




from __future__ import annotations

import numpy as np




def 							_UpperCAmelCase  (UpperCamelCase_ : np.ndarray       ):
				'''simple docstring'''
				_lowerCAmelCase					,    _lowerCAmelCase    :    int										=      np.shape(UpperCamelCase_       )
				if rows != columns:
								_lowerCAmelCase    :    Optional[int]										=      (
								    """'table' has to be of square shaped array but got a """
								    F"{rows}x{columns} array:\n{table}"
								)
								raise ValueError(UpperCamelCase_       )

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


if __name__ == "__main__":
	import doctest

	doctest.testmod()







from math import loga




def 							_UpperCAmelCase  (UpperCamelCase_ : int       ):
				'''simple docstring'''
				if a < 0:
								raise ValueError("""Input value must be a positive integer"""       )
				elif isinstance(UpperCamelCase_      ,    UpperCamelCase_       ):
								raise TypeError("""Input value must be a 'int' type"""       )
				return 0 if (a == 0) else int(loga(a & -a       )       )


if __name__ == "__main__":
	import doctest

	doctest.testmod()

