vect_dot = k.args[0].dot(self)
outvec += vect_dot * v * k.args[1]
return outvec
from diofant.vector.deloperator import Del
if not isinstance(other, Vector) and not isinstance(other, Del):
raise TypeError(str(other) + " is not a vector, dyadic or " +
"del operator")
# Check if the other is a del operator
if isinstance(other, Del):
def directional_derivative(field):
field = express(field, other.system, variables=True)
out = self.dot(other._i) * df(field, other._x)
out += self.dot(other._j) * df(field, other._y)
out += self.dot(other._k) * df(field, other._z)
if out == 0 and isinstance(field, Vector):
out = Vector.zero
return out
return directional_derivative
if isinstance(self, VectorZero) or isinstance(other, VectorZero):
return Integer(0)
v1 = express(self, other._sys)
v2 = express(other, other._sys)
dotproduct = Integer(0)
for x in other._sys.base_vectors():
dotproduct += (v1.components.get(x, 0) *
v2.components.get(x, 0))
return dotproduct
def as_base_exp(self):
if self.exp == 0:
return self, Integer(1)
return self.func(1), Mul(*self.args)
