


"""simple docstring"""






import numpy as np





def 				A_       ( snake_case__       ,					snake_case__       ,					snake_case__ = 1E-12       ,					snake_case__ = 1_00       ,					)					->  tuple[float, np.ndarray]:
			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__		)
			_UpperCamelCase					:Optional[int]					     =	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.

			_UpperCamelCase					:Optional[Any]					     =	False
			_UpperCamelCase					:Union[str, Any]					     =	0
			_UpperCamelCase					:Union[str, Any]					     =	0
			_UpperCamelCase					:Optional[Any]					     =	1E12

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

						# Check convergence.
						_UpperCamelCase					:Optional[int]					     =	np.abs(lambda_ - lambda_previous		) / lambda_
						iterations += 1

						if error <= error_tol or iterations >= max_iterations:
									_UpperCamelCase					:Union[str, Any]					     =	True

						_UpperCamelCase					:Optional[Any]					     =	lambda_

			if is_complex:
						_UpperCamelCase					:Any					     =	np.real(lambda_		)

			return lambda_, vector





def 				A_       ( )					->  None:
			_UpperCamelCase					:Optional[Any]					     =	np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]		)
			_UpperCamelCase					:int					     =	np.array([41, 4, 20]		)
			_UpperCamelCase					:Tuple					     =	real_input_matrix.astype(np.complexaaa		)
			_UpperCamelCase					:Optional[int]					     =	np.triu(1J * complex_input_matrix       ,					1		)
			complex_input_matrix += imag_matrix
			complex_input_matrix += -1 * imag_matrix.T
			_UpperCamelCase					:List[str]					     =	np.array([41, 4, 20]		).astype(np.complexaaa		)

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

						# Our implementation.
						_UpperCamelCase     ,      _UpperCamelCase					:Union[str, Any]					     =	power_iteration(snake_case__       ,					snake_case__		)

						# Numpy implementation.

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





"""simple docstring"""






from __future__ import annotations

from math import pi
from typing import Protocol

import matplotlib.pyplot as plt
import numpy as np






class      A(  lowerCamelCase__		):

		"""simple docstring"""





		def   _UpperCamelCase( self					,				SCREAMING_SNAKE_CASE__   )      ->	float:

					"""simple docstring"""



					return 0.0





def 				A_       ( snake_case__       ,					snake_case__		)					->  tuple[int | float, int | float]:
			_UpperCamelCase					:Union[str, Any]					     =	min([-20, np.min(fft_results[1 : samplerate // 2 - 1]		)]		)
			_UpperCamelCase					:Optional[int]					     =	max([20, np.max(fft_results[1 : samplerate // 2 - 1]		)]		)
			return lowest, highest





def 				A_       ( snake_case__       ,					snake_case__		)					->  None:
			_UpperCamelCase					:List[str]					     =	5_12
			_UpperCamelCase					:int					     =	[1] + [0] * (size - 1)
			_UpperCamelCase					:Union[str, Any]					     =	[filter_type.process(snake_case__		) for item in inputs]

			_UpperCamelCase					:Any					     =	[0] * (samplerate - size)  # zero-padding
			outputs += filler
			_UpperCamelCase					:int					     =	np.abs(np.fft.fft(snake_case__		)		)
			_UpperCamelCase					:str					     =	20 * np.logaa(snake_case__		)

			# Frequencies on log scale from 24 to nyquist frequency
			plt.xlim(24       ,					samplerate / 2 - 1		)
			plt.xlabel('''Frequency (Hz)'''		)
			plt.xscale('''log'''		)

			# Display within reasonable bounds
			_UpperCamelCase					:Tuple					     =	get_bounds(snake_case__       ,					snake_case__		)
			plt.ylim(max([-80, bounds[0]]		)       ,					min([80, bounds[1]]		)		)
			plt.ylabel('''Gain (dB)'''		)

			plt.plot(snake_case__		)
			plt.show()





def 				A_       ( snake_case__       ,					snake_case__		)					->  None:
			_UpperCamelCase					:Tuple					     =	5_12
			_UpperCamelCase					:Union[str, Any]					     =	[1] + [0] * (size - 1)
			_UpperCamelCase					:List[Any]					     =	[filter_type.process(snake_case__		) for item in inputs]

			_UpperCamelCase					:str					     =	[0] * (samplerate - size)  # zero-padding
			outputs += filler
			_UpperCamelCase					:Tuple					     =	np.angle(np.fft.fft(snake_case__		)		)

			# Frequencies on log scale from 24 to nyquist frequency
			plt.xlim(24       ,					samplerate / 2 - 1		)
			plt.xlabel('''Frequency (Hz)'''		)
			plt.xscale('''log'''		)

			plt.ylim(-2 * pi       ,					2 * pi		)
			plt.ylabel('''Phase shift (Radians)'''		)
			plt.plot(np.unwrap(snake_case__       ,					-2 * pi		)		)
			plt.show()

