

import numpy as np







def       A_				(   _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   )
						SCREAMING_SNAKE_CASE_:			Optional[Any]  		=	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.

						SCREAMING_SNAKE_CASE_:			Dict  		=	False
						SCREAMING_SNAKE_CASE_:			Optional[Any]  		=	0
						SCREAMING_SNAKE_CASE_:			List[str]  		=	0
						SCREAMING_SNAKE_CASE_:			Optional[Any]  		=	1e12

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

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

												if error <= error_tol or iterations >= max_iterations:
																		SCREAMING_SNAKE_CASE_:			Tuple  		=	True

												SCREAMING_SNAKE_CASE_:			List[str]  		=	lambda_

						if is_complex:
												SCREAMING_SNAKE_CASE_:			str  		=	np.real(lambda_   )

						return lambda_, vector







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

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

												# Our implementation.
												SCREAMING_SNAKE_CASE_		,   SCREAMING_SNAKE_CASE_:			Any  		=	power_iteration(_UpperCAmelCase    ,     _UpperCAmelCase   )

												# Numpy implementation.

												# Get eigenvalues and eigenvectors using built-in numpy
												# eigh (eigh used for symmetric or hermetian matrices).
												SCREAMING_SNAKE_CASE_		,   SCREAMING_SNAKE_CASE_:			Optional[int]  		=	np.linalg.eigh(_UpperCAmelCase   )
												# Last eigenvalue is the maximum one.
												SCREAMING_SNAKE_CASE_:			Optional[int]  		=	eigen_values[-1]
												# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
												SCREAMING_SNAKE_CASE_:			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()









import doctest
from collections import deque

import numpy as np



class   __lowercase    :




			"""simple docstring"""







			def __init__(			self  :					Union[str, Any]):
									SCREAMING_SNAKE_CASE_:			int  		=	[2, 1, 2, -1]
									SCREAMING_SNAKE_CASE_:			Optional[Any]  		=	[1, 2, 3, 4]




			def 	_SCREAMING_SNAKE_CASE      (			self  :					Tuple):
									SCREAMING_SNAKE_CASE_:			Any  		=	len(self.first_signal)
									SCREAMING_SNAKE_CASE_:			Dict  		=	len(self.second_signal)

									SCREAMING_SNAKE_CASE_:			Union[str, Any]  		=	max(lowerCAmelCase__       ,						lowerCAmelCase__)

									# create a zero matrix of max_length x max_length
									SCREAMING_SNAKE_CASE_:			List[Any]  		=	[[0] * max_length for i in range(lowerCAmelCase__)]

									# fills the smaller signal with zeros to make both signals of same length
									if length_first_signal < length_second_signal:
															self.first_signal += [0] * (max_length - length_first_signal)
									elif length_first_signal > length_second_signal:
															self.second_signal += [0] * (max_length - length_second_signal)
									for i in range(lowerCAmelCase__):
															SCREAMING_SNAKE_CASE_:			Tuple  		=	deque(self.second_signal)
															rotated_signal.rotate(lowerCAmelCase__)
															for j, item in enumerate(lowerCAmelCase__):
																					matrix[i][j] += item

        # multiply the matrix with the first signal
									SCREAMING_SNAKE_CASE_:			Optional[Any]  		=	np.matmul(np.transpose(lowerCAmelCase__)       ,						np.transpose(self.first_signal))

									# rounding-off to two decimal places
									return [round(lowerCAmelCase__       ,						2) for i in final_signal]


if __name__ == "__main__":
						doctest.testmod()






