




"""simple docstring"""




import numpy as np







def      __UpperCAmelCase     ( __UpperCamelCase	,				__UpperCamelCase	,				__UpperCamelCase = 1e-12	,				__UpperCamelCase = 1_00	,				):
			assert np.shape(__UpperCamelCase	)[0] == np.shape(__UpperCamelCase	)[1]
			# Ensure proper dimensionality.
			assert np.shape(__UpperCamelCase	)[0] == np.shape(__UpperCamelCase	)[0]
			# Ensure inputs are either both complex or both real
			assert np.iscomplexobj(__UpperCamelCase	) == np.iscomplexobj(__UpperCamelCase	)
			__lowercase			:  Tuple								=       np.iscomplexobj(__UpperCamelCase	)
			if is_complex:
						# Ensure complex input_matrix is Hermitian
						assert np.array_equal(__UpperCamelCase	,				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			:  Tuple								=       False
			__lowercase			:  str								=       0
			__lowercase			:  Dict								=       0
			__lowercase			:  Any								=       1e12

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

						# Check convergence.
						__lowercase			:  Union[str, Any]								=       np.abs(lambda_ - lambda_previous	) / lambda_
						iterations += 1

						if error <= error_tol or iterations >= max_iterations:
									__lowercase			:  Optional[int]								=       True

						__lowercase			:  Dict								=       lambda_

			if is_complex:
						__lowercase			:  Optional[Any]								=       np.real(lambda_	)

			return lambda_, vector







def      __UpperCAmelCase     ( ):
			__lowercase			:  str								=       np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]	)
			__lowercase			:  Any								=       np.array([41, 4, 20]	)
			__lowercase			:  int								=       real_input_matrix.astype(np.complexaaa	)
			__lowercase			:  Union[str, Any]								=       np.triu(1J * complex_input_matrix	,				1	)
			complex_input_matrix += imag_matrix
			complex_input_matrix += -1 * imag_matrix.T
			__lowercase			:  Dict								=       np.array([41, 4, 20]	).astype(np.complexaaa	)

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

						# Our implementation.
						__lowercase			,__lowercase			:  List[Any]								=       power_iteration(__UpperCamelCase	,				__UpperCamelCase	)

						# Numpy implementation.

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


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			test_power_iteration()






"""simple docstring"""
def 					a__						(						)	-> Union[str, Any]:
      UpperCAmelCase__			:							Dict							=    []
      UpperCAmelCase__			:							Tuple							=    1

      while len(lowerCAmelCase    ) < 1E6:
            constant.append(str(lowerCAmelCase    )    )
            i += 1

      UpperCAmelCase__			:							Any							=    """""".join(lowerCAmelCase    )

      return (
          int(constant[0]    )
          * int(constant[9]    )
          * int(constant[99]    )
          * int(constant[9_99]    )
          * int(constant[99_99]    )
          * int(constant[9_99_99]    )
          * int(constant[99_99_99]    )
      )


if __name__ == "__main__":
    print(solution())

