
from typing import Any

import numpy as np


def 						UpperCAmelCase			(	a_						)		->	List[str]:







				"""simple docstring"""



				return np.array_equal(__lowerCamelCase		,   matrix.conjugate().T						)


def 						UpperCAmelCase			(	a_		,   a_						)		->	Union[str, Any]:







				"""simple docstring"""



				__A   			= v.conjugate().T
				__A   			= v_star.dot(__lowerCamelCase						)
				assert isinstance(__lowerCamelCase		,   np.ndarray						)
				return (v_star_dot.dot(__lowerCamelCase						)) / (v_star.dot(__lowerCamelCase						))


def 						UpperCAmelCase			(	)		->	Any:







				"""simple docstring"""



				__A   			= np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]						)
				__A   			= np.array([[1], [2], [3]]						)
				assert is_hermitian(__lowerCamelCase						), F'''{a} is not hermitian.'''
				print(rayleigh_quotient(__lowerCamelCase		,   __lowerCamelCase						)						)

				__A   			= np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]						)
				assert is_hermitian(__lowerCamelCase						), F'''{a} is not hermitian.'''
				assert rayleigh_quotient(__lowerCamelCase		,   __lowerCamelCase						) == float(3						)


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					tests()


from __future__ import annotations

__lowerCamelCase							=       list[list[int]]

# assigning initial values to the grid
__lowerCamelCase							=       [
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
__lowerCamelCase							=       [
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]


def 			UpperCamelCase					(    __lowerCamelCase     : Matrix				,					__lowerCamelCase     : int				,					__lowerCamelCase     : int				,					__lowerCamelCase     : int    ):
       for i in range(9    ):
              if grid[row][i] == n or grid[i][column] == n:
                     return False

       for i in range(3    ):
              for j in range(3    ):
                     if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
                            return False

       return True


def 			UpperCamelCase					(    __lowerCamelCase     : Matrix    ):
       for i in range(9    ):
              for j in range(9    ):
                     if grid[i][j] == 0:
                            return i, j
       return None


def 			UpperCamelCase					(    __lowerCamelCase     : Matrix    ):
       if location := find_empty_location(__lowerCamelCase    ):
              snake_case				,					snake_case       :   Union[str, Any]							=      location
       else:
              # If the location is ``None``, then the grid is solved.
              return grid

       for digit in range(1				,					10    ):
              if is_safe(__lowerCamelCase				,					__lowerCamelCase				,					__lowerCamelCase				,					__lowerCamelCase    ):
                     snake_case       :   List[Any]							=      digit

                     if sudoku(__lowerCamelCase    ) is not None:
                            return grid

                     snake_case       :   Union[str, Any]							=      0

       return None


def 			UpperCamelCase					(    __lowerCamelCase     : Matrix    ):
       for row in grid:
              for cell in row:
                     print(__lowerCamelCase				,					end=" "    )
              print()


if __name__ == "__main__":
   # make a copy of grid so that you can compare with the unmodified grid
   for example_grid in (initial_grid, no_solution):
      print("""\nExample grid:\n""" + """=""" * 20)
      print_solution(example_grid)
      print("""\nExample grid solution:""")
      __lowerCamelCase							=       sudoku(example_grid)
      if solution is not None:
         print_solution(solution)
      else:
         print("""Cannot find a solution.""")





