from __future__ import annotations

import numpy as np







def     _a				(      lowercase__  :			np.ndarray ):







     '''simple docstring'''


     __A				:				Any			=		np.shape(lowercase__ )
     if rows != columns:
          __A				:				Dict			=		(
              '\'table\' has to be of square shaped array but got a '
              f'''{rows}x{columns} array:\n{table}'''
          )
          raise ValueError(lowercase__ )

     __A				:				Optional[Any]			=		np.zeros((rows, columns) )
     __A				:				Optional[Any]			=		np.zeros((rows, columns) )
     for i in range(lowercase__ ):
          for j in range(lowercase__ ):
               __A				:				Union[str, Any]			=		sum(lower[i][k] * upper[k][j] for k in range(lowercase__ ) )
               if upper[j][j] == 0:
                    raise ArithmeticError('No LU decomposition exists' )
               __A				:				Optional[int]			=		(table[i][j] - total) / upper[j][j]
          __A				:				Union[str, Any]			=		1
          for j in range(lowercase__     ,  lowercase__ ):
               __A				:				Optional[Any]			=		sum(lower[i][k] * upper[k][j] for k in range(lowercase__ ) )
               __A				:				Optional[int]			=		table[i][j] - total
     return lower, upper


if __name__ == "__main__":
      import doctest

      doctest.testmod()




import math







def     _a				(      lowercase__  :			int ):







     '''simple docstring'''


     assert isinstance(lowercase__     ,  lowercase__ ) and (
         number >= 0
     ), "'number' must been an int and positive"

     if 1 < number < 4:
          # 2 and 3 are primes
          return True
     elif number < 2 or not number % 2:
          # Negatives, 0, 1 and all even numbers are not primes
          return False

     SCREAMING_SNAKE_CASE__				:				Tuple			=		range(3     ,  int(math.sqrt(lowercase__ ) + 1 )     ,  2 )
     return not any(not number % i for i in odd_numbers )







def     _a				(      lowercase__  :			List[str]     ,  lowercase__  :			Any=1     ,  **lowercase__  :			Any ):







     '''simple docstring'''


     SCREAMING_SNAKE_CASE__				:				int			=		factor * value
     SCREAMING_SNAKE_CASE__				:				Dict			=		value

     while not is_prime(lowercase__ ):
          value += 1 if not ("desc" in kwargs and kwargs["desc"] is True) else -1

     if value == first_value_val:
          return next_prime(value + 1     ,  **lowercase__ )
     return value



