






import numpy as np
from numpy import floataa
from numpy.typing import NDArray
def    a__    (	A		,					A       ):
   lowercase__    :						Optional[Any]       =			np.shape(A_       )

   lowercase__    :						NDArray[floataa]       =			np.zeros((rows, 1)		,					dtype=A_       )
   for row in reversed(range(A_       )       ):
      lowercase__    :						Tuple       =			0
      for col in range(row + 1		,					A_       ):
         total += coefficients[row, col] * x[col]

      lowercase__    :						int       =			(vector[row] - total) / coefficients[row, row]

   return x
def    a__    (	A		,					A       ):
   lowercase__    :						List[str]       =			np.shape(A_       )
   if rows != columns:
      return np.array(()		,					dtype=A_       )

   # augmented matrix
   lowercase__    :						NDArray[floataa]       =			np.concatenate((coefficients, vector)		,					axis=1       )
   lowercase__    :						List[str]       =			augmented_mat.astype("""float64"""       )

   # scale the matrix leaving it triangular
   for row in range(rows - 1       ):
      lowercase__    :						Dict       =			augmented_mat[row, row]
      for col in range(row + 1		,					A_       ):
         lowercase__    :						int       =			augmented_mat[col, row] / pivot
         augmented_mat[col, :] -= factor * augmented_mat[row, :]

   lowercase__    :						Optional[Any]       =			retroactive_resolution(
       augmented_mat[:, 0:columns]		,					augmented_mat[:, columns : columns + 1]       )

   return x


if __name__ == "__main__":
 import doctest

 doctest.testmod()










"""simple docstring"""


from __future__ import annotations




def      snake_case    (A_ :list[int | float]      ,     A_ :int      ,     A_ :int ):







  '''simple docstring'''

  if len(A_ ) == 0:
    raise ValueError('find_min() arg is an empty sequence' )
  if (
      left >= len(A_ )
      or left < -len(A_ )
      or right >= len(A_ )
      or right < -len(A_ )
  ):
    raise IndexError('list index out of range' )
  if left == right:
    return nums[left]
  a	:     Optional[Any]							=  (left + right) >> 1  # the middle
  a	:     List[str]							=  find_min(A_      ,     A_      ,     A_ )  # find min in range[left, mid]
  a	:     Union[str, Any]							=  find_min(A_      ,     mid + 1      ,     A_ )  # find min in range[mid + 1, right]

  return left_min if left_min <= right_min else right_min


if __name__ == "__main__":
   import doctest

   doctest.testmod(verbose=True)




