






"""simple docstring"""






import numpy as np



def  __lowerCamelCase					(					lowerCAmelCase__       ,lowerCAmelCase__       ,lowerCAmelCase__ = 1E-1_2       ,lowerCAmelCase__ = 100       ,):
				assert np.shape(lowerCAmelCase__				)[0] == np.shape(lowerCAmelCase__				)[1]
				# Ensure proper dimensionality.
				assert np.shape(lowerCAmelCase__				)[0] == np.shape(lowerCAmelCase__				)[0]
				# Ensure inputs are either both complex or both real
				assert np.iscomplexobj(lowerCAmelCase__				) == np.iscomplexobj(lowerCAmelCase__				)
				A__							 =       np.iscomplexobj(lowerCAmelCase__				)
				if is_complex:
								# Ensure complex input_matrix is Hermitian
								assert np.array_equal(lowerCAmelCase__       ,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__							 =       1E1_2

				while not convergence:
								# Multiple matrix by the vector.
								A__							 =       np.dot(lowerCAmelCase__       ,lowerCAmelCase__				)
								# Normalize the resulting output vector.
								A__							 =       w / np.linalg.norm(lowerCAmelCase__				)
								# 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(lowerCAmelCase__       ,np.dot(lowerCAmelCase__       ,lowerCAmelCase__				)				)

								# 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					(					):
				A__							 =       np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]				)
				A__							 =       np.array([41, 4, 20]				)
				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([41, 4, 20]				).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(lowerCAmelCase__       ,lowerCAmelCase__				)

								# Numpy implementation.

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


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					test_power_iteration()










"""simple docstring"""






from __future__ import annotations



def  __lowerCamelCase					(					lowerCAmelCase__       ,lowerCAmelCase__ = None				):
				A__							 =       word_bank or []
				# create a table
				A__							 =       len(lowerCAmelCase__				) + 1

				A__							 =       []
				for _ in range(lowerCAmelCase__				):
								table.append([]				)
				# seed value
				A__							 =       [[]]  # because empty string has empty combination

				# iterate through the indices
				for i in range(lowerCAmelCase__				):
								# condition
								if table[i] != []:
												for word in word_bank:
																# slice condition
																if target[i : i + len(lowerCAmelCase__				)] == word:
																				A__							 =       [
																				    [word, *way] for way in table[i]
																				]
																				# adds the word to every combination the current position holds
																				# now,push that combination to the table[i+len(word)]
																				table[i + len(lowerCAmelCase__				)] += new_combinations

    # combinations are in reverse order so reverse for better output
				for combination in table[len(lowerCAmelCase__				)]:
								combination.reverse()

				return table[len(lowerCAmelCase__				)]


if __name__ == "__main__":
					print(all_construct('''jwajalapa''', ['''jwa''', '''j''', '''w''', '''a''', '''la''', '''lapa''']))
					print(all_construct('''rajamati''', ['''s''', '''raj''', '''amat''', '''raja''', '''ma''', '''i''', '''t''']))
					print(
					    all_construct(
					        '''hexagonosaurus''',
					        ['''h''', '''ex''', '''hex''', '''ag''', '''ago''', '''ru''', '''auru''', '''rus''', '''go''', '''no''', '''o''', '''s'''],
					    )
					)


