"""simple docstring"""



from typing import Any

import numpy as np

def    _A(		lowerCAmelCase       ):
							assert np.shape(lowerCAmelCase       )[0] == np.shape(lowerCAmelCase       )[1]

							# If matrix not symmetric, exit right away.
							if np.allclose(lowerCAmelCase ,     matrix.T       ) is False:
														return False

							# Get eigenvalues and eignevectors for a symmetric matrix.
							A__							,       A__					:    str														= np.linalg.eigh(lowerCAmelCase       )

							# Check sign of all eigenvalues.
							# np.all returns a value of type np.bool_
							return bool(np.all(eigen_values > 0       )       )

def    _A(		lowerCAmelCase       ):
							A__					:    str														= np.random.randn(lowerCAmelCase ,     lowerCAmelCase       )
							A__					:    Dict														= np.dot(lowerCAmelCase ,     random_matrix.T       )
							assert _is_matrix_spd(lowerCAmelCase       )
							return spd_matrix

def    _A(		lowerCAmelCase ,     lowerCAmelCase ,     lowerCAmelCase = 1000 ,     lowerCAmelCase = 1E-8 ,     ):
							assert np.shape(lowerCAmelCase       )[0] == np.shape(lowerCAmelCase       )[1]
							assert np.shape(lowerCAmelCase       )[0] == np.shape(lowerCAmelCase       )[0]
							assert _is_matrix_spd(lowerCAmelCase       )

							# Initialize solution guess, residual, search direction.
							A__					:    str														= np.zeros((np.shape(lowerCAmelCase       )[0], 1)       )
							A__					:    int														= np.copy(lowerCAmelCase       )
							A__					:    str														= np.copy(lowerCAmelCase       )

							# Set initial errors in solution guess and residual.
							A__					:    Tuple														= 1E9
							A__					:    Dict														= 1E9
							A__					:    Optional[Any]														= 1E9

							# Set iteration counter to threshold number of iterations.
							A__					:    Any														= 0

							while error > tol:
														# Save this value so we only calculate the matrix-vector product once.
														A__					:    Any														= np.dot(lowerCAmelCase ,     lowerCAmelCase       )

														# The main algorithm.

														# Update search direction magnitude.
														A__					:    List[Any]														= np.dot(ra.T ,     lowerCAmelCase       ) / np.dot(pa.T ,     lowerCAmelCase       )
														# Update solution guess.
														A__					:    Dict														= xa + alpha * pa
														# Calculate new residual.
														A__					:    Any														= ra - alpha * w
														# Calculate new Krylov subspace scale.
														A__					:    Optional[int]														= np.dot(r.T ,     lowerCAmelCase       ) / np.dot(ra.T ,     lowerCAmelCase       )
														# Calculate new A conjuage search direction.
														A__					:    Any														= r + beta * pa

														# Calculate errors.
														A__					:    Any														= np.linalg.norm(r - ra       )
														A__					:    Optional[Any]														= np.linalg.norm(x - xa       )
														A__					:    Any														= np.maximum(lowerCAmelCase ,     lowerCAmelCase       )

														# Update variables.
														A__					:    Tuple														= np.copy(lowerCAmelCase       )
														A__					:    Any														= np.copy(lowerCAmelCase       )
														A__					:    Optional[int]														= np.copy(lowerCAmelCase       )

														# Update number of iterations.
														iterations += 1
														if iterations > max_iterations:
																					break

							return x

def    _A(		):
							A__					:    List[str]														= 3
							A__					:    Tuple														= _create_spd_matrix(lowerCAmelCase       )
							A__					:    Optional[int]														= np.random.randn(lowerCAmelCase ,     1       )
							A__					:    List[Any]														= np.dot(lowerCAmelCase ,     lowerCAmelCase       )

							# Numpy solution.
							A__					:    int														= np.linalg.solve(lowerCAmelCase ,     lowerCAmelCase       )

							# Our implementation.
							A__					:    Union[str, Any]														= conjugate_gradient(lowerCAmelCase ,     lowerCAmelCase       )

							# Ensure both solutions are close to x_true (and therefore one another).
							assert np.linalg.norm(x_numpy - x_true       ) <= 1E-6
							assert np.linalg.norm(x_conjugate_gradient - x_true       ) <= 1E-6


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				test_conjugate_gradient()






"""simple docstring"""



from dataclasses import dataclass

_UpperCamelCase         =		"0.1"
_UpperCamelCase         =		"Lucia Harcekova"
@dataclass
class 	__UpperCAmelCase						:







			'''simple docstring'''
			_UpperCamelCase     :	int
			_UpperCamelCase     :	int
			_UpperCamelCase     :	str





			def __repr__( self       ):







										'''simple docstring'''


										return F'''({self.offset}, {self.length}, {self.indicator})'''





class 	__UpperCAmelCase						:







			'''simple docstring'''





			def __init__( self      ,			snake_case_ = 13      ,			snake_case_ = 6       ):







										'''simple docstring'''


										A__					:    str														= window_size
										A__					:    Optional[int]														= lookahead_buffer_size
										A__					:    Optional[Any]														= self.window_size - self.lookahead_buffer_size





			def 		lowerCamelCase				( self      ,			snake_case_       ):







										'''simple docstring'''


										A__					:    str														= []
										A__					:    Any														= """"""

										# while there are still characters in text to compress
										while text:
																	# find the next encoding phrase
																	# - triplet with offset, length, indicator (the next encoding character)
																	A__					:    Optional[Any]														= self._find_encoding_token(snake_case_      ,			snake_case_       )

																	# update the search buffer:
																	# - add new characters from text into it
																	# - check if size exceed the max search buffer size, if so, drop the
																	#   oldest elements
																	search_buffer += text[: token.length + 1]
																	if len(snake_case_       ) > self.search_buffer_size:
																								A__					:    Any														= search_buffer[-self.search_buffer_size :]

																	# update the text
																	A__					:    Optional[Any]														= text[token.length + 1 :]

																	# append the token to output
																	output.append(snake_case_       )

										return output





			def 		lowerCamelCase				( self      ,			snake_case_       ):







										'''simple docstring'''


										A__					:    List[Any]														= """"""

										for token in tokens:
																	for _ in range(token.length       ):
																								output += output[-token.offset]
																	output += token.indicator

										return output





			def 		lowerCamelCase				( self      ,			snake_case_      ,			snake_case_       ):







										'''simple docstring'''


										if not text:
																	raise ValueError("""We need some text to work with."""       )

										# Initialise result parameters to default values
										A__							,       A__					:    List[Any]														= 0, 0

										if not search_buffer:
																	return Token(snake_case_      ,			snake_case_      ,			text[length]       )

										for i, character in enumerate(snake_case_       ):
																	A__					:    Tuple														= len(snake_case_       ) - i
																	if character == text[0]:
																								A__					:    Tuple														= self._match_length_from_index(snake_case_      ,			snake_case_      ,			0      ,			snake_case_       )
																								# if the found length is bigger than the current or if it's equal,
																								# which means it's offset is smaller: update offset and length
																								if found_length >= length:
																															A__							,       A__					:    List[str]														= found_offset, found_length

										return Token(snake_case_      ,			snake_case_      ,			text[length]       )





			def 		lowerCamelCase				( self      ,			snake_case_      ,			snake_case_      ,			snake_case_      ,			snake_case_       ):







										'''simple docstring'''


										if not text or text[text_index] != window[window_index]:
																	return 0
										return 1 + self._match_length_from_index(
										    snake_case_      ,			window + text[text_index]      ,			text_index + 1      ,			window_index + 1       )


if __name__ == "__main__":
				from doctest import testmod

				testmod()
				# Initialize compressor class
				_UpperCamelCase         =		LZaaCompressor(window_size=13, lookahead_buffer_size=6)

				# Example
				_UpperCamelCase         =		"cabracadabrarrarrad"
				_UpperCamelCase         =		lzaa_compressor.compress(TEXT)
				print(lzaa_compressor.compress("ababcbababaa"))
				_UpperCamelCase         =		lzaa_compressor.decompress(compressed_text)
				assert decompressed_text == TEXT, "The LZ77 algorithm returned the invalid result."





