




"""simple docstring"""

from __future__ import annotations

a							      =     [
    [-1, 0],  # left
    [0, -1],  # down
    [1, 0],  # right
    [0, 1],  # up
]






def 		_snake_case	(   _snake_case				:   list[list[int]]							,    _snake_case				:   list[int]							,    _snake_case				:   list[int]							,    _snake_case				:   int							,    _snake_case				:   list[list[int]]							,    )       ->   List[str]:






      '''simple docstring'''


      _A     =  [
          [0 for col in range(len(grid[0]				)				)] for row in range(len(__lowerCamelCase				)				)
      ]  # the reference grid
      _A     =  1
      _A     =  [
          [0 for col in range(len(grid[0]				)				)] for row in range(len(__lowerCamelCase				)				)
      ]  # the action grid

      _A     =  init[0]
      _A     =  init[1]
      _A     =  0
      _A     =  g + heuristic[x][y]  # cost from starting cell to destination cell
      _A     =  [[f, g, x, y]]

      _A     =  False  # flag that is set when search is complete
      _A     =  False  # flag set if we can't find expand

      while not found and not resign:
            if len(__lowerCamelCase				) == 0:
                  raise ValueError('Algorithm is unable to find solution'				)
            else:  # to choose the least costliest action so as to move closer to the goal
                  cell.sort()
                  cell.reverse()
                  _A     =  cell.pop()
                  _A     =  next_cell[2]
                  _A     =  next_cell[3]
                  _A     =  next_cell[1]

                  if x == goal[0] and y == goal[1]:
                        _A     =  True
                  else:
                        for i in range(len(__lowerCamelCase				)				):  # to try out different valid actions
                              _A     =  x + DIRECTIONS[i][0]
                              _A     =  y + DIRECTIONS[i][1]
                              if xa >= 0 and xa < len(__lowerCamelCase				) and ya >= 0 and ya < len(grid[0]				):
                                    if closed[xa][ya] == 0 and grid[xa][ya] == 0:
                                          _A     =  g + cost
                                          _A     =  ga + heuristic[xa][ya]
                                          cell.append([fa, ga, xa, ya]				)
                                          _A     =  1
                                          _A     =  i
      _A     =  []
      _A     =  goal[0]
      _A     =  goal[1]
      invpath.append([x, y]				)  # we get the reverse path from here
      while x != init[0] or y != init[1]:
            _A     =  x - DIRECTIONS[action[x][y]][0]
            _A     =  y - DIRECTIONS[action[x][y]][1]
            _A     =  xa
            _A     =  ya
            invpath.append([x, y]				)

      _A     =  []
      for i in range(len(__lowerCamelCase				)				):
            path.append(invpath[len(__lowerCamelCase				) - 1 - i]				)
      return path, action


if __name__ == "__main__":
       a							      =     [
           [0, 1, 0, 0, 0, 0],
           [0, 1, 0, 0, 0, 0],  # 0 are free path whereas 1's are obstacles
           [0, 1, 0, 0, 0, 0],
           [0, 1, 0, 0, 1, 0],
           [0, 0, 0, 0, 1, 0],
       ]

       a							      =     [0, 0]
       # all coordinates are given in format [y,x]
       a							      =     [len(grid) - 1, len(grid[0]) - 1]
       a							      =     1

       # the cost map which pushes the path closer to the goal
       a							      =     [[0 for row in range(len(grid[0]))] for col in range(len(grid))]
       for i in range(len(grid)):
              for j in range(len(grid[0])):
                     a							      =     abs(i - goal[0]) + abs(j - goal[1])
                     if grid[i][j] == 1:
                            # added extra penalty in the heuristic map
                            a							      =     99

       a      ,		a							      =     search(grid, init, goal, cost, heuristic)

       print('''ACTION MAP''')
       for i in range(len(action)):
              print(action[i])

       for i in range(len(path)):
              print(path[i])





'''simple docstring'''






from typing import Any

import numpy as np





def lowerCamelCase__     (		__lowerCamelCase			:				np.ndarray				):







   '''simple docstring'''




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





def lowerCamelCase__     (		__lowerCamelCase			:				np.ndarray			,       __lowerCamelCase			:				np.ndarray				):







   '''simple docstring'''




   _UpperCAmelCase  :  str				       =v.conjugate().T
   _UpperCAmelCase  :  Optional[int]				       =v_star.dot(__lowerCamelCase				)
   assert isinstance(__lowerCamelCase			,       np.ndarray				)
   return (v_star_dot.dot(__lowerCamelCase				)) / (v_star.dot(__lowerCamelCase				))





def lowerCamelCase__     (		):







   '''simple docstring'''




   _UpperCAmelCase  :  Optional[Any]				       =np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]				)
   _UpperCAmelCase  :  List[str]				       =np.array([[1], [2], [3]]				)
   assert is_hermitian(__lowerCamelCase				), f"{a} is not hermitian."
   print(rayleigh_quotient(__lowerCamelCase			,       __lowerCamelCase				)				)

   _UpperCAmelCase  :  List[str]				       =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()

