



from __future__ import annotations







def     UpperCAmelCase_		(					snake_case__    ,			snake_case__					)	->     bool:



				"""simple docstring"""





				lowerCAmelCase__		  =  get_failure_array(snake_case__					)

				# 2) Step through text searching for pattern
				lowerCAmelCase__  ,   lowerCAmelCase__		  =  0, 0  # index into text, pattern
				while i < len(snake_case__					):
								if pattern[j] == text[i]:
												if j == (len(snake_case__					) - 1):
																return True
												j += 1

								# if this is a prefix in our pattern
								# just go back far enough to continue
								elif j > 0:
												lowerCAmelCase__		  =  failure[j - 1]
												continue
								i += 1
				return False







def     UpperCAmelCase_		(					snake_case__					)	->     list[int]:



				"""simple docstring"""





				lowerCAmelCase__		  =  [0]
				lowerCAmelCase__		  =  0
				lowerCAmelCase__		  =  1
				while j < len(snake_case__					):
								if pattern[i] == pattern[j]:
												i += 1
								elif i > 0:
												lowerCAmelCase__		  =  failure[i - 1]
												continue
								j += 1
								failure.append(snake_case__					)
				return failure


if __name__ == "__main__":
	# Test 1)
	_lowerCAmelCase	:   str    =   "abc1abc12"
	_lowerCAmelCase	:   Union[str, Any]    =   "alskfjaldsabc1abc1abc12k23adsfabcabc"
	_lowerCAmelCase	:   int    =   "alskfjaldsk23adsfabcabc"
	assert kmp(pattern, texta) and not kmp(pattern, texta)

	# Test 2)
	_lowerCAmelCase	:   str    =   "ABABX"
	_lowerCAmelCase	:   Any    =   "ABABZABABYABABX"
	assert kmp(pattern, text)

	# Test 3)
	_lowerCAmelCase	:   str    =   "AAAB"
	_lowerCAmelCase	:   Tuple    =   "ABAAAAAB"
	assert kmp(pattern, text)

	# Test 4)
	_lowerCAmelCase	:   List[str]    =   "abcdabcy"
	_lowerCAmelCase	:   List[Any]    =   "abcxabcdabxabcdabcdabcy"
	assert kmp(pattern, text)

	# Test 5)
	_lowerCAmelCase	:   Dict    =   "aabaabaaa"
	assert get_failure_array(pattern) == [0, 1, 0, 1, 2, 3, 4, 5, 2]





import numpy as np







def     UpperCAmelCase_		(					snake_case__    ,			snake_case__    ,			snake_case__ = 1E-12    ,			snake_case__ = 100    ,			)	->     tuple[float, np.ndarray]:



				"""simple docstring"""





				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__					)
				lowerCAmelCase__		  =  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.

				lowerCAmelCase__		  =  False
				lowerCAmelCase__		  =  0
				lowerCAmelCase__		  =  0
				lowerCAmelCase__		  =  1E12

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

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

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

								lowerCAmelCase__		  =  lambda_

				if is_complex:
								lowerCAmelCase__		  =  np.real(lambda_					)

				return lambda_, vector







def     UpperCAmelCase_		(					)	->     None:



				"""simple docstring"""





				lowerCAmelCase__		  =  np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]					)
				lowerCAmelCase__		  =  np.array([41, 4, 20]					)
				lowerCAmelCase__		  =  real_input_matrix.astype(np.complexaaa					)
				lowerCAmelCase__		  =  np.triu(1j * complex_input_matrix    ,			1					)
				complex_input_matrix += imag_matrix
				complex_input_matrix += -1 * imag_matrix.T
				lowerCAmelCase__		  =  np.array([41, 4, 20]					).astype(np.complexaaa					)

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

								# Our implementation.
								lowerCAmelCase__  ,   lowerCAmelCase__		  =  power_iteration(snake_case__    ,			snake_case__					)

								# Numpy implementation.

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