
def   __lowerCAmelCase    (				UpperCAmelCase__					:   Tuple      ,					UpperCAmelCase__					:   Tuple							)	->     bool:
							lowerCamelCase_					=       len(snake_case__							)
							lowerCamelCase_					=       [[False] * (required_sum + 1) for _ in range(arr_len + 1							)]

							# for each arr value, a sum of zero(0) can be formed by not taking any element
							# hence True/1
							for i in range(arr_len + 1							):
														lowerCamelCase_					=       True

							# sum is not zero and set is empty then false
							for i in range(1      ,					required_sum + 1							):
														lowerCamelCase_					=       False

							for i in range(1      ,					arr_len + 1							):
														for j in range(1      ,					required_sum + 1							):
																					if arr[i - 1] > j:
																												lowerCamelCase_					=       subset[i - 1][j]
																					if arr[i - 1] <= j:
																												lowerCamelCase_					=       subset[i - 1][j] or subset[i - 1][j - arr[i - 1]]

							return subset[arr_len][required_sum]


if __name__ == "__main__":
		import doctest

		doctest.testmod()










import numpy as np







def     UpperCAmelCase_		(					snake_case__    ,			snake_case__    ,			snake_case__ = 1E-12    ,			snake_case__ = 100    ,			)	->     tuple[float, np.ndarray]:



				"""simple docstring"""





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

				lowerCAmelCase__		  =  False
				lowerCAmelCase__		  =  0
				lowerCAmelCase__		  =  0
				lowerCAmelCase__		  =  1E12

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

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

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

								lowerCAmelCase__		  =  lambda_

				if is_complex:
								lowerCAmelCase__		  =  np.real(lambda_					)

				return lambda_, vector







def     UpperCAmelCase_		(					)	->     None:



				"""simple docstring"""





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

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

								# Our implementation.
								lowerCAmelCase__  ,   lowerCAmelCase__		  =  power_iteration(snake_case__    ,			snake_case__					)

								# Numpy implementation.

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


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	test_power_iteration()
