def _eval_expand_func(self, **hints):
n, z = self.args
if n.is_Integer and n.is_nonnegative:
if z.is_Add:
coeff = z.args[0]
if coeff.is_Integer:
e = -(n + 1)
if coeff > 0:
tail = Add(
*[Pow(z - i, e) for i in range(1,
int(coeff) + 1)])
else:
tail = -Add(
*[Pow(z + i, e) for i in range(0, int(-coeff))])
return polygamma(n,
z - coeff) + (-1)**n * factorial(n) * tail
elif z.is_Mul:
coeff, z = z.as_two_terms()
if coeff.is_Integer and coeff.is_positive:
tail = [
polygamma(n, z + Rational(i, coeff))
for i in range(0, int(coeff))
]
if n == 0:
return Add(*tail) / coeff + log(coeff)
else:
return Add(*tail) / coeff**(n + 1)
z *= coeff
return polygamma(n, z)
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(z, Rational(3, 2))
if re(z).is_positive:
return ot * sqrt(z) * (besseli(-ot, tt * a) - besseli(ot, tt * a))
else:
return ot * (Pow(a, ot) * besseli(-ot, tt * a) -
z * Pow(a, -ot) * besseli(ot, tt * a))
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = tt * Pow(z, Rational(3, 2))
if re(z).is_positive:
return z / sqrt(3) * (besseli(-tt, a) + besseli(tt, a))
else:
a = Pow(z, Rational(3, 2))
b = Pow(a, tt)
c = Pow(a, -tt)
return sqrt(ot) * (b * besseli(-tt, tt * a) +
z**2 * c * besseli(tt, tt * a))
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = tt * Pow(z, Rational(3, 2))
if re(z).is_positive:
return z / 3 * (besseli(tt, a) - besseli(-tt, a))
else:
a = Pow(z, Rational(3, 2))
b = Pow(a, tt)
c = Pow(a, -tt)
return ot * (z**2 * c * besseli(tt, tt * a) -
b * besseli(-ot, tt * a))
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(z, Rational(3, 2))
if re(z).is_positive:
return sqrt(z) / sqrt(3) * (besseli(-ot, tt * a) +
besseli(ot, tt * a))
else:
b = Pow(a, ot)
c = Pow(a, -ot)
return sqrt(ot) * (b * besseli(-ot, tt * a) +
z * c * besseli(ot, tt * a))
def exp(arg, **kwargs):
"""
The exponential function, :math:`e^x`.
See Also
========
diofant.functions.elementary.exponential.log
"""
return Pow(S.Exp1, arg, **kwargs)
def _sqrt(d):
# remove squares from square root since both will be represented
# in the results; a similar thing is happening in roots() but
# must be duplicated here because not all quadratics are binomials
co = []
other = []
for di in Mul.make_args(d):
if di.is_Pow and di.exp.is_Integer and di.exp % 2 == 0:
co.append(Pow(di.base, di.exp//2))
else:
other.append(di)
if co:
d = Mul(*other)
co = Mul(*co)
return co*sqrt(d)
return sqrt(d)
def test_core_power():
for c in (Pow, Pow(x, 4)):
check(c)
def factorint(n,
limit=None,
use_trial=True,
use_rho=True,
use_pm1=True,
verbose=False,
visual=None):
r"""
Given a positive integer ``n``, ``factorint(n)`` returns a dict containing
the prime factors of ``n`` as keys and their respective multiplicities
as values. For example:
>>> from diofant.ntheory import factorint
>>> factorint(2000) # 2000 = (2**4) * (5**3)
{2: 4, 5: 3}
>>> factorint(65537) # This number is prime
{65537: 1}
For input less than 2, factorint behaves as follows:
- ``factorint(1)`` returns the empty factorization, ``{}``
- ``factorint(0)`` returns ``{0:1}``
- ``factorint(-n)`` adds ``-1:1`` to the factors and then factors ``n``
Partial Factorization:
If ``limit`` (> 3) is specified, the search is stopped after performing
trial division up to (and including) the limit (or taking a
corresponding number of rho/p-1 steps). This is useful if one has
a large number and only is interested in finding small factors (if
any). Note that setting a limit does not prevent larger factors
from being found early; it simply means that the largest factor may
be composite. Since checking for perfect power is relatively cheap, it is
done regardless of the limit setting.
This number, for example, has two small factors and a huge
semi-prime factor that cannot be reduced easily:
>>> from diofant.ntheory import isprime
>>> a = 1407633717262338957430697921446883
>>> f = factorint(a, limit=10000)
>>> f == {991: 1, int(202916782076162456022877024859): 1, 7: 1}
True
>>> isprime(max(f))
False
This number has a small factor and a residual perfect power whose
base is greater than the limit:
>>> factorint(3*101**7, limit=5)
{3: 1, 101: 7}
Visual Factorization:
If ``visual`` is set to ``True``, then it will return a visual
factorization of the integer. For example:
>>> from diofant import pprint
>>> pprint(factorint(4200, visual=True), use_unicode=False)
3 1 2 1
2 *3 *5 *7
Note that this is achieved by using the evaluate=False flag in Mul
and Pow. If you do other manipulations with an expression where
evaluate=False, it may evaluate. Therefore, you should use the
visual option only for visualization, and use the normal dictionary
returned by visual=False if you want to perform operations on the
factors.
You can easily switch between the two forms by sending them back to
factorint:
>>> from diofant import Mul, Pow
>>> regular = factorint(1764); regular
{2: 2, 3: 2, 7: 2}
>>> pprint(factorint(regular), use_unicode=False)
2 2 2
2 *3 *7
>>> visual = factorint(1764, visual=True); pprint(visual, use_unicode=False)
2 2 2
2 *3 *7
>>> print(factorint(visual))
{2: 2, 3: 2, 7: 2}
If you want to send a number to be factored in a partially factored form
you can do so with a dictionary or unevaluated expression:
>>> factorint(factorint({4: 2, 12: 3})) # twice to toggle to dict form
{2: 10, 3: 3}
>>> factorint(Mul(4, 12, evaluate=False))
{2: 4, 3: 1}
The table of the output logic is:
====== ====== ======= =======
Visual
------ ----------------------
Input True False other
====== ====== ======= =======
dict mul dict mul
n mul dict dict
mul mul dict dict
====== ====== ======= =======
Notes
=====
The function switches between multiple algorithms. Trial division
quickly finds small factors (of the order 1-5 digits), and finds
all large factors if given enough time. The Pollard rho and p-1
algorithms are used to find large factors ahead of time; they
will often find factors of the order of 10 digits within a few
seconds:
>>> factors = factorint(12345678910111213141516)
>>> for base, exp in sorted(factors.items()):
... print('%s %s' % (base, exp))
...
2 2
2507191691 1
1231026625769 1
Any of these methods can optionally be disabled with the following
boolean parameters:
- ``use_trial``: Toggle use of trial division
- ``use_rho``: Toggle use of Pollard's rho method
- ``use_pm1``: Toggle use of Pollard's p-1 method
``factorint`` also periodically checks if the remaining part is
a prime number or a perfect power, and in those cases stops.
If ``verbose`` is set to ``True``, detailed progress is printed.
See Also
========
smoothness, smoothness_p, divisors
"""
factordict = {}
if visual and not isinstance(n, Mul) and not isinstance(n, dict):
factordict = factorint(n,
limit=limit,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose,
visual=False)
elif isinstance(n, Mul):
factordict = {
int(k): int(v)
for k, v in list(n.as_powers_dict().items())
}
elif isinstance(n, dict):
factordict = n
if factordict and (isinstance(n, Mul) or isinstance(n, dict)):
# check it
for k in list(factordict.keys()):
if isprime(k):
continue
e = factordict.pop(k)
d = factorint(k,
limit=limit,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose,
visual=False)
for k, v in d.items():
if k in factordict:
factordict[k] += v * e
else:
factordict[k] = v * e
if visual or (type(n) is dict and visual is not True
and visual is not False):
if factordict == {}:
return S.One
if -1 in factordict:
factordict.pop(-1)
args = [S.NegativeOne]
else:
args = []
args.extend(
[Pow(*i, evaluate=False) for i in sorted(factordict.items())])
return Mul(*args, evaluate=False)
elif isinstance(n, dict) or isinstance(n, Mul):
return factordict
assert use_trial or use_rho or use_pm1
n = as_int(n)
if limit:
limit = int(limit)
# special cases
if n < 0:
factors = factorint(-n,
limit=limit,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose,
visual=False)
factors[-1] = 1
return factors
if limit and limit < 2:
if n == 1:
return {}
return {n: 1}
elif n < 10:
# doing this we are assured of getting a limit > 2
# when we have to compute it later
return [{
0: 1
}, {}, {
2: 1
}, {
3: 1
}, {
2: 2
}, {
5: 1
}, {
2: 1,
3: 1
}, {
7: 1
}, {
2: 3
}, {
3: 2
}][n]
factors = {}
# do simplistic factorization
if verbose:
sn = str(n)
if len(sn) > 50:
print('Factoring %s' % sn[:5] + '..(%i other digits)..' %
(len(sn) - 10) + sn[-5:])
else:
print('Factoring', n)
if use_trial:
# this is the preliminary factorization for small factors
small = 2**15
fail_max = 600
small = min(small, limit or small)
if verbose:
print(trial_int_msg % (2, small, fail_max))
n, next_p = _factorint_small(factors, n, small, fail_max)
else:
next_p = 2
if factors and verbose:
for k in sorted(factors):
print(factor_msg % (k, factors[k]))
if next_p == 0:
if n > 1:
factors[int(n)] = 1
if verbose:
print(complete_msg)
return factors
# continue with more advanced factorization methods
# first check if the simplistic run didn't finish
# because of the limit and check for a perfect
# power before exiting
try:
if limit and next_p > limit:
if verbose:
print('Exceeded limit:', limit)
_check_termination(factors, n, limit, use_trial, use_rho, use_pm1,
verbose)
if n > 1:
factors[int(n)] = 1
return factors
else:
# Before quitting (or continuing on)...
# ...do a Fermat test since it's so easy and we need the
# square root anyway. Finding 2 factors is easy if they are
# "close enough." This is the big root equivalent of dividing by
# 2, 3, 5.
sqrt_n = integer_nthroot(n, 2)[0]
a = sqrt_n + 1
a2 = a**2
b2 = a2 - n
for i in range(3):
b, fermat = integer_nthroot(b2, 2)
if fermat:
break
b2 += 2 * a + 1 # equiv to (a+1)**2 - n
a += 1
if fermat:
if verbose:
print(fermat_msg)
if limit:
limit -= 1
for r in [a - b, a + b]:
facs = factorint(r,
limit=limit,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose)
factors.update(facs)
raise StopIteration
# ...see if factorization can be terminated
_check_termination(factors, n, limit, use_trial, use_rho, use_pm1,
verbose)
except StopIteration:
if verbose:
print(complete_msg)
return factors
# these are the limits for trial division which will
# be attempted in parallel with pollard methods
low, high = next_p, 2 * next_p
limit = limit or sqrt_n
# add 1 to make sure limit is reached in primerange calls
limit += 1
while 1:
try:
high_ = high
if limit < high_:
high_ = limit
# Trial division
if use_trial:
if verbose:
print(trial_msg % (low, high_))
ps = sieve.primerange(low, high_)
n, found_trial = _trial(factors, n, ps, verbose)
if found_trial:
_check_termination(factors, n, limit, use_trial, use_rho,
use_pm1, verbose)
else:
found_trial = False
if high > limit:
if verbose:
print('Exceeded limit:', limit)
if n > 1:
factors[int(n)] = 1
raise StopIteration
# Only used advanced methods when no small factors were found
if not found_trial:
if (use_pm1 or use_rho):
high_root = max(int(math.log(high_**0.7)), low, 3)
# Pollard p-1
if use_pm1:
if verbose:
print(pm1_msg % (high_root, high_))
c = pollard_pm1(n, B=high_root, seed=high_)
if c:
# factor it and let _trial do the update
ps = factorint(c,
limit=limit - 1,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose)
n, _ = _trial(factors, n, ps, verbose=False)
_check_termination(factors, n, limit, use_trial,
use_rho, use_pm1, verbose)
# Pollard rho
if use_rho:
max_steps = high_root
if verbose:
print(rho_msg % (1, max_steps, high_))
c = pollard_rho(n,
retries=1,
max_steps=max_steps,
seed=high_)
if c:
# factor it and let _trial do the update
ps = factorint(c,
limit=limit - 1,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose)
n, _ = _trial(factors, n, ps, verbose=False)
_check_termination(factors, n, limit, use_trial,
use_rho, use_pm1, verbose)
except StopIteration:
if verbose:
print(complete_msg)
return factors
low, high = high, high * 2
def _eval_rewrite_as_besselj(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(-z, Rational(3, 2))
if re(z).is_negative:
return ot * sqrt(-z) * (besselj(-ot, tt * a) + besselj(ot, tt * a))
def _eval_rewrite_as_besselj(self, z):
tt = Rational(2, 3)
a = tt * Pow(-z, Rational(3, 2))
if re(z).is_negative:
return -z / sqrt(3) * (besselj(-tt, a) + besselj(tt, a))
def _eval_rewrite_as_besselj(self, z):
tt = Rational(2, 3)
a = Pow(-z, Rational(3, 2))
if re(z).is_negative:
return z / 3 * (besselj(-tt, tt * a) - besselj(tt, tt * a))