




from __future__ import annotations

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


def    A						(__A   :    Union[str, Any]   ,				__A   :    Dict   ,				__A   :    Dict   ,				__A   :    int   ,				)   ->      list[float]:







     """simple docstring"""


     UpperCAmelCase_     ,       UpperCAmelCase_          =							coefficient_matrix.shape
     UpperCAmelCase_     ,       UpperCAmelCase_          =							constant_matrix.shape

     if rowsa != colsa:
          UpperCAmelCase_          =							F"""Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}"""
          raise ValueError(__snake_case       )

     if colsa != 1:
          UpperCAmelCase_          =							F"""Constant matrix must be nx1 but received {rowsa}x{colsa}"""
          raise ValueError(__snake_case       )

     if rowsa != rowsa:
          UpperCAmelCase_          =							(
              '''Coefficient and constant matrices dimensions must be nxn and nx1 but '''
              F"""received {rowsa}x{colsa} and {rowsa}x{colsa}"""
          )
          raise ValueError(__snake_case       )

     if len(__snake_case       ) != rowsa:
          UpperCAmelCase_          =							(
              '''Number of initial values must be equal to number of rows in coefficient '''
              F"""matrix but received {len(__snake_case       )} and {rowsa}"""
          )
          raise ValueError(__snake_case       )

     if iterations <= 0:
          raise ValueError('''Iterations must be at least 1'''       )

     UpperCAmelCase_          =							np.concatenate(
         (coefficient_matrix, constant_matrix)   ,				axis=1       )

     UpperCAmelCase_     ,       UpperCAmelCase_          =							table.shape

     strictly_diagonally_dominant(__snake_case       )

     # Iterates the whole matrix for given number of times
     for _ in range(__snake_case       ):
          UpperCAmelCase_          =							[]
          for row in range(__snake_case       ):
               UpperCAmelCase_          =							0
               for col in range(__snake_case       ):
                    if col == row:
                         UpperCAmelCase_          =							table[row][col]
                    elif col == cols - 1:
                         UpperCAmelCase_          =							table[row][col]
                    else:
                         temp += (-1) * table[row][col] * init_val[col]
               UpperCAmelCase_          =							(temp + val) / denom
               new_val.append(__snake_case       )
          UpperCAmelCase_          =							new_val

     return [float(__snake_case       ) for i in new_val]


def    A						(__A   :    Union[str, Any]       )   ->      bool:







     """simple docstring"""


     UpperCAmelCase_     ,       UpperCAmelCase_          =							table.shape

     UpperCAmelCase_          =							True

     for i in range(0   ,				__snake_case       ):
          UpperCAmelCase_          =							0
          for j in range(0   ,				cols - 1       ):
               if i == j:
                    continue
               else:
                    total += table[i][j]

          if table[i][i] <= total:
               raise ValueError('''Coefficient matrix is not strictly diagonally dominant'''       )

     return is_diagonally_dominant


# Test Cases
if __name__ == "__main__":
    import doctest

    doctest.testmod()










from math import isqrt
def 				UpperCAmelCase_					(							__snake_case						)     ->			list[int]:







 """simple docstring"""


 _lowercase							      =[True] * max_number
 for i in range(2     ,						isqrt(max_number - 1						) + 1						):
  if is_prime[i]:
   for j in range(i**2     ,						__snake_case     ,						__snake_case						):
    _lowercase							      =False

 return [i for i in range(2     ,						__snake_case						) if is_prime[i]]
def 				UpperCAmelCase_					(							__snake_case = 10**8						)     ->			int:







 """simple docstring"""


 _lowercase							      =calculate_prime_numbers(max_number // 2						)

 _lowercase							      =0
 _lowercase							      =0
 _lowercase							      =len(__snake_case						) - 1
 while left <= right:
  while prime_numbers[left] * prime_numbers[right] >= max_number:
   right -= 1
  semiprimes_count += right - left + 1
  left += 1

 return semiprimes_count


if __name__ == "__main__":
  print(f'''{solution() = }''')

