


'''simple docstring'''




import numpy as np





def 	a_					( lowerCamelCase	:     np.ndarray	, lowerCamelCase	:     np.ndarray	, lowerCamelCase	:     float = 1e-12	, lowerCamelCase	:     int = 100	, ):
   assert np.shape(lowerCamelCase				)[0] == np.shape(lowerCamelCase				)[1]
   # Ensure proper dimensionality.
   assert np.shape(lowerCamelCase				)[0] == np.shape(lowerCamelCase				)[0]
   # Ensure inputs are either both complex or both real
   assert np.iscomplexobj(lowerCamelCase				) == np.iscomplexobj(lowerCamelCase				)
   lowerCAmelCase     =			np.iscomplexobj(lowerCamelCase				)
   if is_complex:
      # Ensure complex input_matrix is Hermitian
      assert np.array_equal(lowerCamelCase	, input_matrix.conj().T				)

   # Set convergence to False. Will define convergence when we exceed max_iterations
   # or when we have small changes from one iteration to next.

   lowerCAmelCase     =			False
   lowerCAmelCase     =			0
   lowerCAmelCase     =			0
   lowerCAmelCase     =			1e12

   while not convergence:
      # Multiple matrix by the vector.
      lowerCAmelCase     =			np.dot(lowerCamelCase	, lowerCamelCase				)
      # Normalize the resulting output vector.
      lowerCAmelCase     =			w / np.linalg.norm(lowerCamelCase				)
      # Find rayleigh quotient
      # (faster than usual b/c we know vector is normalized already)
      lowerCAmelCase     =			vector.conj().T if is_complex else vector.T
      lowerCAmelCase     =			np.dot(lowerCamelCase	, np.dot(lowerCamelCase	, lowerCamelCase				)				)

      # Check convergence.
      lowerCAmelCase     =			np.abs(lambda_ - lambda_previous				) / lambda_
      iterations += 1

      if error <= error_tol or iterations >= max_iterations:
         lowerCAmelCase     =			True

      lowerCAmelCase     =			lambda_

   if is_complex:
      lowerCAmelCase     =			np.real(lambda_				)

   return lambda_, vector





def 	a_					( ):
   lowerCAmelCase     =			np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]				)
   lowerCAmelCase     =			np.array([41, 4, 20]				)
   lowerCAmelCase     =			real_input_matrix.astype(np.complexaaa				)
   lowerCAmelCase     =			np.triu(1J * complex_input_matrix	, 1				)
   complex_input_matrix += imag_matrix
   complex_input_matrix += -1 * imag_matrix.T
   lowerCAmelCase     =			np.array([41, 4, 20]				).astype(np.complexaaa				)

   for problem_type in ["real", "complex"]:
      if problem_type == "real":
         lowerCAmelCase     =			real_input_matrix
         lowerCAmelCase     =			real_vector
      elif problem_type == "complex":
         lowerCAmelCase     =			complex_input_matrix
         lowerCAmelCase     =			complex_vector

      # Our implementation.
      lowerCAmelCase     ,  lowerCAmelCase     =			power_iteration(lowerCamelCase	, lowerCamelCase				)

      # Numpy implementation.

      # Get eigenvalues and eigenvectors using built-in numpy
      # eigh (eigh used for symmetric or hermetian matrices).
      lowerCAmelCase     ,  lowerCAmelCase     =			np.linalg.eigh(lowerCamelCase				)
      # Last eigenvalue is the maximum one.
      lowerCAmelCase     =			eigen_values[-1]
      # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
      lowerCAmelCase     =			eigen_vectors[:, -1]

      # Check our implementation and numpy gives close answers.
      assert np.abs(eigen_value - eigen_value_max				) <= 1e-6
      # Take absolute values element wise of each eigenvector.
      # as they are only unique to a minus sign.
      assert np.linalg.norm(np.abs(lowerCamelCase				) - np.abs(lowerCamelCase				)				) <= 1e-6


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       test_power_iteration()













from __future__ import annotations

UpperCAmelCase__				   =		list[list[int]]

# assigning initial values to the grid
UpperCAmelCase__				   =		[
    [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
UpperCAmelCase__				   =		[
    [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 						_a    (      a :Matrix	,							a :int	,							a :int	,							a :int )							->			bool:
 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 						_a    (      a :Matrix )							->			tuple[int, int] | None:
 for i in range(9 ):
  for j in range(9 ):
   if grid[i][j] == 0:
    return i, j
 return None
def 						_a    (      a :Matrix )							->			Matrix | None:
 if location := find_empty_location(a ):
  a  ,     a		  =      location
 else:
  # If the location is ``None``, then the grid is solved.
  return grid

 for digit in range(1	,							10 ):
  if is_safe(a	,							a	,							a	,							a ):
   a		  =      digit

   if sudoku(a ) is not None:
    return grid

   a		  =      0

 return None
def 						_a    (      a :Matrix )							->			None:
 for row in grid:
  for cell in row:
   print(a	,							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:")
          UpperCAmelCase__				   =		sudoku(example_grid)
          if solution is not None:
               print_solution(solution)
          else:
               print("Cannot find a solution.")


