

'''simple docstring'''






import numpy as np


def 	_A  (					A__  ,       A__  ,       A__ = 1e-12  ,       A__ = 100  ,       ):






 """simple docstring"""



 assert np.shape(A__   )[0] == np.shape(A__   )[1]
 # Ensure proper dimensionality.
 assert np.shape(A__   )[0] == np.shape(A__   )[0]
 # Ensure inputs are either both complex or both real
 assert np.iscomplexobj(A__   ) == np.iscomplexobj(A__   )
 __lowercase													=				np.iscomplexobj(A__   )
 if is_complex:
  # Ensure complex input_matrix is Hermitian
  assert np.array_equal(A__  ,       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.

 __lowercase													=				False
 __lowercase													=				0
 __lowercase													=				0
 __lowercase													=				1e12

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

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

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

  __lowercase													=				lambda_

 if is_complex:
  __lowercase													=				np.real(lambda_   )

 return lambda_, vector


def 	_A  (					):






 """simple docstring"""



 __lowercase													=				np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]   )
 __lowercase													=				np.array([41, 4, 20]   )
 __lowercase													=				real_input_matrix.astype(np.complexaaa   )
 __lowercase													=				np.triu(1j * complex_input_matrix  ,       1   )
 complex_input_matrix += imag_matrix
 complex_input_matrix += -1 * imag_matrix.T
 __lowercase													=				np.array([41, 4, 20]   ).astype(np.complexaaa   )

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

  # Our implementation.
  __lowercase						,			__lowercase													=				power_iteration(A__  ,       A__   )

  # Numpy implementation.

  # Get eigenvalues and eigenvectors using built-in numpy
  # eigh (eigh used for symmetric or hermetian matrices).
  __lowercase						,			__lowercase													=				np.linalg.eigh(A__   )
  # Last eigenvalue is the maximum one.
  __lowercase													=				eigen_values[-1]
  # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
  __lowercase													=				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(A__   ) - np.abs(A__   )   ) <= 1e-6


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 test_power_iteration()




'''simple docstring'''






from collections.abc import Callable

import numpy as np


def 	_A  (					A__  ,       A__  ,       A__  ,       A__  ,       A__   ):






 """simple docstring"""



 __lowercase													=				int(np.ceil((x_end - xa) / step_size   )   )
 __lowercase													=				np.zeros((n + 1,)   )
 __lowercase													=				ya
 __lowercase													=				xa

 for k in range(A__   ):
  __lowercase													=				y[k] + step_size * ode_func(A__  ,       y[k]   )
  x += step_size

 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()

