


'''simple docstring'''



from __future__ import annotations

from math import gcd






def   lowerCamelCase							(_SCREAMING_SNAKE_CASE      :   int						,      _SCREAMING_SNAKE_CASE      :   int = 2						,      _SCREAMING_SNAKE_CASE      :   int = 1						,      _SCREAMING_SNAKE_CASE      :   int = 3						,      ):
					# A value less than 2 can cause an infinite loop in the algorithm.
					if num < 2:
										raise ValueError('The input value cannot be less than 2'   )

					# Because of the relationship between ``f(f(x))`` and ``f(x)``, this
					# algorithm struggles to find factors that are divisible by two.
					# As a workaround, we specifically check for two and even inputs.
					#   See: https://math.stackexchange.com/a/2856214/165820
					if num > 2 and num % 2 == 0:
										return 2

					# Pollard's Rho algorithm requires a function that returns pseudorandom
					# values between 0 <= X < ``num``.  It doesn't need to be random in the
					# sense that the output value is cryptographically secure or difficult
					# to calculate, it only needs to be random in the sense that all output
					# values should be equally likely to appear.
					# For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
					# However, the success of Pollard's algorithm isn't guaranteed and is
					# determined in part by the initial seed and the chosen random function.
					# To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
					# where ``C`` is a value that we can modify between each attempt.
					def rand_fn(_SCREAMING_SNAKE_CASE      :   int						,      _SCREAMING_SNAKE_CASE      :   int						,      _SCREAMING_SNAKE_CASE      :   int   ) -> int:
										return (pow(_SCREAMING_SNAKE_CASE						,      2   ) + step) % modulus

					for _ in range(_SCREAMING_SNAKE_CASE   ):
										# These track the position within the cycle detection logic.
										__a			:       int					    =  seed
										__a			:       Tuple					    =  seed

										while True:
															# At each iteration, the tortoise moves one step and the hare moves two.
															__a			:       List[str]					    =  rand_fn(_SCREAMING_SNAKE_CASE						,      _SCREAMING_SNAKE_CASE						,      _SCREAMING_SNAKE_CASE   )
															__a			:       Optional[int]					    =  rand_fn(_SCREAMING_SNAKE_CASE						,      _SCREAMING_SNAKE_CASE						,      _SCREAMING_SNAKE_CASE   )
															__a			:       Optional[Any]					    =  rand_fn(_SCREAMING_SNAKE_CASE						,      _SCREAMING_SNAKE_CASE						,      _SCREAMING_SNAKE_CASE   )

															# At some point both the tortoise and the hare will enter a cycle whose
															# length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
															# the tortoise and hare will end up on the same value modulo ``p``.
															# We can detect when this happens because the position difference between
															# the tortoise and the hare will share a common divisor with ``num``.
															__a			:       List[str]					    =  gcd(hare - tortoise						,      _SCREAMING_SNAKE_CASE   )

															if divisor == 1:
																				# No common divisor yet, just keep searching.
																				continue
															else:
																				# We found a common divisor!
																				if divisor == num:
																									# Unfortunately, the divisor is ``num`` itself and is useless.
																									break
																				else:
																									# The divisor is a nontrivial factor of ``num``!
																									return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
										__a			:       List[str]					    =  hare

										# The new step value for the random function can just be incremented.
										# At first the results will be similar to what the old function would
										# have produced, but the value will quickly diverge after a bit.
										step += 1

					# We haven't found a divisor within the requested number of attempts.
					# We were unlucky or ``num`` itself is actually prime.
					return None


if __name__ == "__main__":
	import argparse

	__lowercase	:						Any       =	argparse.ArgumentParser()
	parser.add_argument(
	    'num',
	    type=int,
	    help='The value to find a divisor of',
	)
	parser.add_argument(
	    '--attempts',
	    type=int,
	    default=3,
	    help='The number of attempts before giving up',
	)
	__lowercase	:						List[str]       =	parser.parse_args()

	__lowercase	:						List[str]       =	pollard_rho(args.num, attempts=args.attempts)
	if divisor is None:
		print(f'''{args.num} is probably prime''')
	else:
		__lowercase	:						List[str]       =	args.num // divisor
		print(f'''{args.num} = {divisor} * {quotient}''')






'''simple docstring'''
import numpy as np
from PIL import Image


def        _lowerCAmelCase					(						_UpperCamelCase						:     np.ndarray							, _UpperCamelCase						:     int							, _UpperCamelCase						:     int				)      ->			np.ndarray:
     """simple docstring"""




     _SCREAMING_SNAKE_CASE							    =np.array(_UpperCamelCase				)
     if arr.shape[0] != arr.shape[1]:
          raise ValueError('The input array is not a square matrix'				)
     _SCREAMING_SNAKE_CASE							    =0
     _SCREAMING_SNAKE_CASE							    =0
     _SCREAMING_SNAKE_CASE							    =0
     _SCREAMING_SNAKE_CASE							    =0

     # compute the shape of the output matrix
     _SCREAMING_SNAKE_CASE							    =(arr.shape[0] - size) // stride + 1
     # initialize the output matrix with zeros of shape maxpool_shape
     _SCREAMING_SNAKE_CASE							    =np.zeros((maxpool_shape, maxpool_shape)				)

     while i < arr.shape[0]:
          if i + size > arr.shape[0]:
               # if the end of the matrix is reached, break
               break
          while j < arr.shape[1]:
               # if the end of the matrix is reached, break
               if j + size > arr.shape[1]:
                    break
               # compute the maximum of the pooling matrix
               _SCREAMING_SNAKE_CASE							    =np.max(arr[i : i + size, j : j + size]				)
               # shift the pooling matrix by stride of column pixels
               j += stride
               mat_j += 1

          # shift the pooling matrix by stride of row pixels
          i += stride
          mat_i += 1

          # reset the column index to 0
          _SCREAMING_SNAKE_CASE							    =0
          _SCREAMING_SNAKE_CASE							    =0

     return updated_arr


def        _lowerCAmelCase					(						_UpperCamelCase						:     np.ndarray							, _UpperCamelCase						:     int							, _UpperCamelCase						:     int				)      ->			np.ndarray:
     """simple docstring"""




     _SCREAMING_SNAKE_CASE							    =np.array(_UpperCamelCase				)
     if arr.shape[0] != arr.shape[1]:
          raise ValueError('The input array is not a square matrix'				)
     _SCREAMING_SNAKE_CASE							    =0
     _SCREAMING_SNAKE_CASE							    =0
     _SCREAMING_SNAKE_CASE							    =0
     _SCREAMING_SNAKE_CASE							    =0

     # compute the shape of the output matrix
     _SCREAMING_SNAKE_CASE							    =(arr.shape[0] - size) // stride + 1
     # initialize the output matrix with zeros of shape avgpool_shape
     _SCREAMING_SNAKE_CASE							    =np.zeros((avgpool_shape, avgpool_shape)				)

     while i < arr.shape[0]:
          # if the end of the matrix is reached, break
          if i + size > arr.shape[0]:
               break
          while j < arr.shape[1]:
               # if the end of the matrix is reached, break
               if j + size > arr.shape[1]:
                    break
               # compute the average of the pooling matrix
               _SCREAMING_SNAKE_CASE							    =int(np.average(arr[i : i + size, j : j + size]				)				)
               # shift the pooling matrix by stride of column pixels
               j += stride
               mat_j += 1

          # shift the pooling matrix by stride of row pixels
          i += stride
          mat_i += 1
          # reset the column index to 0
          _SCREAMING_SNAKE_CASE							    =0
          _SCREAMING_SNAKE_CASE							    =0

     return updated_arr


# Main Function
if __name__ == "__main__":
     from doctest import testmod

     testmod(name="avgpooling", verbose=True)

     # Loading the image
     lowerCamelCase :      Optional[Any]					 =     Image.open("path_to_image")

     # Converting the image to numpy array and maxpooling, displaying the result
     # Ensure that the image is a square matrix

     Image.fromarray(maxpooling(np.array(image), size=3, stride=2)).show()

     # Converting the image to numpy array and averagepooling, displaying the result
     # Ensure that the image is a square matrix

     Image.fromarray(avgpooling(np.array(image), size=3, stride=2)).show()
