






import numpy as np
from numpy import floataa
from numpy.typing import NDArray





def 						A_					(						lowercase_	,  lowercase_			)				->NDArray[floataa]:







 """simple docstring"""

 SCREAMING_SNAKE_CASE    ,   SCREAMING_SNAKE_CASE				    =		np.shape(lowercase_			)

 SCREAMING_SNAKE_CASE				    =		np.zeros((rows, 1)	,  dtype=lowercase_			)
 for row in reversed(range(lowercase_			)			):
  SCREAMING_SNAKE_CASE				    =		0
  for col in range(row + 1	,  lowercase_			):
   total += coefficients[row, col] * x[col]

  SCREAMING_SNAKE_CASE				    =		(vector[row] - total) / coefficients[row, row]

 return x





def 						A_					(						lowercase_	,  lowercase_			)				->NDArray[floataa]:







 """simple docstring"""

 SCREAMING_SNAKE_CASE    ,   SCREAMING_SNAKE_CASE				    =		np.shape(lowercase_			)
 if rows != columns:
  return np.array(()	,  dtype=lowercase_			)

 # 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	,  lowercase_			):
   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()












from __future__ import annotations

import re


def __A       ( __lowerCAmelCase      )->    list[str]:





						"""simple docstring"""



						def alphanum_key(__lowerCAmelCase      ):
												return [int(__lowerCAmelCase      ) if s.isdigit() else s.lower() for s in re.split('([0-9]+)'					,						__lowerCAmelCase      )]

						return sorted(__lowerCAmelCase					,						key=__lowerCAmelCase      )


if __name__ == "__main__":
		import doctest

		doctest.testmod()
