

import numpy as np



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							np.ndarray	,       SCREAMING_SNAKE_CASE_:							np.ndarray	,       SCREAMING_SNAKE_CASE_:							float = 1e-12	,       SCREAMING_SNAKE_CASE_:							int = 1_0_0	,       )			->      tuple[float, np.ndarray]:




				'''simple docstring'''





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

				A__        =			False
				A__        =			0
				A__        =			0
				A__        =			1e12

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

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

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

								A__        =			lambda_

				if is_complex:
								A__        =			np.real(lambda_      )

				return lambda_, vector



def 					lowerCAmelCase__							(	)			->      None:




				'''simple docstring'''





				A__        =			np.array([[4_1, 4, 2_0], [4, 2_6, 3_0], [2_0, 3_0, 5_0]]      )
				A__        =			np.array([4_1, 4, 2_0]      )
				A__        =			real_input_matrix.astype(np.complexaaa      )
				A__        =			np.triu(1j * complex_input_matrix	,       1      )
				complex_input_matrix += imag_matrix
				complex_input_matrix += -1 * imag_matrix.T
				A__        =			np.array([4_1, 4, 2_0]      ).astype(np.complexaaa      )

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

								# Our implementation.
								A__    ,				A__        =			power_iteration(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )

								# Numpy implementation.

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


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	test_power_iteration()




from __future__ import annotations

from collections.abc import Iterator
from typing import Any

class 					a__						:
			"""simple docstring"""







			def __init__(    self      ,      lowercase       )  ->						int:
							'''simple docstring'''






							A__        =			data
							A__        =			None

class 					a__						:
			"""simple docstring"""







			def __init__(    self       )  ->						List[Any]:
							'''simple docstring'''






							A__        =			None
							A__        =			None


			def __iter__(    self       )  ->						Iterator[Any]:
							'''simple docstring'''






							A__        =			self.head
							while self.head:
											yield node.data
											A__        =			node.next
											if node == self.head:
															break


			def __len__(    self       )  ->						int:
							'''simple docstring'''






							return sum(1 for _ in self       )


			def __repr__(    self       )  ->						List[Any]:
							'''simple docstring'''






							return "->".join(str(lowercase       ) for item in iter(self       )       )


			def      UpperCamelCase		(    self      ,      lowercase       )  ->						None:
							'''simple docstring'''






							self.insert_nth(len(self       )      ,      lowercase       )


			def      UpperCamelCase		(    self      ,      lowercase       )  ->						None:
							'''simple docstring'''






							self.insert_nth(0      ,      lowercase       )


			def      UpperCamelCase		(    self      ,      lowercase      ,      lowercase       )  ->						None:
							'''simple docstring'''






							if index < 0 or index > len(self       ):
											raise IndexError("list index out of range."       )
							A__        =			Node(lowercase       )
							if self.head is None:
											A__        =			new_node  # first node points itself
											A__        =			A__        =			new_node
							elif index == 0:  # insert at head
											A__        =			self.head
											A__        =			A__        =			new_node
							else:
											A__        =			self.head
											for _ in range(index - 1       ):
															A__        =			temp.next
											A__        =			temp.next
											A__        =			new_node
											if index == len(self       ) - 1:  # insert at tail
															A__        =			new_node


			def      UpperCamelCase		(    self       )  ->						List[Any]:
							'''simple docstring'''






							return self.delete_nth(0       )


			def      UpperCamelCase		(    self       )  ->						Any:
							'''simple docstring'''






							return self.delete_nth(len(self       ) - 1       )


			def      UpperCamelCase		(    self      ,      lowercase = 0       )  ->						Any:
							'''simple docstring'''






							if not 0 <= index < len(self       ):
											raise IndexError("list index out of range."       )
							A__        =			self.head
							if self.head == self.tail:  # just one node
											A__        =			A__        =			None
							elif index == 0:  # delete head node
											A__        =			self.tail.next.next
											A__        =			self.head.next
							else:
											A__        =			self.head
											for _ in range(index - 1       ):
															A__        =			temp.next
											A__        =			temp.next
											A__        =			temp.next.next
											if index == len(self       ) - 1:  # delete at tail
															A__        =			temp
							return delete_node.data


			def      UpperCamelCase		(    self       )  ->						bool:
							'''simple docstring'''






							return len(self       ) == 0



def 					lowerCAmelCase__							(	)			->      None:




				'''simple docstring'''





				A__        =			CircularLinkedList()
				assert len(SCREAMING_SNAKE_CASE_      ) == 0
				assert circular_linked_list.is_empty() is True
				assert str(SCREAMING_SNAKE_CASE_      ) == ""

				try:
								circular_linked_list.delete_front()
								raise AssertionError  # This should not happen
				except IndexError:
								assert True  # This should happen

				try:
								circular_linked_list.delete_tail()
								raise AssertionError  # This should not happen
				except IndexError:
								assert True  # This should happen

				try:
								circular_linked_list.delete_nth(-1      )
								raise AssertionError
				except IndexError:
								assert True

				try:
								circular_linked_list.delete_nth(0      )
								raise AssertionError
				except IndexError:
								assert True

				assert circular_linked_list.is_empty() is True
				for i in range(5      ):
								assert len(SCREAMING_SNAKE_CASE_      ) == i
								circular_linked_list.insert_nth(SCREAMING_SNAKE_CASE_	,       i + 1      )
				assert str(SCREAMING_SNAKE_CASE_      ) == "->".join(str(SCREAMING_SNAKE_CASE_      ) for i in range(1	,       6      )      )

				circular_linked_list.insert_tail(6      )
				assert str(SCREAMING_SNAKE_CASE_      ) == "->".join(str(SCREAMING_SNAKE_CASE_      ) for i in range(1	,       7      )      )
				circular_linked_list.insert_head(0      )
				assert str(SCREAMING_SNAKE_CASE_      ) == "->".join(str(SCREAMING_SNAKE_CASE_      ) for i in range(0	,       7      )      )

				assert circular_linked_list.delete_front() == 0
				assert circular_linked_list.delete_tail() == 6
				assert str(SCREAMING_SNAKE_CASE_      ) == "->".join(str(SCREAMING_SNAKE_CASE_      ) for i in range(1	,       6      )      )
				assert circular_linked_list.delete_nth(2      ) == 3

				circular_linked_list.insert_nth(2	,       3      )
				assert str(SCREAMING_SNAKE_CASE_      ) == "->".join(str(SCREAMING_SNAKE_CASE_      ) for i in range(1	,       6      )      )

				assert circular_linked_list.is_empty() is False


if __name__ == "__main__":
	import doctest

	doctest.testmod()

