from typing import Any

import numpy as np
def 							lowerCAmelCase(	__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							=   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 							lowerCAmelCase(	__lowerCamelCase						):
					__a							=   np.random.randn(__lowerCamelCase      ,				__lowerCamelCase						)
					__a							=   np.dot(__lowerCamelCase      ,				random_matrix.T						)
					assert _is_matrix_spd(__lowerCamelCase						)
					return spd_matrix
def 							lowerCAmelCase(	__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							=   np.zeros((np.shape(__lowerCamelCase						)[0], 1)						)
					__a							=   np.copy(__lowerCamelCase						)
					__a							=   np.copy(__lowerCamelCase						)

					# Set initial errors in solution guess and residual.
					__a							=   1e9
					__a							=   1e9
					__a							=   1e9

					# Set iteration counter to threshold number of iterations.
					__a							=   0

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

										# The main algorithm.

										# Update search direction magnitude.
										__a							=   np.dot(ra.T      ,				__lowerCamelCase						) / np.dot(pa.T      ,				__lowerCamelCase						)
										# Update solution guess.
										__a							=   xa + alpha * pa
										# Calculate new residual.
										__a							=   ra - alpha * w
										# Calculate new Krylov subspace scale.
										__a							=   np.dot(r.T      ,				__lowerCamelCase						) / np.dot(ra.T      ,				__lowerCamelCase						)
										# Calculate new A conjuage search direction.
										__a							=   r + beta * pa

										# Calculate errors.
										__a							=   np.linalg.norm(r - ra						)
										__a							=   np.linalg.norm(x - xa						)
										__a							=   np.maximum(__lowerCamelCase      ,				__lowerCamelCase						)

										# Update variables.
										__a							=   np.copy(__lowerCamelCase						)
										__a							=   np.copy(__lowerCamelCase						)
										__a							=   np.copy(__lowerCamelCase						)

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

					return x
def 							lowerCAmelCase(	):
					__a							=   3
					__a							=   _create_spd_matrix(__lowerCamelCase						)
					__a							=   np.random.randn(__lowerCamelCase      ,				1						)
					__a							=   np.dot(__lowerCamelCase      ,				__lowerCamelCase						)

					# Numpy solution.
					__a							=   np.linalg.solve(__lowerCamelCase      ,				__lowerCamelCase						)

					# Our implementation.
					__a							=   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 functools import lru_cache






@lru_cache
def 		_lowercase		(	lowercase_  ):



		'''simple docstring'''







		if not isinstance(lowercase_						,							lowercase_  ):
				raise TypeError("k must be an integer."  )
		if k < 0:
				raise ValueError("k integer must be greater or equal to zero."  )
		if k == 0:
				return 0
		elif k == 1:
				return 1
		else:
				return fibonacci(k - 1  ) + fibonacci(k - 2  )






def 		_lowercase		(	lowercase_						,							lowercase_  ):



		'''simple docstring'''







		_snake_case	       =       len(lowercase_  )
		# Find m such that F_m >= n where F_i is the i_th fibonacci number.
		_snake_case	       =       0
		while True:
				if fibonacci(lowercase_  ) >= len_list:
						_snake_case	       =       i
						break
				i += 1
		_snake_case	       =       0
		while fibb_k > 0:
				_snake_case	       =       min(
				    offset + fibonacci(fibb_k - 1  )						,							len_list - 1  )  # Prevent out of range
				_snake_case	       =       arr[index_k]
				if item_k_1 == val:
						return index_k
				elif val < item_k_1:
						fibb_k -= 1
				elif val > item_k_1:
						offset += fibonacci(fibb_k - 1  )
						fibb_k -= 2
		else:
				return -1


if __name__ == "__main__":
						import doctest

						doctest.testmod()





