

from typing import Any

import numpy as np







def      snake_case__							(  lowerCAmelCase_     ):





     """simple docstring"""







     assert np.shape(lowerCAmelCase_     )[0] == np.shape(lowerCAmelCase_     )[1]

     # If matrix not symmetric, exit right away.
     if np.allclose(lowerCAmelCase_,       matrix.T     ) is False:
          return False

     # Get eigenvalues and eignevectors for a symmetric matrix.
     SCREAMING_SNAKE_CASE		,    SCREAMING_SNAKE_CASE 		=np.linalg.eigh(lowerCAmelCase_     )

     # Check sign of all eigenvalues.
     # np.all returns a value of type np.bool_
     return bool(np.all(eigen_values > 0     )     )







def      snake_case__							(  lowerCAmelCase_     ):





     """simple docstring"""







     SCREAMING_SNAKE_CASE 		=np.random.randn(lowerCAmelCase_,       lowerCAmelCase_     )
     SCREAMING_SNAKE_CASE 		=np.dot(lowerCAmelCase_,       random_matrix.T     )
     assert _is_matrix_spd(lowerCAmelCase_     )
     return spd_matrix







def      snake_case__							(  lowerCAmelCase_,       lowerCAmelCase_,       lowerCAmelCase_ = 1000,       lowerCAmelCase_ = 1e-8,       ):





     """simple docstring"""







     assert np.shape(lowerCAmelCase_     )[0] == np.shape(lowerCAmelCase_     )[1]
     assert np.shape(lowerCAmelCase_     )[0] == np.shape(lowerCAmelCase_     )[0]
     assert _is_matrix_spd(lowerCAmelCase_     )

     # Initialize solution guess, residual, search direction.
     SCREAMING_SNAKE_CASE 		=np.zeros((np.shape(lowerCAmelCase_     )[0], 1)     )
     SCREAMING_SNAKE_CASE 		=np.copy(lowerCAmelCase_     )
     SCREAMING_SNAKE_CASE 		=np.copy(lowerCAmelCase_     )

     # Set initial errors in solution guess and residual.
     SCREAMING_SNAKE_CASE 		=1e9
     SCREAMING_SNAKE_CASE 		=1e9
     SCREAMING_SNAKE_CASE 		=1e9

     # Set iteration counter to threshold number of iterations.
     SCREAMING_SNAKE_CASE 		=0

     while error > tol:
          # Save this value so we only calculate the matrix-vector product once.
          SCREAMING_SNAKE_CASE 		=np.dot(lowerCAmelCase_,       lowerCAmelCase_     )

          # The main algorithm.

          # Update search direction magnitude.
          SCREAMING_SNAKE_CASE 		=np.dot(ra.T,       lowerCAmelCase_     ) / np.dot(pa.T,       lowerCAmelCase_     )
          # Update solution guess.
          SCREAMING_SNAKE_CASE 		=xa + alpha * pa
          # Calculate new residual.
          SCREAMING_SNAKE_CASE 		=ra - alpha * w
          # Calculate new Krylov subspace scale.
          SCREAMING_SNAKE_CASE 		=np.dot(r.T,       lowerCAmelCase_     ) / np.dot(ra.T,       lowerCAmelCase_     )
          # Calculate new A conjuage search direction.
          SCREAMING_SNAKE_CASE 		=r + beta * pa

          # Calculate errors.
          SCREAMING_SNAKE_CASE 		=np.linalg.norm(r - ra     )
          SCREAMING_SNAKE_CASE 		=np.linalg.norm(x - xa     )
          SCREAMING_SNAKE_CASE 		=np.maximum(lowerCAmelCase_,       lowerCAmelCase_     )

          # Update variables.
          SCREAMING_SNAKE_CASE 		=np.copy(lowerCAmelCase_     )
          SCREAMING_SNAKE_CASE 		=np.copy(lowerCAmelCase_     )
          SCREAMING_SNAKE_CASE 		=np.copy(lowerCAmelCase_     )

          # Update number of iterations.
          iterations += 1
          if iterations > max_iterations:
               break

     return x







def      snake_case__							(  ):





     """simple docstring"""







     SCREAMING_SNAKE_CASE 		=3
     SCREAMING_SNAKE_CASE 		=_create_spd_matrix(lowerCAmelCase_     )
     SCREAMING_SNAKE_CASE 		=np.random.randn(lowerCAmelCase_,       1     )
     SCREAMING_SNAKE_CASE 		=np.dot(lowerCAmelCase_,       lowerCAmelCase_     )

     # Numpy solution.
     SCREAMING_SNAKE_CASE 		=np.linalg.solve(lowerCAmelCase_,       lowerCAmelCase_     )

     # Our implementation.
     SCREAMING_SNAKE_CASE 		=conjugate_gradient(lowerCAmelCase_,       lowerCAmelCase_     )

     # Ensure both solutions are close to x_true (and therefore one another).
     assert np.linalg.norm(x_numpy - x_true     ) <= 1e-6
     assert np.linalg.norm(x_conjugate_gradient - x_true     ) <= 1e-6


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 test_conjugate_gradient()




from math import cos, sin







def      snake_case__							(  lowerCAmelCase_     ):





     """simple docstring"""







     SCREAMING_SNAKE_CASE 		=float(lowerCAmelCase_     )
     return [[scaling_factor * int(x == y     ) for x in range(2     )] for y in range(2     )]







def      snake_case__							(  lowerCAmelCase_     ):





     """simple docstring"""







     SCREAMING_SNAKE_CASE		,    SCREAMING_SNAKE_CASE 		=cos(lowerCAmelCase_     ), sin(lowerCAmelCase_     )
     return [[c, -s], [s, c]]







def      snake_case__							(  lowerCAmelCase_     ):





     """simple docstring"""







     SCREAMING_SNAKE_CASE		,    SCREAMING_SNAKE_CASE 		=cos(lowerCAmelCase_     ), sin(lowerCAmelCase_     )
     SCREAMING_SNAKE_CASE 		=c * s
     return [[c * c, cs], [cs, s * s]]







def      snake_case__							(  lowerCAmelCase_     ):





     """simple docstring"""







     SCREAMING_SNAKE_CASE		,    SCREAMING_SNAKE_CASE 		=cos(lowerCAmelCase_     ), sin(lowerCAmelCase_     )
     SCREAMING_SNAKE_CASE 		=c * s
     return [[2 * c - 1, 2 * cs], [2 * cs, 2 * s - 1]]


print(f'    {scaling(5) = }')
print(f'  {rotation(45) = }')
print(f'{projection(45) = }')



print(f'{reflection(45) = }')

