code: 
  
  
  import numpy as np
  from numpy import floataa
  from numpy.typing import NDArray
  
  
  
  
  
  
  
  def 							UpperCAmelCase				(     UpperCAmelCase				,UpperCAmelCase      )->		NDArray[floataa]:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  					SCREAMING_SNAKE_CASE_      ,SCREAMING_SNAKE_CASE_	      =       np.shape(UpperCAmelCase      )
  
  					SCREAMING_SNAKE_CASE_	      =       np.zeros((rows, 1)				,dtype=UpperCAmelCase      )
  					for row in reversed(range(UpperCAmelCase      )      ):
  										SCREAMING_SNAKE_CASE_	      =       0
  										for col in range(row + 1				,UpperCAmelCase      ):
  															total += coefficients[row, col] * x[col]
  
  										SCREAMING_SNAKE_CASE_	      =       (vector[row] - total) / coefficients[row, row]
  
  					return x
  
  
  
  
  
  
  
  def 							UpperCAmelCase				(     UpperCAmelCase				,UpperCAmelCase      )->		NDArray[floataa]:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  					SCREAMING_SNAKE_CASE_      ,SCREAMING_SNAKE_CASE_	      =       np.shape(UpperCAmelCase      )
  					if rows != columns:
  										return np.array(()				,dtype=UpperCAmelCase      )
  
  					# augmented matrix
  					SCREAMING_SNAKE_CASE_	      =       np.concatenate((coefficients, vector)				,axis=1      )
  					SCREAMING_SNAKE_CASE_	      =       augmented_mat.astype('''float64'''      )
  
  					# scale the matrix leaving it triangular
  					for row in range(rows - 1      ):
  										SCREAMING_SNAKE_CASE_	      =       augmented_mat[row, row]
  										for col in range(row + 1				,UpperCAmelCase      ):
  															SCREAMING_SNAKE_CASE_	      =       augmented_mat[col, row] / pivot
  															augmented_mat[col, :] -= factor * augmented_mat[row, :]
  
  					SCREAMING_SNAKE_CASE_	      =       retroactive_resolution(
  					    augmented_mat[:, 0:columns]				,augmented_mat[:, columns : columns + 1]      )
  
  					return x
  
  
  if __name__ == "__main__":
  		import doctest
  
  		doctest.testmod()
  
  
  
style_context: 
  
  
  def 							UpperCAmelCase				(     UpperCAmelCase      )->		bool:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  					if not isinstance(UpperCAmelCase				,UpperCAmelCase      ):
  										SCREAMING_SNAKE_CASE_	      =       f'''Input value of [number={number}] must be an integer'''
  										raise TypeError(UpperCAmelCase      )
  					if number < 0 or number % 2 == 1:
  										return False
  					SCREAMING_SNAKE_CASE_	      =       int(number**0.5      )
  					return number == number_sqrt * (number_sqrt + 1)
  
  
  if __name__ == "__main__":
  		import doctest
  
  		doctest.testmod()
  
  
  