




'''simple docstring'''





from __future__ import annotations





def UpperCAmelCase    (       UpperCAmelCase_   ,	UpperCAmelCase_   ,	UpperCAmelCase_						):
    if len(UpperCAmelCase_						) == 0:
        raise ValueError("""find_min() arg is an empty sequence"""						)
    if (
        left >= len(UpperCAmelCase_						)
        or left < -len(UpperCAmelCase_						)
        or right >= len(UpperCAmelCase_						)
        or right < -len(UpperCAmelCase_						)
    ):
        raise IndexError("""list index out of range"""						)
    if left == right:
        return nums[left]
    _a				:							Any							       =      (left + right) >> 1  # the middle
    _a				:							Dict							       =      find_min(UpperCAmelCase_   ,	UpperCAmelCase_   ,	UpperCAmelCase_						)  # find min in range[left, mid]
    _a				:							str							       =      find_min(UpperCAmelCase_   ,	mid + 1   ,	UpperCAmelCase_						)  # 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)





"""simple docstring"""




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



def 						UpperCAmelCase__ (					A__						,     A__ )   ->   NDArray[floataa]:




   """simple docstring"""
   lowerCamelCase__       ,							lowerCamelCase__						       =      np.shape(A__ )

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

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

   return x



def 						UpperCAmelCase__ (					A__						,     A__ )   ->   NDArray[floataa]:




   """simple docstring"""
   lowerCamelCase__       ,							lowerCamelCase__						       =      np.shape(A__ )
   if rows != columns:
      return np.array(()						,     dtype=A__ )

   # augmented matrix
   lowerCamelCase__						       =      np.concatenate((coefficients, vector)						,     axis=1 )
   lowerCamelCase__						       =      augmented_mat.astype("float64" )

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

   lowerCamelCase__						       =      retroactive_resolution(
       augmented_mat[:, 0:columns]						,     augmented_mat[:, columns : columns + 1] )

   return x


if __name__ == "__main__":
     import doctest

     doctest.testmod()




