






"""simple docstring"""


def 			snake_case  (A_      :list[int]      ):







  '''simple docstring'''

  if len(A_      ) > 1:
    a     :			Optional[Any]				=   len(A_      ) // 2  # Finds the middle of the array
    a     :			Optional[Any]				=   arr[
        :middle_length
    ]  # Creates an array of the elements in the first half.
    a     :			Any				=   arr[
        middle_length:
    ]  # Creates an array of the elements in the second half.
    a     :			List[Any]				=   len(A_      )
    a     :			List[Any]				=   len(A_      )
    merge(A_      )  # Starts sorting the left.
    merge(A_      )  # Starts sorting the right
    a     :			int				=   0  # Left Counter
    a     :			Optional[Any]				=   0  # Right Counter
    a     :			Tuple				=   0  # Position Counter
    while (
        left_index < left_size and right_index < right_size
    ):  # Runs until the lowers size of the left and right are sorted.
      if left_array[left_index] < right_array[right_index]:
        a     :			Union[str, Any]				=   left_array[left_index]
        left_index += 1
      else:
        a     :			Optional[Any]				=   right_array[right_index]
        right_index += 1
      index += 1
    while (
        left_index < left_size
    ):  # Adds the left over elements in the left half of the array
      a     :			List[str]				=   left_array[left_index]
      left_index += 1
      index += 1
    while (
        right_index < right_size
    ):  # Adds the left over elements in the right half of the array
      a     :			List[Any]				=   right_array[right_index]
      right_index += 1
      index += 1
  return arr


if __name__ == "__main__":
   import doctest

   doctest.testmod()





from typing import Any

import numpy as np
def 							lowerCAmelCase(	__lowerCamelCase						):
					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.
					__a					,      __a							=   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 							lowerCAmelCase(	__lowerCamelCase						):
					__a							=   np.random.randn(__lowerCamelCase      ,				__lowerCamelCase						)
					__a							=   np.dot(__lowerCamelCase      ,				random_matrix.T						)
					assert _is_matrix_spd(__lowerCamelCase						)
					return spd_matrix
def 							lowerCAmelCase(	__lowerCamelCase      ,				__lowerCamelCase      ,				__lowerCamelCase = 1000      ,				__lowerCamelCase = 1e-8      ,				):
					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.
					__a							=   np.zeros((np.shape(__lowerCamelCase						)[0], 1)						)
					__a							=   np.copy(__lowerCamelCase						)
					__a							=   np.copy(__lowerCamelCase						)

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

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

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

										# The main algorithm.

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

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

										# Update variables.
										__a							=   np.copy(__lowerCamelCase						)
										__a							=   np.copy(__lowerCamelCase						)
										__a							=   np.copy(__lowerCamelCase						)

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

					return x
def 							lowerCAmelCase(	):
					__a							=   3
					__a							=   _create_spd_matrix(__lowerCamelCase						)
					__a							=   np.random.randn(__lowerCamelCase      ,				1						)
					__a							=   np.dot(__lowerCamelCase      ,				__lowerCamelCase						)

					# Numpy solution.
					__a							=   np.linalg.solve(__lowerCamelCase      ,				__lowerCamelCase						)

					# Our implementation.
					__a							=   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()




