




from __future__ import annotations

import numpy as np



def 					UpperCAmelCase__     (   lowerCamelCase      ):
  lowercase		,			lowercase			:Tuple							=  np.shape(lowerCamelCase      )
  if rows != columns:
    lowercase			:Dict							=  (
        "'table' has to be of square shaped array but got a "
        F"{rows}x{columns} array:\n{table}"
    )
    raise ValueError(lowerCamelCase      )

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


if __name__ == "__main__":
     import doctest

     doctest.testmod()









import math
import sys



def 					UpperCAmelCase__     (   lowerCamelCase      ):
  if number != int(lowerCamelCase      ):
    raise ValueError("the value of input must be a natural number"      )
  if number < 0:
    raise ValueError("the value of input must not be a negative number"      )
  if number == 0:
    return 1
  lowercase			:Dict							=  [-1] * (number + 1)
  lowercase			:Optional[Any]							=  0
  for i in range(1,	number + 1      ):
    lowercase			:Optional[int]							=  sys.maxsize
    lowercase			:Any							=  int(math.sqrt(lowerCamelCase      )      )
    for j in range(1,	root + 1      ):
      lowercase			:Any							=  1 + answers[i - (j**2)]
      lowercase			:Dict							=  min(lowerCamelCase,	lowerCamelCase      )
    lowercase			:Union[str, Any]							=  answer
  return answers[number]


if __name__ == "__main__":
     import doctest

     doctest.testmod()



