'''simple docstring'''

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


def __a			(						__lowerCamelCase      :    NDArray[floataa]			,		__lowerCamelCase      :    NDArray[floataa]					)						->				NDArray[floataa]:
      '''simple docstring'''


      lowercase_			,				lowercase_     			=   np.shape(__lowerCamelCase					)

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

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

      return x


def __a			(						__lowerCamelCase      :    NDArray[floataa]			,		__lowerCamelCase      :    NDArray[floataa]					)						->				NDArray[floataa]:
      '''simple docstring'''


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

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

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

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

      return x


if __name__ == "__main__":
       import doctest

       doctest.testmod()







'''simple docstring'''

from hashlib import mda, shaaaa

lowerCAmelCase_			:			Union[str, Any]    	=   (shaaaa, mda)





class 				lowercase    :


   def __init__( self					:						Optional[Any] ,		__lowerCAmelCase					:						int = 8)   ->       None:
         lowercase_     			=   0B0
         lowercase_     			=   size


   def __UpperCAmelCase							( self					:						Dict ,		__lowerCAmelCase					:						str)   ->       None:
         lowercase_     			=   self.hash_(__lowerCAmelCase)
         self.bitarray |= h


   def __UpperCAmelCase							( self					:						Union[str, Any] ,		__lowerCAmelCase					:						str)   ->       bool:
         lowercase_     			=   self.hash_(__lowerCAmelCase)
         return (h & self.bitarray) == h


   def __contains__( self					:						str ,		__lowerCAmelCase					:						str)   ->       bool:
         return self.exists(__lowerCAmelCase)


   def __UpperCAmelCase							( self					:						Union[str, Any] ,		__lowerCAmelCase					:						int)   ->       str:
         lowercase_     			=   bin(__lowerCAmelCase)[2:]
         return res.zfill(self.size)


   @property
   def __UpperCAmelCase							( self					:						Dict)   ->       str:
         return self.format_bin(self.bitarray)


   def __UpperCAmelCase							( self					:						Any ,		__lowerCAmelCase					:						str)   ->       int:
         lowercase_     			=   0B0
         for func in HASH_FUNCTIONS:
               lowercase_     			=   (
                   int.from_bytes(func(value.encode()).digest() ,		"little") % self.size
               )
               res |= 2**position
         return res


   def __UpperCAmelCase							( self					:						Union[str, Any] ,		__lowerCAmelCase					:						str)   ->       str:
         return self.format_bin(self.hash_(__lowerCAmelCase))


   @property
   def __UpperCAmelCase							( self					:						str)   ->       float:
         lowercase_     			=   bin(self.bitarray).count("1")
         return (n_ones / self.size) ** len(__lowerCAmelCase)






