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"""Imported from the recipes section of the itertools documentation.
All functions taken from the recipes section of the itertools library docs
[1]_.
Some backward-compatible usability improvements have been made.
.. [1] http://docs.python.org/library/itertools.html#recipes
"""
import math
import operator
from collections import deque
from collections.abc import Sized
from functools import partial, reduce
from itertools import (
chain,
combinations,
compress,
count,
cycle,
groupby,
islice,
product,
repeat,
starmap,
tee,
zip_longest,
)
from random import randrange, sample, choice
from sys import hexversion
__all__ = [
'all_equal',
'batched',
'before_and_after',
'consume',
'convolve',
'dotproduct',
'first_true',
'factor',
'flatten',
'grouper',
'iter_except',
'iter_index',
'matmul',
'ncycles',
'nth',
'nth_combination',
'padnone',
'pad_none',
'pairwise',
'partition',
'polynomial_eval',
'polynomial_from_roots',
'polynomial_derivative',
'powerset',
'prepend',
'quantify',
'reshape',
'random_combination_with_replacement',
'random_combination',
'random_permutation',
'random_product',
'repeatfunc',
'roundrobin',
'sieve',
'sliding_window',
'subslices',
'sum_of_squares',
'tabulate',
'tail',
'take',
'totient',
'transpose',
'triplewise',
'unique',
'unique_everseen',
'unique_justseen',
]
_marker = object()
# zip with strict is available for Python 3.10+
try:
zip(strict=True)
except TypeError:
_zip_strict = zip
else:
_zip_strict = partial(zip, strict=True)
# math.sumprod is available for Python 3.12+
_sumprod = getattr(math, 'sumprod', lambda x, y: dotproduct(x, y))
def take(n, iterable):
"""Return first *n* items of the iterable as a list.
>>> take(3, range(10))
[0, 1, 2]
If there are fewer than *n* items in the iterable, all of them are
returned.
>>> take(10, range(3))
[0, 1, 2]
"""
return list(islice(iterable, n))
def tabulate(function, start=0):
"""Return an iterator over the results of ``func(start)``,
``func(start + 1)``, ``func(start + 2)``...
*func* should be a function that accepts one integer argument.
If *start* is not specified it defaults to 0. It will be incremented each
time the iterator is advanced.
>>> square = lambda x: x ** 2
>>> iterator = tabulate(square, -3)
>>> take(4, iterator)
[9, 4, 1, 0]
"""
return map(function, count(start))
def tail(n, iterable):
"""Return an iterator over the last *n* items of *iterable*.
>>> t = tail(3, 'ABCDEFG')
>>> list(t)
['E', 'F', 'G']
"""
# If the given iterable has a length, then we can use islice to get its
# final elements. Note that if the iterable is not actually Iterable,
# either islice or deque will throw a TypeError. This is why we don't
# check if it is Iterable.
if isinstance(iterable, Sized):
yield from islice(iterable, max(0, len(iterable) - n), None)
else:
yield from iter(deque(iterable, maxlen=n))
def consume(iterator, n=None):
"""Advance *iterable* by *n* steps. If *n* is ``None``, consume it
entirely.
Efficiently exhausts an iterator without returning values. Defaults to
consuming the whole iterator, but an optional second argument may be
provided to limit consumption.
>>> i = (x for x in range(10))
>>> next(i)
0
>>> consume(i, 3)
>>> next(i)
4
>>> consume(i)
>>> next(i)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
StopIteration
If the iterator has fewer items remaining than the provided limit, the
whole iterator will be consumed.
>>> i = (x for x in range(3))
>>> consume(i, 5)
>>> next(i)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
StopIteration
"""
# Use functions that consume iterators at C speed.
if n is None:
# feed the entire iterator into a zero-length deque
deque(iterator, maxlen=0)
else:
# advance to the empty slice starting at position n
next(islice(iterator, n, n), None)
def nth(iterable, n, default=None):
"""Returns the nth item or a default value.
>>> l = range(10)
>>> nth(l, 3)
3
>>> nth(l, 20, "zebra")
'zebra'
"""
return next(islice(iterable, n, None), default)
def all_equal(iterable, key=None):
"""
Returns ``True`` if all the elements are equal to each other.
>>> all_equal('aaaa')
True
>>> all_equal('aaab')
False
A function that accepts a single argument and returns a transformed version
of each input item can be specified with *key*:
>>> all_equal('AaaA', key=str.casefold)
True
>>> all_equal([1, 2, 3], key=lambda x: x < 10)
True
"""
return len(list(islice(groupby(iterable, key), 2))) <= 1
def quantify(iterable, pred=bool):
"""Return the how many times the predicate is true.
>>> quantify([True, False, True])
2
"""
return sum(map(pred, iterable))
def pad_none(iterable):
"""Returns the sequence of elements and then returns ``None`` indefinitely.
>>> take(5, pad_none(range(3)))
[0, 1, 2, None, None]
Useful for emulating the behavior of the built-in :func:`map` function.
See also :func:`padded`.
"""
return chain(iterable, repeat(None))
padnone = pad_none
def ncycles(iterable, n):
"""Returns the sequence elements *n* times
>>> list(ncycles(["a", "b"], 3))
['a', 'b', 'a', 'b', 'a', 'b']
"""
return chain.from_iterable(repeat(tuple(iterable), n))
def dotproduct(vec1, vec2):
"""Returns the dot product of the two iterables.
>>> dotproduct([10, 10], [20, 20])
400
"""
return sum(map(operator.mul, vec1, vec2))
def flatten(listOfLists):
"""Return an iterator flattening one level of nesting in a list of lists.
>>> list(flatten([[0, 1], [2, 3]]))
[0, 1, 2, 3]
See also :func:`collapse`, which can flatten multiple levels of nesting.
"""
return chain.from_iterable(listOfLists)
def repeatfunc(func, times=None, *args):
"""Call *func* with *args* repeatedly, returning an iterable over the
results.
If *times* is specified, the iterable will terminate after that many
repetitions:
>>> from operator import add
>>> times = 4
>>> args = 3, 5
>>> list(repeatfunc(add, times, *args))
[8, 8, 8, 8]
If *times* is ``None`` the iterable will not terminate:
>>> from random import randrange
>>> times = None
>>> args = 1, 11
>>> take(6, repeatfunc(randrange, times, *args)) # doctest:+SKIP
[2, 4, 8, 1, 8, 4]
"""
if times is None:
return starmap(func, repeat(args))
return starmap(func, repeat(args, times))
def _pairwise(iterable):
"""Returns an iterator of paired items, overlapping, from the original
>>> take(4, pairwise(count()))
[(0, 1), (1, 2), (2, 3), (3, 4)]
On Python 3.10 and above, this is an alias for :func:`itertools.pairwise`.
"""
a, b = tee(iterable)
next(b, None)
return zip(a, b)
try:
from itertools import pairwise as itertools_pairwise
except ImportError:
pairwise = _pairwise
else:
def pairwise(iterable):
return itertools_pairwise(iterable)
pairwise.__doc__ = _pairwise.__doc__
class UnequalIterablesError(ValueError):
def __init__(self, details=None):
msg = 'Iterables have different lengths'
if details is not None:
msg += (': index 0 has length {}; index {} has length {}').format(
*details
)
super().__init__(msg)
def _zip_equal_generator(iterables):
for combo in zip_longest(*iterables, fillvalue=_marker):
for val in combo:
if val is _marker:
raise UnequalIterablesError()
yield combo
def _zip_equal(*iterables):
# Check whether the iterables are all the same size.
try:
first_size = len(iterables[0])
for i, it in enumerate(iterables[1:], 1):
size = len(it)
if size != first_size:
raise UnequalIterablesError(details=(first_size, i, size))
# All sizes are equal, we can use the built-in zip.
return zip(*iterables)
# If any one of the iterables didn't have a length, start reading
# them until one runs out.
except TypeError:
return _zip_equal_generator(iterables)
def grouper(iterable, n, incomplete='fill', fillvalue=None):
"""Group elements from *iterable* into fixed-length groups of length *n*.
>>> list(grouper('ABCDEF', 3))
[('A', 'B', 'C'), ('D', 'E', 'F')]
The keyword arguments *incomplete* and *fillvalue* control what happens for
iterables whose length is not a multiple of *n*.
When *incomplete* is `'fill'`, the last group will contain instances of
*fillvalue*.
>>> list(grouper('ABCDEFG', 3, incomplete='fill', fillvalue='x'))
[('A', 'B', 'C'), ('D', 'E', 'F'), ('G', 'x', 'x')]
When *incomplete* is `'ignore'`, the last group will not be emitted.
>>> list(grouper('ABCDEFG', 3, incomplete='ignore', fillvalue='x'))
[('A', 'B', 'C'), ('D', 'E', 'F')]
When *incomplete* is `'strict'`, a subclass of `ValueError` will be raised.
>>> it = grouper('ABCDEFG', 3, incomplete='strict')
>>> list(it) # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
UnequalIterablesError
"""
args = [iter(iterable)] * n
if incomplete == 'fill':
return zip_longest(*args, fillvalue=fillvalue)
if incomplete == 'strict':
return _zip_equal(*args)
if incomplete == 'ignore':
return zip(*args)
else:
raise ValueError('Expected fill, strict, or ignore')
def roundrobin(*iterables):
"""Yields an item from each iterable, alternating between them.
>>> list(roundrobin('ABC', 'D', 'EF'))
['A', 'D', 'E', 'B', 'F', 'C']
This function produces the same output as :func:`interleave_longest`, but
may perform better for some inputs (in particular when the number of
iterables is small).
"""
# Algorithm credited to George Sakkis
iterators = map(iter, iterables)
for num_active in range(len(iterables), 0, -1):
iterators = cycle(islice(iterators, num_active))
yield from map(next, iterators)
def partition(pred, iterable):
"""
Returns a 2-tuple of iterables derived from the input iterable.
The first yields the items that have ``pred(item) == False``.
The second yields the items that have ``pred(item) == True``.
>>> is_odd = lambda x: x % 2 != 0
>>> iterable = range(10)
>>> even_items, odd_items = partition(is_odd, iterable)
>>> list(even_items), list(odd_items)
([0, 2, 4, 6, 8], [1, 3, 5, 7, 9])
If *pred* is None, :func:`bool` is used.
>>> iterable = [0, 1, False, True, '', ' ']
>>> false_items, true_items = partition(None, iterable)
>>> list(false_items), list(true_items)
([0, False, ''], [1, True, ' '])
"""
if pred is None:
pred = bool
t1, t2, p = tee(iterable, 3)
p1, p2 = tee(map(pred, p))
return (compress(t1, map(operator.not_, p1)), compress(t2, p2))
def powerset(iterable):
"""Yields all possible subsets of the iterable.
>>> list(powerset([1, 2, 3]))
[(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]
:func:`powerset` will operate on iterables that aren't :class:`set`
instances, so repeated elements in the input will produce repeated elements
in the output.
>>> seq = [1, 1, 0]
>>> list(powerset(seq))
[(), (1,), (1,), (0,), (1, 1), (1, 0), (1, 0), (1, 1, 0)]
For a variant that efficiently yields actual :class:`set` instances, see
:func:`powerset_of_sets`.
"""
s = list(iterable)
return chain.from_iterable(combinations(s, r) for r in range(len(s) + 1))
def unique_everseen(iterable, key=None):
"""
Yield unique elements, preserving order.
>>> list(unique_everseen('AAAABBBCCDAABBB'))
['A', 'B', 'C', 'D']
>>> list(unique_everseen('ABBCcAD', str.lower))
['A', 'B', 'C', 'D']
Sequences with a mix of hashable and unhashable items can be used.
The function will be slower (i.e., `O(n^2)`) for unhashable items.
Remember that ``list`` objects are unhashable - you can use the *key*
parameter to transform the list to a tuple (which is hashable) to
avoid a slowdown.
>>> iterable = ([1, 2], [2, 3], [1, 2])
>>> list(unique_everseen(iterable)) # Slow
[[1, 2], [2, 3]]
>>> list(unique_everseen(iterable, key=tuple)) # Faster
[[1, 2], [2, 3]]
Similarly, you may want to convert unhashable ``set`` objects with
``key=frozenset``. For ``dict`` objects,
``key=lambda x: frozenset(x.items())`` can be used.
"""
seenset = set()
seenset_add = seenset.add
seenlist = []
seenlist_add = seenlist.append
use_key = key is not None
for element in iterable:
k = key(element) if use_key else element
try:
if k not in seenset:
seenset_add(k)
yield element
except TypeError:
if k not in seenlist:
seenlist_add(k)
yield element
def unique_justseen(iterable, key=None):
"""Yields elements in order, ignoring serial duplicates
>>> list(unique_justseen('AAAABBBCCDAABBB'))
['A', 'B', 'C', 'D', 'A', 'B']
>>> list(unique_justseen('ABBCcAD', str.lower))
['A', 'B', 'C', 'A', 'D']
"""
if key is None:
return map(operator.itemgetter(0), groupby(iterable))
return map(next, map(operator.itemgetter(1), groupby(iterable, key)))
def unique(iterable, key=None, reverse=False):
"""Yields unique elements in sorted order.
>>> list(unique([[1, 2], [3, 4], [1, 2]]))
[[1, 2], [3, 4]]
*key* and *reverse* are passed to :func:`sorted`.
>>> list(unique('ABBcCAD', str.casefold))
['A', 'B', 'c', 'D']
>>> list(unique('ABBcCAD', str.casefold, reverse=True))
['D', 'c', 'B', 'A']
The elements in *iterable* need not be hashable, but they must be
comparable for sorting to work.
"""
return unique_justseen(sorted(iterable, key=key, reverse=reverse), key=key)
def iter_except(func, exception, first=None):
"""Yields results from a function repeatedly until an exception is raised.
Converts a call-until-exception interface to an iterator interface.
Like ``iter(func, sentinel)``, but uses an exception instead of a sentinel
to end the loop.
>>> l = [0, 1, 2]
>>> list(iter_except(l.pop, IndexError))
[2, 1, 0]
Multiple exceptions can be specified as a stopping condition:
>>> l = [1, 2, 3, '...', 4, 5, 6]
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
[7, 6, 5]
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
[4, 3, 2]
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
[]
"""
try:
if first is not None:
yield first()
while 1:
yield func()
except exception:
pass
def first_true(iterable, default=None, pred=None):
"""
Returns the first true value in the iterable.
If no true value is found, returns *default*
If *pred* is not None, returns the first item for which
``pred(item) == True`` .
>>> first_true(range(10))
1
>>> first_true(range(10), pred=lambda x: x > 5)
6
>>> first_true(range(10), default='missing', pred=lambda x: x > 9)
'missing'
"""
return next(filter(pred, iterable), default)
def random_product(*args, repeat=1):
"""Draw an item at random from each of the input iterables.
>>> random_product('abc', range(4), 'XYZ') # doctest:+SKIP
('c', 3, 'Z')
If *repeat* is provided as a keyword argument, that many items will be
drawn from each iterable.
>>> random_product('abcd', range(4), repeat=2) # doctest:+SKIP
('a', 2, 'd', 3)
This equivalent to taking a random selection from
``itertools.product(*args, **kwarg)``.
"""
pools = [tuple(pool) for pool in args] * repeat
return tuple(choice(pool) for pool in pools)
def random_permutation(iterable, r=None):
"""Return a random *r* length permutation of the elements in *iterable*.
If *r* is not specified or is ``None``, then *r* defaults to the length of
*iterable*.
>>> random_permutation(range(5)) # doctest:+SKIP
(3, 4, 0, 1, 2)
This equivalent to taking a random selection from
``itertools.permutations(iterable, r)``.
"""
pool = tuple(iterable)
r = len(pool) if r is None else r
return tuple(sample(pool, r))
def random_combination(iterable, r):
"""Return a random *r* length subsequence of the elements in *iterable*.
>>> random_combination(range(5), 3) # doctest:+SKIP
(2, 3, 4)
This equivalent to taking a random selection from
``itertools.combinations(iterable, r)``.
"""
pool = tuple(iterable)
n = len(pool)
indices = sorted(sample(range(n), r))
return tuple(pool[i] for i in indices)
def random_combination_with_replacement(iterable, r):
"""Return a random *r* length subsequence of elements in *iterable*,
allowing individual elements to be repeated.
>>> random_combination_with_replacement(range(3), 5) # doctest:+SKIP
(0, 0, 1, 2, 2)
This equivalent to taking a random selection from
``itertools.combinations_with_replacement(iterable, r)``.
"""
pool = tuple(iterable)
n = len(pool)
indices = sorted(randrange(n) for i in range(r))
return tuple(pool[i] for i in indices)
def nth_combination(iterable, r, index):
"""Equivalent to ``list(combinations(iterable, r))[index]``.
The subsequences of *iterable* that are of length *r* can be ordered
lexicographically. :func:`nth_combination` computes the subsequence at
sort position *index* directly, without computing the previous
subsequences.
>>> nth_combination(range(5), 3, 5)
(0, 3, 4)
``ValueError`` will be raised If *r* is negative or greater than the length
of *iterable*.
``IndexError`` will be raised if the given *index* is invalid.
"""
pool = tuple(iterable)
n = len(pool)
if (r < 0) or (r > n):
raise ValueError
c = 1
k = min(r, n - r)
for i in range(1, k + 1):
c = c * (n - k + i) // i
if index < 0:
index += c
if (index < 0) or (index >= c):
raise IndexError
result = []
while r:
c, n, r = c * r // n, n - 1, r - 1
while index >= c:
index -= c
c, n = c * (n - r) // n, n - 1
result.append(pool[-1 - n])
return tuple(result)
def prepend(value, iterator):
"""Yield *value*, followed by the elements in *iterator*.
>>> value = '0'
>>> iterator = ['1', '2', '3']
>>> list(prepend(value, iterator))
['0', '1', '2', '3']
To prepend multiple values, see :func:`itertools.chain`
or :func:`value_chain`.
"""
return chain([value], iterator)
def convolve(signal, kernel):
"""Convolve the iterable *signal* with the iterable *kernel*.
>>> signal = (1, 2, 3, 4, 5)
>>> kernel = [3, 2, 1]
>>> list(convolve(signal, kernel))
[3, 8, 14, 20, 26, 14, 5]
Note: the input arguments are not interchangeable, as the *kernel*
is immediately consumed and stored.
"""
# This implementation intentionally doesn't match the one in the itertools
# documentation.
kernel = tuple(kernel)[::-1]
n = len(kernel)
window = deque([0], maxlen=n) * n
for x in chain(signal, repeat(0, n - 1)):
window.append(x)
yield _sumprod(kernel, window)
def before_and_after(predicate, it):
"""A variant of :func:`takewhile` that allows complete access to the
remainder of the iterator.
>>> it = iter('ABCdEfGhI')
>>> all_upper, remainder = before_and_after(str.isupper, it)
>>> ''.join(all_upper)
'ABC'
>>> ''.join(remainder) # takewhile() would lose the 'd'
'dEfGhI'
Note that the first iterator must be fully consumed before the second
iterator can generate valid results.
"""
it = iter(it)
transition = []
def true_iterator():
for elem in it:
if predicate(elem):
yield elem
else:
transition.append(elem)
return
# Note: this is different from itertools recipes to allow nesting
# before_and_after remainders into before_and_after again. See tests
# for an example.
remainder_iterator = chain(transition, it)
return true_iterator(), remainder_iterator
def triplewise(iterable):
"""Return overlapping triplets from *iterable*.
>>> list(triplewise('ABCDE'))
[('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E')]
"""
for (a, _), (b, c) in pairwise(pairwise(iterable)):
yield a, b, c
def sliding_window(iterable, n):
"""Return a sliding window of width *n* over *iterable*.
>>> list(sliding_window(range(6), 4))
[(0, 1, 2, 3), (1, 2, 3, 4), (2, 3, 4, 5)]
If *iterable* has fewer than *n* items, then nothing is yielded:
>>> list(sliding_window(range(3), 4))
[]
For a variant with more features, see :func:`windowed`.
"""
it = iter(iterable)
window = deque(islice(it, n - 1), maxlen=n)
for x in it:
window.append(x)
yield tuple(window)
def subslices(iterable):
"""Return all contiguous non-empty subslices of *iterable*.
>>> list(subslices('ABC'))
[['A'], ['A', 'B'], ['A', 'B', 'C'], ['B'], ['B', 'C'], ['C']]
This is similar to :func:`substrings`, but emits items in a different
order.
"""
seq = list(iterable)
slices = starmap(slice, combinations(range(len(seq) + 1), 2))
return map(operator.getitem, repeat(seq), slices)
def polynomial_from_roots(roots):
"""Compute a polynomial's coefficients from its roots.
>>> roots = [5, -4, 3] # (x - 5) * (x + 4) * (x - 3)
>>> polynomial_from_roots(roots) # x^3 - 4 * x^2 - 17 * x + 60
[1, -4, -17, 60]
"""
factors = zip(repeat(1), map(operator.neg, roots))
return list(reduce(convolve, factors, [1]))
def iter_index(iterable, value, start=0, stop=None):
"""Yield the index of each place in *iterable* that *value* occurs,
beginning with index *start* and ending before index *stop*.
>>> list(iter_index('AABCADEAF', 'A'))
[0, 1, 4, 7]
>>> list(iter_index('AABCADEAF', 'A', 1)) # start index is inclusive
[1, 4, 7]
>>> list(iter_index('AABCADEAF', 'A', 1, 7)) # stop index is not inclusive
[1, 4]
The behavior for non-scalar *values* matches the built-in Python types.
>>> list(iter_index('ABCDABCD', 'AB'))
[0, 4]
>>> list(iter_index([0, 1, 2, 3, 0, 1, 2, 3], [0, 1]))
[]
>>> list(iter_index([[0, 1], [2, 3], [0, 1], [2, 3]], [0, 1]))
[0, 2]
See :func:`locate` for a more general means of finding the indexes
associated with particular values.
"""
seq_index = getattr(iterable, 'index', None)
if seq_index is None:
# Slow path for general iterables
it = islice(iterable, start, stop)
for i, element in enumerate(it, start):
if element is value or element == value:
yield i
else:
# Fast path for sequences
stop = len(iterable) if stop is None else stop
i = start - 1
try:
while True:
yield (i := seq_index(value, i + 1, stop))
except ValueError:
pass
def sieve(n):
"""Yield the primes less than n.
>>> list(sieve(30))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
"""
if n > 2:
yield 2
start = 3
data = bytearray((0, 1)) * (n // 2)
limit = math.isqrt(n) + 1
for p in iter_index(data, 1, start, limit):
yield from iter_index(data, 1, start, p * p)
data[p * p : n : p + p] = bytes(len(range(p * p, n, p + p)))
start = p * p
yield from iter_index(data, 1, start)
def _batched(iterable, n, *, strict=False):
"""Batch data into tuples of length *n*. If the number of items in
*iterable* is not divisible by *n*:
* The last batch will be shorter if *strict* is ``False``.
* :exc:`ValueError` will be raised if *strict* is ``True``.
>>> list(batched('ABCDEFG', 3))
[('A', 'B', 'C'), ('D', 'E', 'F'), ('G',)]
On Python 3.13 and above, this is an alias for :func:`itertools.batched`.
"""
if n < 1:
raise ValueError('n must be at least one')
it = iter(iterable)
while batch := tuple(islice(it, n)):
if strict and len(batch) != n:
raise ValueError('batched(): incomplete batch')
yield batch
if hexversion >= 0x30D00A2:
from itertools import batched as itertools_batched
def batched(iterable, n, *, strict=False):
return itertools_batched(iterable, n, strict=strict)
else:
batched = _batched
batched.__doc__ = _batched.__doc__
def transpose(it):
"""Swap the rows and columns of the input matrix.
>>> list(transpose([(1, 2, 3), (11, 22, 33)]))
[(1, 11), (2, 22), (3, 33)]
The caller should ensure that the dimensions of the input are compatible.
If the input is empty, no output will be produced.
"""
return _zip_strict(*it)
def reshape(matrix, cols):
"""Reshape the 2-D input *matrix* to have a column count given by *cols*.
>>> matrix = [(0, 1), (2, 3), (4, 5)]
>>> cols = 3
>>> list(reshape(matrix, cols))
[(0, 1, 2), (3, 4, 5)]
"""
return batched(chain.from_iterable(matrix), cols)
def matmul(m1, m2):
"""Multiply two matrices.
>>> list(matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]))
[(49, 80), (41, 60)]
The caller should ensure that the dimensions of the input matrices are
compatible with each other.
"""
n = len(m2[0])
return batched(starmap(_sumprod, product(m1, transpose(m2))), n)
def factor(n):
"""Yield the prime factors of n.
>>> list(factor(360))
[2, 2, 2, 3, 3, 5]
"""
for prime in sieve(math.isqrt(n) + 1):
while not n % prime:
yield prime
n //= prime
if n == 1:
return
if n > 1:
yield n
def polynomial_eval(coefficients, x):
"""Evaluate a polynomial at a specific value.
Example: evaluating x^3 - 4 * x^2 - 17 * x + 60 at x = 2.5:
>>> coefficients = [1, -4, -17, 60]
>>> x = 2.5
>>> polynomial_eval(coefficients, x)
8.125
"""
n = len(coefficients)
if n == 0:
return x * 0 # coerce zero to the type of x
powers = map(pow, repeat(x), reversed(range(n)))
return _sumprod(coefficients, powers)
def sum_of_squares(it):
"""Return the sum of the squares of the input values.
>>> sum_of_squares([10, 20, 30])
1400
"""
return _sumprod(*tee(it))
def polynomial_derivative(coefficients):
"""Compute the first derivative of a polynomial.
Example: evaluating the derivative of x^3 - 4 * x^2 - 17 * x + 60
>>> coefficients = [1, -4, -17, 60]
>>> derivative_coefficients = polynomial_derivative(coefficients)
>>> derivative_coefficients
[3, -8, -17]
"""
n = len(coefficients)
powers = reversed(range(1, n))
return list(map(operator.mul, coefficients, powers))
def totient(n):
"""Return the count of natural numbers up to *n* that are coprime with *n*.
>>> totient(9)
6
>>> totient(12)
4
"""
# The itertools docs use unique_justseen instead of set; see
# https://github.com/more-itertools/more-itertools/issues/823
for p in set(factor(n)):
n = n // p * (p - 1)
return n
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