
def 		lowerCAmelCase_   (	lowerCamelCase   ):
				try:
								__magic_name__	:	int								=float(lowerCamelCase   )
				except ValueError:
								raise ValueError("""Please enter a valid number"""   )
				__magic_name__	:	List[Any]								=decimal - int(lowerCamelCase   )
				if fractional_part == 0:
								return int(lowerCamelCase   ), 1
				else:
								__magic_name__	:	Union[str, Any]								=len(str(lowerCamelCase   ).split("""."""   )[1]   )
								__magic_name__	:	int								=int(decimal * (10**number_of_frac_digits)   )
								__magic_name__	:	Any								=10**number_of_frac_digits
								__magic_name__				, __magic_name__	:	str								=denominator, numerator
								while True:
												__magic_name__	:	int								=dividend % divisor
												if remainder == 0:
																break
												__magic_name__				, __magic_name__	:	int								=divisor, remainder
								__magic_name__				, __magic_name__	:	str								=numerator / divisor, denominator / divisor
								return int(lowerCamelCase   ), int(lowerCamelCase   )


if __name__ == "__main__":
							print(F"""{decimal_to_fraction(2) = }""")
							print(F"""{decimal_to_fraction(89.0) = }""")
							print(F"""{decimal_to_fraction("67") = }""")
							print(F"""{decimal_to_fraction("45.0") = }""")
							print(F"""{decimal_to_fraction(1.5) = }""")
							print(F"""{decimal_to_fraction("6.25") = }""")
							print(F"""{decimal_to_fraction("78td") = }""")










import numpy as np




def 		lowerCAmelCase_   (	lowerCamelCase	,     lowerCamelCase	,     lowerCamelCase = 1E-12	,     lowerCamelCase = 100	,     ):
				assert np.shape(lowerCamelCase   )[0] == np.shape(lowerCamelCase   )[1]
				# Ensure proper dimensionality.
				assert np.shape(lowerCamelCase   )[0] == np.shape(lowerCamelCase   )[0]
				# Ensure inputs are either both complex or both real
				assert np.iscomplexobj(lowerCamelCase   ) == np.iscomplexobj(lowerCamelCase   )
				__magic_name__	:	List[Any]								=np.iscomplexobj(lowerCamelCase   )
				if is_complex:
								# Ensure complex input_matrix is Hermitian
								assert np.array_equal(lowerCamelCase	,     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.

				__magic_name__	:	List[Any]								=False
				__magic_name__	:	List[Any]								=0
				__magic_name__	:	List[str]								=0
				__magic_name__	:	str								=1E12

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

								# Check convergence.
								__magic_name__	:	List[str]								=np.abs(lambda_ - lambda_previous   ) / lambda_
								iterations += 1

								if error <= error_tol or iterations >= max_iterations:
												__magic_name__	:	int								=True

								__magic_name__	:	Optional[Any]								=lambda_

				if is_complex:
								__magic_name__	:	Union[str, Any]								=np.real(lambda_   )

				return lambda_, vector




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

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

								# Our implementation.
								__magic_name__				, __magic_name__	:	Any								=power_iteration(lowerCamelCase	,     lowerCamelCase   )

								# Numpy implementation.

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


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							test_power_iteration()




