
from __future__ import annotations

from collections import deque





class 							UpperCAmelCase      :




	'''simple docstring'''





	def __init__(       self					: Optional[Any]							,A					: list[str] ):
					__A   			= []
					self.adlist.append(
					    {"value": "", "next_states": [], "fail_state": 0, "output": []} )

					for keyword in keywords:
									self.add_keyword(A )
					self.set_fail_transitions()






	def 				UpperCamelCase_  (       self					: Dict							,A					: int							,A					: str ):
					for state in self.adlist[current_state]["next_states"]:
									if char == self.adlist[state]["value"]:
													return state
					return None






	def 				UpperCamelCase_  (       self					: str							,A					: str ):
					__A   			= 0
					for character in keyword:
									__A   			= self.find_next_state(A							,A )
									if next_state is None:
													self.adlist.append(
													    {
													        "value": character,
													        "next_states": [],
													        "fail_state": 0,
													        "output": [],
													    } )
													self.adlist[current_state]["next_states"].append(len(self.adlist ) - 1 )
													__A   			= len(self.adlist ) - 1
									else:
													__A   			= next_state
					self.adlist[current_state]["output"].append(A )






	def 				UpperCamelCase_  (       self					: str ):
					__A   			= deque()
					for node in self.adlist[0]["next_states"]:
									q.append(A )
									__A   			= 0
					while q:
									__A   			= q.popleft()
									for child in self.adlist[r]["next_states"]:
													q.append(A )
													__A   			= self.adlist[r]["fail_state"]
													while (
													    self.find_next_state(A							,self.adlist[child]["value"] ) is None
													    and state != 0
													):
																	__A   			= self.adlist[state]["fail_state"]
													__A   			= self.find_next_state(
													    A							,self.adlist[child]["value"] )
													if self.adlist[child]["fail_state"] is None:
																	__A   			= 0
													__A   			= (
													    self.adlist[child]["output"]
													    + self.adlist[self.adlist[child]["fail_state"]]["output"]
													)







	def 				UpperCamelCase_  (       self					: Optional[int]							,A					: str ):
					__A   			= {}  # returns a dict with keywords and list of its occurrences
					__A   			= 0
					for i in range(len(A ) ):
									while (
									    self.find_next_state(A							,string[i] ) is None
									    and current_state != 0
									):
													__A   			= self.adlist[current_state]["fail_state"]
									__A   			= self.find_next_state(A							,string[i] )
									if next_state is None:
													__A   			= 0
									else:
													__A   			= next_state
													for key in self.adlist[current_state]["output"]:
																	if key not in result:
																					__A   			= []
																	result[key].append(i - len(A ) + 1 )
					return result


if __name__ == "__main__":
					import doctest

					doctest.testmod()


import numpy as np


def 						UpperCAmelCase			(	a_		,   a_		,   a_ = 1E-12		,   a_ = 1_0_0		,   )		->	tuple[float, np.ndarray]:







				"""simple docstring"""



				assert np.shape(a_						)[0] == np.shape(a_						)[1]
				# Ensure proper dimensionality.
				assert np.shape(a_						)[0] == np.shape(a_						)[0]
				# Ensure inputs are either both complex or both real
				assert np.iscomplexobj(a_						) == np.iscomplexobj(a_						)
				__A   			= np.iscomplexobj(a_						)
				if is_complex:
								# Ensure complex input_matrix is Hermitian
								assert np.array_equal(a_		,   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(a_		,   a_						)
								# Normalize the resulting output vector.
								__A   			= w / np.linalg.norm(a_						)
								# 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(a_		,   np.dot(a_		,   a_						)						)

								# 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 						UpperCAmelCase			(	)		->	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(a_		,   a_						)

								# Numpy implementation.

								# Get eigenvalues and eigenvectors using built-in numpy
								# eigh (eigh used for symmetric or hermetian matrices).
								__A			,							__A   			= np.linalg.eigh(a_						)
								# 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(a_						) - np.abs(a_						)						) <= 1E-6


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					test_power_iteration()
