def rotozoom(self, surface, angle, size):
"""
Return Surface rotated and resized by the given angle and size.
"""
if not angle:
width = int(surface.getWidth()*size)
height = int(surface.getHeight()*size)
return self.scale(surface, (width, height))
theta = angle*self.deg_rad
width_i = int(surface.getWidth()*size)
height_i = int(surface.getHeight()*size)
cos_theta = _fabs( _cos(theta) )
sin_theta = _fabs( _sin(theta) )
width_f = int( _ceil((width_i*cos_theta)+(height_i*sin_theta)) )
if width_f % 2:
width_f += 1
height_f = int( _ceil((width_i*sin_theta)+(height_i*cos_theta)) )
if height_f % 2:
height_f += 1
surf = Surface((width_f,height_f), BufferedImage.TYPE_INT_ARGB)
at = AffineTransform()
at.translate(width_f/2, height_f/2)
at.rotate(-theta)
g2d = surf.createGraphics()
ot = g2d.getTransform()
g2d.setTransform(at)
g2d.setRenderingHint(RenderingHints.KEY_INTERPOLATION, RenderingHints.VALUE_INTERPOLATION_BILINEAR)
g2d.drawImage(surface, -width_i//2, -height_i//2, width_i, height_i, None)
g2d.setTransform(ot)
g2d.dispose()
return surf
def rotozoom(self, surface, angle, size):
"""
Return Surface rotated and resized by the given angle and size.
"""
if not angle:
width = int(surface.get_width()*size)
height = int(surface.get_height()*size)
return self.scale(surface, (width, height))
theta = angle*self.deg_rad
width_i = int(surface.get_width()*size)
height_i = int(surface.get_height()*size)
cos_theta = _fabs( _cos(theta) )
sin_theta = _fabs( _sin(theta) )
width_f = int( _ceil((width_i*cos_theta)+(height_i*sin_theta)) )
if width_f % 2:
width_f += 1
height_f = int( _ceil((width_i*sin_theta)+(height_i*cos_theta)) )
if height_f % 2:
height_f += 1
surf = Surface((width_f,height_f))
surf.saveContext()
surf.translate(width_f/2.0, height_f/2.0)
surf.rotate(-theta)
surf.drawImage(surface.canvas, 0, 0, surface.get_width(), surface.get_height(), -width_i/2, -height_i/2, width_i, height_i)
surf.restoreContext()
return surf
def sample(self, population, k):
if isinstance(population, _Set):
population = tuple(population)
if not isinstance(population, _Sequence):
raise TypeError('Population must be a sequence or set. For dicts, use list(d).')
randbelow = self._randbelow
n = len(population)
if not 0 <= k <= n:
raise ValueError('Sample larger than population')
result = [None]*k
setsize = 21
if k > 5:
setsize += 4**_ceil(_log(k*3, 4))
if n <= setsize:
pool = list(population)
for i in range(k):
j = randbelow(n - i)
result[i] = pool[j]
pool[j] = pool[n - i - 1]
else:
selected = set()
selected_add = selected.add
for i in range(k):
j = randbelow(n)
while j in selected:
j = randbelow(n)
selected_add(j)
result[i] = population[j]
return result
def sample(self, population, k):
"""Chooses k unique random elements from a population sequence."""
# https://github.com/python/cpython/blob/2.7/Lib/random.py#L275
n = len(population)
if not 0 <= k <= n:
raise ValueError("sample larger than population")
random = self.random
_int = int
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize or hasattr(population, "keys"):
# An n-length list is smaller than a k-length set, or this is a
# mapping type so the other algorithm wouldn't work.
pool = list(population)
for i in xrange(k): # invariant: non-selected at [0,n-i)
j = _int(random() * (n-i))
result[i] = pool[j]
pool[j] = pool[n-i-1] # move non-selected item into vacancy
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = _int(random() * n)
while j in selected:
j = _int(random() * n)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError): # handle (at least) sets
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result
def sample(self, population, k, generator=None):
# This function exactly parallels the code in Random.py.
# Comments are therefore omitted, to save space.
n = len(population)
if not 0 <= k <= n:
raise ValueError('sample larger than population')
randrange = self.randrange
result = [None] * k
setsize = 21
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4))
if n <= setsize or hasattr(population, 'keys'):
pool = list(population)
for i in xrange(k):
j = randrange(n-i, generator=generator)
result[i] = pool[j]
pool[j] = pool[n-i-1]
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = randrange(n, generator=generator)
while j in selected:
j = randrange(n, generator=generator)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError):
if isinstance(population, list):
raise
return self.sample(tuple(population), k, generator)
return result
def round(space, number, ndigits=0):
"""round(number[, ndigits]) -> floating point number
Round a number to a given precision in decimal digits (default 0 digits).
This always returns a floating point number. Precision may be negative."""
# Algortithm copied directly from CPython
f = 1.0
if ndigits < 0:
i = -ndigits
else:
i = ndigits
while i > 0:
f = f*10.0
i -= 1
if ndigits < 0:
number /= f
else:
number *= f
if number >= 0.0:
number = _floor(number + 0.5)
else:
number = _ceil(number - 0.5)
if ndigits < 0:
number *= f
else:
number /= f
return space.wrap(number)
def sample(self, population, k):
n = len(population)
if not 0 <= k <= n:
raise ValueError, 'sample larger than population'
random = self.random
_int = int
result = [None] * k
setsize = 21
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4))
if n <= setsize or hasattr(population, 'keys'):
pool = list(population)
for i in xrange(k):
j = _int(random() * (n - i))
result[i] = pool[j]
pool[j] = pool[n - i - 1]
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = _int(random() * n)
while j in selected:
j = _int(random() * n)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError):
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result
def dup_revert(f, n, K):
"""
Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.
This function computes first ``2**n`` terms of a polynomial that
is a result of inversion of a polynomial modulo ``x**n``. This is
useful to efficiently compute series expansion of ``1/f``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.densetools import dup_revert
>>> f = [-QQ(1,720), QQ(0), QQ(1,24), QQ(0), -QQ(1,2), QQ(0), QQ(1)]
>>> dup_revert(f, 8, QQ)
[61/720, 0/1, 5/24, 0/1, 1/2, 0/1, 1/1]
"""
g = [K.revert(dup_TC(f, K))]
h = [K.one, K.zero, K.zero]
N = int(_ceil(_log(n, 2)))
for i in xrange(1, N + 1):
a = dup_mul_ground(g, K(2), K)
b = dup_mul(f, dup_sqr(g, K), K)
g = dup_rem(dup_sub(a, b, K), h, K)
h = dup_lshift(h, dup_degree(h), K)
return g
def dup_revert(f, n, K):
"""
Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.
This function computes first ``2**n`` terms of a polynomial that
is a result of inversion of a polynomial modulo ``x**n``. This is
useful to efficiently compute series expansion of ``1/f``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1
>>> R.dup_revert(f, 8)
61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1
"""
g = [K.revert(dup_TC(f, K))]
h = [K.one, K.zero, K.zero]
N = int(_ceil(_log(n, 2)))
for i in range(1, N + 1):
a = dup_mul_ground(g, K(2), K)
b = dup_mul(f, dup_sqr(g, K), K)
g = dup_rem(dup_sub(a, b, K), h, K)
h = dup_lshift(h, dup_degree(h), K)
return g
def sample(self, population, k):
"""Chooses k unique random elements from a population sequence.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
selection in the sample.
To choose a sample in a range of integers, use xrange as an argument.
This is especially fast and space efficient for sampling from a
large population: sample(xrange(10000000), 60)
"""
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
n = len(population)
if not 0 <= k <= n:
raise ValueError("sample larger than population")
random = self.random
_int = int
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize or hasattr(population, "keys"):
# An n-length list is smaller than a k-length set, or this is a
# mapping type so the other algorithm wouldn't work.
pool = list(population)
for i in xrange(k): # invariant: non-selected at [0,n-i)
j = _int(random() * (n-i))
result[i] = pool[j]
pool[j] = pool[n-i-1] # move non-selected item into vacancy
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = _int(random() * n)
while j in selected:
j = _int(random() * n)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError): # handle (at least) sets
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result
def sample(self, population: Iterable[T], k: int) -> List[T]:
"""Chooses k unique random elements from a population sequence or set.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
selection in the sample.
To choose a sample in a range of integers, use range as an argument.
This is especially fast and space efficient for sampling from a
large population: sample(range(10000000), 60)
"""
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
if isinstance(population, _Sequence):
populationseq = population
elif isinstance(population, _Set):
populationseq = list(population)
else:
raise TypeError("Population must be a sequence or set. For dicts, use list(d).")
randbelow = self._randbelow
n = len(populationseq)
if not (0 <= k and k <= n):
raise ValueError("Sample larger than population")
result = [cast(T, None)] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize:
# An n-length list is smaller than a k-length set
pool = list(populationseq)
for i in range(k): # invariant: non-selected at [0,n-i)
j = randbelow(n-i)
result[i] = pool[j]
pool[j] = pool[n-i-1] # move non-selected item into vacancy
else:
selected = Set[int]()
selected_add = selected.add
for i in range(k):
j = randbelow(n)
while j in selected:
j = randbelow(n)
selected_add(j)
result[i] = populationseq[j]
return result
def is_prime2(num):
'''Tests if a given number is prime. Written with a map.'''
if num == 2:
return True
elif num % 2 == 0 or num <= 1:
return False
root = _ceil(_sqrt(num))
return all(map(lambda div: False if num % div == 0 else True,
range(3, root+1, 2)))
def is_prime3(num):
'''Tests if a given number is prime. Written with reduce.'''
if num == 2:
return True
elif num % 2 == 0 or num <= 1:
return False
root = _ceil(_sqrt(num))
return _reduce(lambda acc, d: False if not acc or num % d == 0 else True,
range(3, root+1, 2), True)
def execute(self, target, *args):
# get func source without first 'def func(...):' line
src = getsource(target)
src = '\n'.join( src.splitlines()[1:] )
# extract benchmark title
if target.func_doc is not None:
self.benchtitle = target.func_doc
else:
self.benchtitle = src.splitlines()[0].strip()
# XXX we ignore args
timer = Timer(src, globals=target.func_globals)
if self.name.startswith('timeit_'):
# from IPython.Magic.magic_timeit
repeat = 3
number = 1
for i in range(1,10):
t = timer.timeit(number)
if t >= 0.2:
number *= (0.2 / t)
number = int(_ceil(number))
break
if t <= 0.02:
# we are not close enough to that 0.2s
number *= 10
else:
# since we are very close to be > 0.2s we'd better adjust number
# so that timing time is not too high
number *= (0.2 / t)
number = int(_ceil(number))
break
self.benchtime = min(timer.repeat(repeat, number)) / number
# 'bench_<smth>'
else:
self.benchtime = timer.timeit(1)
def dup_zz_hensel_lift(p, f, f_list, l, K):
"""
Multifactor Hensel lifting in `Z[x]`.
Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
over `Z[x]` satisfying::
f = lc(f) f_1 ... f_r (mod p)
and a positive integer `l`, returns a list of monic polynomials
`F_1`, `F_2`, ..., `F_r` satisfying::
f = lc(f) F_1 ... F_r (mod p**l)
F_i = f_i (mod p), i = 1..r
References
==========
1. [Gathen99]_
"""
r = len(f_list)
lc = dup_LC(f, K)
if r == 1:
F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
return [ dup_trunc(F, p**l, K) ]
m = p
k = r // 2
d = int(_ceil(_log(l, 2)))
g = gf_from_int_poly([lc], p)
for f_i in f_list[:k]:
g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)
h = gf_from_int_poly(f_list[k], p)
for f_i in f_list[k + 1:]:
h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)
s, t, _ = gf_gcdex(g, h, p, K)
g = gf_to_int_poly(g, p)
h = gf_to_int_poly(h, p)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
for _ in range(1, d + 1):
(g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2
return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
+ dup_zz_hensel_lift(p, h, f_list[k:], l, K)
def p010(ceiling): numbers = list(range(2, ceiling)) i = 0 while i < ceiling - 2: jump = numbers[i] numbers[i + jump:ceiling - 2:jump] = _repeat(0, _ceil((len(numbers) - i) / jump) - 1) i += 1 while i < ceiling - 2 and numbers[i] == 0: i += 1 return sum(numbers)
def _size_estimate(self, text=None): #for browsers HTML5Canvas not implemented
if not self.char_size:
self.char_size = self._get_char_size()
self.fontname = ','.join(Font._font_family[0])
self.fontstyle = ''
size = []
for char in text:
try:
size.append(self.char_size[char] * self.fontsize)
except KeyError:
size.append(self.char_size['x'] * self.fontsize)
x = _ceil( sum(size) )
return x
def getrandbits(self, bitlength):
"""Return a Python long with `bitlength` random bits."""
num_bytes = int(_ceil(bitlength / 8.0))
bits_to_zero = (bitlength % 8)
mask = 0xff if not bits_to_zero else 0xff >> (8 - bits_to_zero)
s = array('B', rand_bytes(num_bytes))
s[0] = s[0] & mask
# Alas, int.from_bytes is only available in Python 3. Calling
# `hexlify` is very slightly faster than b.encode('hex'). Also
# note that on Python 2, long(_) is slightly faster than int(_)
# for large numbers.
return long(hexlify(s), 16)
def sample(self, population, k):
"""Chooses k unique random elements from a population sequence.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
selection in the sample.
To choose a sample in a range of integers, use xrange as an argument.
This is especially fast and space efficient for sampling from a
large population: sample(xrange(10000000), 60)
"""
n = len(population)
if not 0 <= k <= n:
raise ValueError("sample larger than population")
random = self.random
_int = int
result = [None] * k
setsize = 21
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4))
if n <= setsize or hasattr(population, "keys"):
pool = list(population)
for i in xrange(k):
j = _int(random() * (n - i))
result[i] = pool[j]
pool[j] = pool[n - i - 1]
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = _int(random() * n)
while j in selected:
j = _int(random() * n)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError):
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result
def check_resampled(freq_new):
# resample
resampled = s.resample(freq_new)
# new number of samples
new_samples = _ceil(samples * freq_new / freq)
# length Y
self.assertEqual(len(resampled), new_samples)
# length X
self.assertEqual(len(resampled.get_times()), len(resampled))
# sampling frequency in Hz
self.assertEqual(resampled.get_sampling_freq(), freq_new)
# duration differs less than 1s from the original
self.assertLess(abs(resampled.get_duration() - samples / freq), 1)
# start timestamp
self.assertEqual(resampled.get_start_time(), start)
# end timestamp
self.assertEqual(resampled.get_end_time(), start + resampled.get_duration())
# start time
self.assertEqual(resampled.get_signal_nature(), nature)
def frange(*args):
'''Generalize the builtin range() by allowing float start,stop and step.
@param args: stop or (start,stop) or (start,stop,step)
@type args: one,two or three integers.
'''
start=0; step=1
if len(args) == 1: stop = args[0]
elif len(args) == 2: start,stop = args
elif len(args) == 3: start,stop,step = args
else: raise TypeError, "frange() requires 1-3 int arguments"
for x in stop,start,step:
if not isinstance(x,int):
break
else:
return range(start,stop,step)
results = []
steps = _int(_ceil((stop-start) / float(step)))
for i in xrange(steps):
results.append(start)
start += step
return results
def ceil(x, options=None):
"""Return the ceiling of *x*. If *options* is set, return the smallest
integer or float from *options* that is greater than or equal to
*x*.
Args:
x (int or float): Number to be tested.
options (iterable): Optional iterable of arbitrary numbers
(ints or floats).
>>> VALID_CABLE_CSA = [1.5, 2.5, 4, 6, 10, 25, 35, 50]
>>> ceil(3.5, options=VALID_CABLE_CSA)
4
>>> ceil(4, options=VALID_CABLE_CSA)
4
"""
if options is None:
return _ceil(x)
options = sorted(options)
i = bisect.bisect_left(options, x)
if i == len(options):
raise ValueError("no ceil options greater than or equal to: %r" % x)
return options[i]
def _write(self, body, path, format):
""" Actually write the file to the cache directory, return its size.
If filesystem block size is known, try to return actual disk space used.
"""
fullpath = pathjoin(self.cachepath, path)
try:
umask_old = os.umask(self.umask)
os.makedirs(dirname(fullpath), 0o777&~self.umask)
except OSError as e:
if e.errno != 17:
raise
finally:
os.umask(umask_old)
fh, tmp_path = mkstemp(dir=self.cachepath, suffix='.' + format.lower())
os.write(fh, body)
os.close(fh)
try:
os.rename(tmp_path, fullpath)
except OSError:
os.unlink(fullpath)
os.rename(tmp_path, fullpath)
os.chmod(fullpath, 0o666&~self.umask)
stat = os.stat(fullpath)
size = stat.st_size
if hasattr(stat, 'st_blksize'):
blocks = _ceil(size / float(stat.st_blksize))
size = int(blocks * stat.st_blksize)
return size
def ceil(x): return int(_ceil(x))
def gf_ddf_shoup(f, p, K):
"""
Kaltofen-Shoup: Deterministic Distinct Degree Factorization
Given a monic square-free polynomial ``f`` in ``GF(p)[x]``, computes
partial distinct degree factorization ``f_1,...,f_d`` of ``f`` where
``deg(f_i) != deg(f_j)`` for ``i != j``. The result is returned as a
list of pairs ``(f_i, e_i)`` where ``deg(f_i) > 0`` and ``e_i > 0``
is an argument to the equal degree factorization routine.
This algorithm is an improved version of Zassenhaus algorithm for
large ``deg(f)`` and modulus ``p`` (especially for ``deg(f) ~ lg(p)``).
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.galoistools import gf_ddf_shoup, gf_from_dict
>>> f = gf_from_dict({6: ZZ(1), 5: ZZ(-1), 4: ZZ(1), 3: ZZ(1), 1: ZZ(-1)}, 3, ZZ)
>>> gf_ddf_shoup(f, 3, ZZ)
[([1, 1, 0], 1), ([1, 1, 0, 1, 2], 2)]
References
==========
1. [Kaltofen98]_
2. [Shoup95]_
3. [Gathen92]_
"""
n = gf_degree(f)
k = int(_ceil(_sqrt(n // 2)))
h = gf_pow_mod([K.one, K.zero], int(p), f, p, K)
U = [[K.one, K.zero], h] + [K.zero] * (k - 1)
for i in xrange(2, k + 1):
U[i] = gf_compose_mod(U[i - 1], h, f, p, K)
h, U = U[k], U[:k]
V = [h] + [K.zero] * (k - 1)
for i in xrange(1, k):
V[i] = gf_compose_mod(V[i - 1], h, f, p, K)
factors = []
for i, v in enumerate(V):
h, j = [K.one], k - 1
for u in U:
g = gf_sub(v, u, p, K)
h = gf_mul(h, g, p, K)
h = gf_rem(h, f, p, K)
g = gf_gcd(f, h, p, K)
f = gf_quo(f, g, p, K)
for u in reversed(U):
h = gf_sub(v, u, p, K)
F = gf_gcd(g, h, p, K)
if F != [K.one]:
factors.append((F, k * (i + 1) - j))
g, j = gf_quo(g, F, p, K), j - 1
if f != [K.one]:
factors.append((f, gf_degree(f)))
return factors
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
fc = f[-1]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1))*2**n*A*b))
C = int((n + 1)**(2*n)*A**(2*n - 1))
gamma = int(_ceil(2*_log(C, 2)))
bound = int(2*gamma*_log(gamma))
a = []
# choose a prime number `p` such that `f` be square free in Z_p
# if there are many factors in Z_p, choose among a few different `p`
# the one with fewer factors
for px in range(3, bound + 1):
if not isprime(px) or b % px == 0:
continue
px = K.convert(px)
F = gf_from_int_poly(f, px)
if not gf_sqf_p(F, px, K):
continue
fsqfx = gf_factor_sqf(F, px, K)[1]
a.append((px, fsqfx))
if len(fsqfx) < 15 or len(a) > 4:
break
p, fsqf = min(a, key=lambda x: len(x[1]))
l = int(_ceil(_log(2*B + 1, p)))
modular = [gf_to_int_poly(ff, p) for ff in fsqf]
g = dup_zz_hensel_lift(p, f, modular, l, K)
sorted_T = range(len(g))
T = set(sorted_T)
factors, s = [], 1
pl = p**l
while 2*s <= len(T):
for S in subsets(sorted_T, s):
# lift the constant coefficient of the product `G` of the factors
# in the subset `S`; if it is does not divide `fc`, `G` does
# not divide the input polynomial
if b == 1:
q = 1
for i in S:
q = q*g[i][-1]
q = q % pl
if not _test_pl(fc, q, pl):
continue
else:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
G = dup_primitive(G, K)[1]
q = G[-1]
if q and fc % q != 0:
continue
H = [b]
S = set(S)
T_S = T - S
if b == 1:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
for i in T_S:
H = dup_mul(H, g[i], K)
H = dup_trunc(H, pl, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm*H_norm <= B:
T = T_S
sorted_T = [i for i in sorted_T if i not in S]
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def _exposure(ret):
ex = len(ret[(~_np.isnan(ret)) & (ret != 0)]) / len(ret)
return _ceil(ex * 100) / 100
def sample(self, population, k):
"""Chooses k unique random elements from a population sequence or set.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
selection in the sample.
To choose a sample in a range of integers, use range as an argument.
This is especially fast and space efficient for sampling from a
large population: sample(range(10000000), 60)
"""
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
# The number of calls to _randbelow() is kept at or near k, the
# theoretical minimum. This is important because running time
# is dominated by _randbelow() and because it extracts the
# least entropy from the underlying random number generators.
# Memory requirements are kept to the smaller of a k-length
# set or an n-length list.
# There are other sampling algorithms that do not require
# auxiliary memory, but they were rejected because they made
# too many calls to _randbelow(), making them slower and
# causing them to eat more entropy than necessary.
if isinstance(population, _Set):
_warn(
'Sampling from a set deprecated\n'
'since Python 3.9 and will be removed in a subsequent version.',
DeprecationWarning, 2)
population = tuple(population)
if not isinstance(population, _Sequence):
raise TypeError(
"Population must be a sequence. For dicts or sets, use sorted(d)."
)
randbelow = self._randbelow
n = len(population)
if not 0 <= k <= n:
raise ValueError("Sample larger than population or is negative")
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4**_ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize:
# An n-length list is smaller than a k-length set
pool = list(population)
for i in range(k): # invariant: non-selected at [0,n-i)
j = randbelow(n - i)
result[i] = pool[j]
pool[j] = pool[n - i -
1] # move non-selected item into vacancy
else:
selected = set()
selected_add = selected.add
for i in range(k):
j = randbelow(n)
while j in selected:
j = randbelow(n)
selected_add(j)
result[i] = population[j]
return result
def sample(self, population, k):
# """Chooses k unique random elements from a population sequence.
# Returns a new list containing elements from the population while
# leaving the original population unchanged. The resulting list is
# in selection order so that all sub-slices will also be valid random
# samples. This allows raffle winners (the sample) to be partitioned
# into grand prize and second place winners (the subslices).
#
# Members of the population need not be hashable or unique. If the
# population contains repeats, then each occurrence is a possible
# selection in the sample.
#
# To choose a sample in a range of integers, use xrange as an argument.
# This is especially fast and space efficient for sampling from a
# large population: sample(xrange(10000000), 60)
# """
# XXX Although the documentation says `population` is "a sequence",
# XXX attempts are made to cater to any iterable with a __len__
# XXX method. This has had mixed success. Examples from both
# XXX sides: sets work fine, and should become officially supported;
# XXX dicts are much harder, and have failed in various subtle
# XXX ways across attempts. Support for mapping types should probably
# XXX be dropped (and users should pass mapping.keys() or .values()
# XXX explicitly).
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
n = len(population)
if not 0 <= k <= n:
raise ValueError, "sample larger than population"
__random = self.random
_int = int
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4**_ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize or hasattr(population, "keys"):
# An n-length list is smaller than a k-length set, or this is a
# mapping type so the other algorithm wouldn't work.
pool = list(population)
for i in xrange(k): # invariant: non-selected at [0,n-i)
j = _int(__random() * (n - i))
result[i] = pool[j]
pool[j] = pool[n - i -
1] # move non-selected item into vacancy
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = _int(__random() * n)
while j in selected:
j = _int(__random() * n)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError): # handle (at least) sets
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result
def chart_eia_sd(market, key, start_dt="2010-01-01", output="chart", **kwargs):
"""
Function for plotting and returning data from the EIA Supply & Demand Balances for
refined oil products such as mogas, diesel, jet and resid.
Parameters
----------
market : ['mogas', 'diesel', 'jet', 'resid']
Refined product type to build the S&D balance for
key : str
EIA API key
start_dt : str | datetime
Starting date for S&D balance data
output : ['chart','data']
Output as a chart or return data as a dataframe, by default 'chart'
Returns
-------
pandas dataframe or a plotly figure object
Examples
--------
>>> import risktools as rt
>>> fig = rt.chart_eia_sd('mogas', up['eia'])
>>> fig.show()
"""
eia = data.open_data("tickers_eia")
eia = eia[eia.sd_category == market]
df = get_eia_df(eia.tick_eia.to_list(), key=key)
df = df.merge(eia[["tick_eia", "category"]],
left_on=["series_id"],
right_on=["tick_eia"]).drop("tick_eia", axis=1)
df.date = pd.to_datetime(df.date)
df = df.set_index("date").sort_index()
# create list of plotly figure objects using repeating calls to the
# five_year_plot function to loop through later to create subplot
figs = []
for c in df.category.unique():
tf = df.loc[df.category == c, "value"]
figs.append(chart_five_year_plot(tf, title=c))
# calc shape of final subplot
n = len(df.category.unique())
m = 2
n = _ceil(n / m)
fig = _make_subplots(
n,
m,
subplot_titles=(" ", " ", " ", " ", " ", " "),
)
# Copy returns figures from five_year_plot to a single subplot figure
a = 1
b = 1
for i, _ in enumerate(figs):
for j, _ in enumerate(figs[i]["data"]):
fig.add_trace(figs[i]["data"][j], row=a, col=b)
fig["layout"]["annotations"][
(a - 1) * 2 + b -
1]["text"] = figs[i]["layout"]["title"]["text"]
# copy xaxis nticks and tickformat to subplots so that it keeps that formatting.
# if they don't exist, pass
try:
fig["layout"][f"xaxis{i+1}"]["nticks"] = figs[i]["layout"][
"xaxis"]["nticks"]
fig["layout"][f"xaxis{i+1}"]["tickformat"] = figs[i]["layout"][
"xaxis"]["tickformat"]
except:
pass
if b == m:
b = 1
a += 1
else:
b += 1
fig.update_layout(showlegend=False)
# return figs
if output == "chart":
return fig
else:
return df
def sample(self, population, k):
"""Chooses k unique random elements from a population sequence.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
selection in the sample.
To choose a sample in a range of integers, use xrange as an argument.
This is especially fast and space efficient for sampling from a
large population: sample(xrange(10000000), 60)
"""
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
n = len(population)
if not 0 <= k <= n:
raise ValueError("sample larger than population")
random = self.random
_int = int
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4**_ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize or hasattr(population, "keys"):
# An n-length list is smaller than a k-length set, or this is a
# mapping type so the other algorithm wouldn't work.
pool = list(population)
for i in xrange(k): # invariant: non-selected at [0,n-i)
j = _int(random() * (n - i))
result[i] = pool[j]
pool[j] = pool[n - i -
1] # move non-selected item into vacancy
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = _int(random() * n)
while j in selected:
j = _int(random() * n)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError): # handle (at least) sets
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result
def sample(self, population: Union[_Set[T], _Sequence[T]],
k: int) -> List[T]:
"""Chooses k unique random elements from a population sequence or set.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
selection in the sample.
To choose a sample in a range of integers, use range as an argument.
This is especially fast and space efficient for sampling from a
large population: sample(range(10000000), 60)
"""
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
if isinstance(population, _Set):
population = list(population)
if not isinstance(population, _Sequence):
raise TypeError(
"Population must be a sequence or set. For dicts, use list(d)."
)
randbelow = self._randbelow
n = len(population)
if not (0 <= k and k <= n):
raise ValueError("Sample larger than population")
result = [cast(T, None)] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4**_ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize:
# An n-length list is smaller than a k-length set
pool = list(population)
for i in range(k): # invariant: non-selected at [0,n-i)
j = randbelow(n - i)
result[i] = pool[j]
pool[j] = pool[n - i -
1] # move non-selected item into vacancy
else:
selected = Set[int]()
selected_add = selected.add
for i in range(k):
j = randbelow(n)
while j in selected:
j = randbelow(n)
selected_add(j)
result[i] = population[j]
return result
def sample(self, population, k):
# """Chooses k unique random elements from a population sequence.
# Returns a new list containing elements from the population while
# leaving the original population unchanged. The resulting list is
# in selection order so that all sub-slices will also be valid random
# samples. This allows raffle winners (the sample) to be partitioned
# into grand prize and second place winners (the subslices).
#
# Members of the population need not be hashable or unique. If the
# population contains repeats, then each occurrence is a possible
# selection in the sample.
#
# To choose a sample in a range of integers, use xrange as an argument.
# This is especially fast and space efficient for sampling from a
# large population: sample(xrange(10000000), 60)
# """
# XXX Although the documentation says `population` is "a sequence",
# XXX attempts are made to cater to any iterable with a __len__
# XXX method. This has had mixed success. Examples from both
# XXX sides: sets work fine, and should become officially supported;
# XXX dicts are much harder, and have failed in various subtle
# XXX ways across attempts. Support for mapping types should probably
# XXX be dropped (and users should pass mapping.keys() or .values()
# XXX explicitly).
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
n = len(population)
if not 0 <= k <= n:
raise ValueError, "sample larger than population"
__random = self.random
_int = int
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize or hasattr(population, "keys"):
# An n-length list is smaller than a k-length set, or this is a
# mapping type so the other algorithm wouldn't work.
pool = list(population)
for i in xrange(k): # invariant: non-selected at [0,n-i)
j = _int(__random() * (n-i))
result[i] = pool[j]
pool[j] = pool[n-i-1] # move non-selected item into vacancy
else:
try:
selected = set()
selected_add = selected.add
for i in xrange(k):
j = _int(__random() * n)
while j in selected:
j = _int(__random() * n)
selected_add(j)
result[i] = population[j]
except (TypeError, KeyError): # handle (at least) sets
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result
def dup_zz_hensel_lift(p, f, f_list, l, K):
"""
Multifactor Hensel lifting in `Z[x]`.
Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
over `Z[x]` satisfying::
f = lc(f) f_1 ... f_r (mod p)
and a positive integer `l`, returns a list of monic polynomials
`F_1`, `F_2`, ..., `F_r` satisfying::
f = lc(f) F_1 ... F_r (mod p**l)
F_i = f_i (mod p), i = 1..r
References
==========
1. [Gathen99]_
"""
r = len(f_list)
lc = dup_LC(f, K)
if r == 1:
F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
return [dup_trunc(F, p**l, K)]
m = p
k = r // 2
d = int(_ceil(_log(l, 2)))
g = gf_from_int_poly([lc], p)
for f_i in f_list[:k]:
# print("g: %s * %s" % (g,f_i))
g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)
# print("g: %s" % g)
h = gf_from_int_poly(f_list[k], p)
for f_i in f_list[k + 1:]:
h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)
s, t, q = gf_gcdex(g, h, p, K)
# print("gcdex %s %s %d = %s %s %s)" % (g,h,p,q,s,t))
g = gf_to_int_poly(g, p)
h = gf_to_int_poly(h, p)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
# print("h: %s" % f_list[k])
for _ in range(1, d + 1):
# print("go %d %s %s %s %s %s" % (m, f, g, h, s, t))
(g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2
# print("go %d %s %s %s %s %s" % (m, f, g, h, s, t))
return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
+ dup_zz_hensel_lift(p, h, f_list[k:], l, K)
def renderArea(self, width_, height_, srs, xmin_, ymin_, xmax_, ymax_, zoom):
"""
"""
merc = Proj(srs)
# use the center to figure out our UTM zone
lon, lat = merc((xmin_ + xmax_)/2, (ymin_ + ymax_)/2, inverse=True)
zone = lon2zone(lon)
hemi = lat2hemi(lat)
utm = Proj(proj='utm', zone=zone, datum='WGS84')
# get to UTM coords
(minlon, minlat), (maxlon, maxlat) = merc(xmin_, ymin_, inverse=1), merc(xmax_, ymax_, inverse=1)
(xmin, ymin), (xmax, ymax) = utm(minlon, minlat), utm(maxlon, maxlat)
# figure out how widely-spaced they should be
pixels = _hypot(width_, height_) # number of pixels across the image
units = _hypot(xmax - xmin, ymax - ymin) # number of UTM units across the image
tick = self.tick * units/pixels # desired tick length in UTM units
count = pixels / self.spacing # approximate number of lines across the image
bound = units / count # too-precise step between lines in UTM units
zeros = int(_ceil(_log(bound) / _log(10))) # this value gets used again to format numbers
step = int(_pow(10, zeros)) # a step that falls right on the 10^n
# and the outer UTM bounds
xbot, xtop = int(xmin - xmin % step), int(xmax - xmax % step) + 2 * step
ybot, ytop = int(ymin - ymin % step), int(ymax - xmax % step) + 2 * step
# start doing things in pixels
img = Image.new('RGBA', (width_, height_), (0xEE, 0xEE, 0xEE, 0x00))
draw = ImageDraw.ImageDraw(img)
xform = transform(width_, height_, xmin_, ymax_, xmax_, ymin_)
lines = []
labels = []
for col in range(xbot, xtop, step):
# set up the verticals
utms = [(col, y) for y in range(ybot, ytop, step/10)]
mercs = [merc(*utm(x, y, inverse=1)) for (x, y) in utms]
lines.append( [xform(x, y) for (x, y) in mercs] )
# and the tick marks
for row in range(ybot, ytop, step/10):
mercs = [merc(*utm(x, y, inverse=1)) for (x, y) in ((col, row), (col - tick, row))]
lines.append( [xform(x, y) for (x, y) in mercs] )
for row in range(ybot, ytop, step):
# set up the horizontals
utms = [(x, row) for x in range(xbot, xtop, step/10)]
mercs = [merc(*utm(x, y, inverse=1)) for (x, y) in utms]
lines.append( [xform(x, y) for (x, y) in mercs] )
# and the tick marks
for col in range(xbot, xtop, step/10):
mercs = [merc(*utm(x, y, inverse=1)) for (x, y) in ((col, row), (col, row - tick))]
lines.append( [xform(x, y) for (x, y) in mercs] )
# set up the intersection labels
for x in range(xbot, xtop, step):
for y in range(ybot, ytop, step):
lon, lat = utm(x, y, inverse=1)
grid = lonlat2grid(lon, lat)
point = xform(*merc(lon, lat))
if self.display == 'utm':
e = ('%07d' % x)[:-zeros]
n = ('%07d' % y)[:-zeros]
text = ' '.join( [grid, e, n] )
elif self.display == 'mgrs':
e, n = Proj(proj='utm', zone=lon2zone(lon), datum='WGS84')(lon, lat)
text = utm2mgrs(round(e), round(n), grid, zeros)
labels.append( (point, text) )
# do the drawing bits
for ((x, y), text) in labels:
x, y = x + 2, y - 18
w, h = self.font.getsize(text)
draw.rectangle((x - 2, y, x + w + 2, y + h), fill=(0xFF, 0xFF, 0xFF, 0x99))
for line in lines:
draw.line(line, fill=(0xFF, 0xFF, 0xFF))
for line in lines:
draw.line([(x-1, y-1) for (x, y) in line], fill=(0x00, 0x00, 0x00))
for ((x, y), text) in labels:
x, y = x + 2, y - 18
draw.text((x, y), text, fill=(0x00, 0x00, 0x00), font=self.font)
return img
def _get_trading_periods(trading_year_days=252):
half_year = _ceil(trading_year_days / 2)
return trading_year_days, half_year
def ceil(i):
return int(_ceil(i))
def ceil(x):
return _ceil(x)
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1)) * 2**n * A * b))
C = int((n + 1)**(2 * n) * A**(2 * n - 1))
gamma = int(_ceil(2 * _log(C, 2)))
bound = int(2 * gamma * _log(gamma))
for p in xrange(3, bound + 1):
if not isprime(p) or b % p == 0:
continue
p = K.convert(p)
F = gf_from_int_poly(f, p)
if gf_sqf_p(F, p, K):
break
l = int(_ceil(_log(2 * B + 1, p)))
modular = []
for ff in gf_factor_sqf(F, p, K)[1]:
modular.append(gf_to_int_poly(ff, p))
g = dup_zz_hensel_lift(p, f, modular, l, K)
T = set(range(len(g)))
factors, s = [], 1
while 2 * s <= len(T):
for S in subsets(T, s):
G, H = [b], [b]
S = set(S)
for i in S:
G = dup_mul(G, g[i], K)
for i in T - S:
H = dup_mul(H, g[i], K)
G = dup_trunc(G, p**l, K)
H = dup_trunc(H, p**l, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm * H_norm <= B:
T = T - S
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def ceil(x):
return _ceil(x)
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
fc = f[-1]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1)) * 2**n * A * b))
C = int((n + 1)**(2 * n) * A**(2 * n - 1))
gamma = int(_ceil(2 * _log(C, 2)))
bound = int(2 * gamma * _log(gamma))
a = []
# choose a prime number `p` such that `f` be square free in Z_p
# if there are many factors in Z_p, choose among a few different `p`
# the one with fewer factors
for px in range(3, bound + 1):
if not isprime(px) or b % px == 0:
continue
px = K.convert(px)
F = gf_from_int_poly(f, px)
if not gf_sqf_p(F, px, K):
continue
fsqfx = gf_factor_sqf(F, px, K)[1]
a.append((px, fsqfx))
if len(fsqfx) < 15 or len(a) > 4:
break
p, fsqf = min(a, key=lambda x: len(x[1]))
l = int(_ceil(_log(2 * B + 1, p)))
modular = [gf_to_int_poly(ff, p) for ff in fsqf]
g = dup_zz_hensel_lift(p, f, modular, l, K)
sorted_T = range(len(g))
T = set(sorted_T)
factors, s = [], 1
pl = p**l
while 2 * s <= len(T):
for S in subsets(sorted_T, s):
# lift the constant coefficient of the product `G` of the factors
# in the subset `S`; if it is does not divide `fc`, `G` does
# not divide the input polynomial
if b == 1:
q = 1
for i in S:
q = q * g[i][-1]
q = q % pl
if not _test_pl(fc, q, pl):
continue
else:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
G = dup_primitive(G, K)[1]
q = G[-1]
if q and fc % q != 0:
continue
H = [b]
S = set(S)
T_S = T - S
if b == 1:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
for i in T_S:
H = dup_mul(H, g[i], K)
H = dup_trunc(H, pl, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm * H_norm <= B:
T = T_S
sorted_T = [i for i in sorted_T if i not in S]
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def gf_ddf_shoup(f, p, K):
"""
Kaltofen-Shoup: Deterministic Distinct Degree Factorization
Given a monic square-free polynomial ``f`` in ``GF(p)[x]``, computes
partial distinct degree factorization ``f_1,...,f_d`` of ``f`` where
``deg(f_i) != deg(f_j)`` for ``i != j``. The result is returned as a
list of pairs ``(f_i, e_i)`` where ``deg(f_i) > 0`` and ``e_i > 0``
is an argument to the equal degree factorization routine.
This algorithm is an improved version of Zassenhaus algorithm for
large ``deg(f)`` and modulus ``p`` (especially for ``deg(f) ~ lg(p)``).
Examples
========
>>> from diofant.polys.domains import ZZ
>>> f = gf_from_dict({6: ZZ(1), 5: ZZ(-1), 4: ZZ(1), 3: ZZ(1), 1: ZZ(-1)}, 3, ZZ)
>>> gf_ddf_shoup(f, 3, ZZ)
[([1, 1, 0], 1), ([1, 1, 0, 1, 2], 2)]
References
==========
.. [1] [Kaltofen98]_
.. [2] [Shoup95]_
.. [3] [Gathen92]_
"""
n = gf_degree(f)
k = int(_ceil(_sqrt(n // 2)))
b = gf_frobenius_monomial_base(f, p, K)
h = gf_frobenius_map([K.one, K.zero], f, b, p, K)
# U[i] = x**(p**i)
U = [[K.one, K.zero], h] + [K.zero] * (k - 1)
for i in range(2, k + 1):
U[i] = gf_frobenius_map(U[i - 1], f, b, p, K)
h, U = U[k], U[:k]
# V[i] = x**(p**(k*(i+1)))
V = [h] + [K.zero] * (k - 1)
for i in range(1, k):
V[i] = gf_compose_mod(V[i - 1], h, f, p, K)
factors = []
for i, v in enumerate(V):
h, j = [K.one], k - 1
for u in U:
g = gf_sub(v, u, p, K)
h = gf_mul(h, g, p, K)
h = gf_rem(h, f, p, K)
g = gf_gcd(f, h, p, K)
f = gf_quo(f, g, p, K)
for u in reversed(U):
h = gf_sub(v, u, p, K)
F = gf_gcd(g, h, p, K)
if F != [K.one]:
factors.append((F, k * (i + 1) - j))
g, j = gf_quo(g, F, p, K), j - 1
if f != [K.one]:
factors.append((f, gf_degree(f)))
return factors
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1))*2**n*A*b))
C = int((n + 1)**(2*n)*A**(2*n - 1))
gamma = int(_ceil(2*_log(C, 2)))
bound = int(2*gamma*_log(gamma))
for p in xrange(3, bound + 1):
if not isprime(p) or b % p == 0:
continue
p = K.convert(p)
F = gf_from_int_poly(f, p)
if gf_sqf_p(F, p, K):
break
l = int(_ceil(_log(2*B + 1, p)))
modular = []
for ff in gf_factor_sqf(F, p, K)[1]:
modular.append(gf_to_int_poly(ff, p))
g = dup_zz_hensel_lift(p, f, modular, l, K)
sorted_T = range(len(g))
T = set(sorted_T)
factors, s = [], 1
while 2*s <= len(T):
for S in subsets(sorted_T, s):
G, H = [b], [b]
S = set(S)
T_S = T - S
for i in S:
G = dup_mul(G, g[i], K)
for i in T_S:
H = dup_mul(H, g[i], K)
G = dup_trunc(G, p**l, K)
H = dup_trunc(H, p**l, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm*H_norm <= B:
T = T_S
sorted_T = [i for i in sorted_T if i not in S]
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def sample(self, population, k, *, counts=None):
"""Chooses k unique random elements from a population sequence.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
selection in the sample.
Repeated elements can be specified one at a time or with the optional
counts parameter. For example:
sample(['red', 'blue'], counts=[4, 2], k=5)
is equivalent to:
sample(['red', 'red', 'red', 'red', 'blue', 'blue'], k=5)
To choose a sample from a range of integers, use range() for the
population argument. This is especially fast and space efficient
for sampling from a large population:
sample(range(10000000), 60)
"""
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
# The number of calls to _randbelow() is kept at or near k, the
# theoretical minimum. This is important because running time
# is dominated by _randbelow() and because it extracts the
# least entropy from the underlying random number generators.
# Memory requirements are kept to the smaller of a k-length
# set or an n-length list.
# There are other sampling algorithms that do not require
# auxiliary memory, but they were rejected because they made
# too many calls to _randbelow(), making them slower and
# causing them to eat more entropy than necessary.
if not isinstance(population, _Sequence):
raise TypeError("Population must be a sequence. "
"For dicts or sets, use sorted(d).")
n = len(population)
if counts is not None:
cum_counts = list(_accumulate(counts))
if len(cum_counts) != n:
raise ValueError(
'The number of counts does not match the population')
total = cum_counts.pop()
if not isinstance(total, int):
raise TypeError('Counts must be integers')
if total <= 0:
raise ValueError('Total of counts must be greater than zero')
selections = self.sample(range(total), k=k)
bisect = _bisect
return [population[bisect(cum_counts, s)] for s in selections]
randbelow = self._randbelow
if not 0 <= k <= n:
raise ValueError("Sample larger than population or is negative")
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4**_ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize:
# An n-length list is smaller than a k-length set.
# Invariant: non-selected at pool[0 : n-i]
pool = list(population)
for i in range(k):
j = randbelow(n - i)
result[i] = pool[j]
pool[j] = pool[n - i -
1] # move non-selected item into vacancy
else:
selected = set()
selected_add = selected.add
for i in range(k):
j = randbelow(n)
while j in selected:
j = randbelow(n)
selected_add(j)
result[i] = population[j]
return result
