



from typing import Any

import numpy as np






def       snake_case      (							A    )							->							bool:




	'''simple docstring'''


	assert np.shape(A    )[0] == np.shape(A    )[1]

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

	# Get eigenvalues and eignevectors for a symmetric matrix.
	lowerCamelCase_							:	Any							     =							np.linalg.eigh(A    )

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






def       snake_case      (							A    )							->							Any:




	'''simple docstring'''


	lowerCamelCase_							:	Tuple							     =							np.random.randn(A		,     A    )
	lowerCamelCase_							:	List[Any]							     =							np.dot(A		,     random_matrix.T    )
	assert _is_matrix_spd(A    )
	return spd_matrix






def       snake_case      (							A		,     A		,     A = 1000		,     A = 1e-8		,     )							->							Any:




	'''simple docstring'''


	assert np.shape(A    )[0] == np.shape(A    )[1]
	assert np.shape(A    )[0] == np.shape(A    )[0]
	assert _is_matrix_spd(A    )

	# Initialize solution guess, residual, search direction.
	lowerCamelCase_							:	List[Any]							     =							np.zeros((np.shape(A    )[0], 1)    )
	lowerCamelCase_							:	List[str]							     =							np.copy(A    )
	lowerCamelCase_							:	Optional[int]							     =							np.copy(A    )

	# Set initial errors in solution guess and residual.
	lowerCamelCase_							:	List[str]							     =							1e9
	lowerCamelCase_							:	int							     =							1e9
	lowerCamelCase_							:	List[str]							     =							1e9

	# Set iteration counter to threshold number of iterations.
	lowerCamelCase_							:	Optional[int]							     =							0

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

		# The main algorithm.

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

		# Calculate errors.
		lowerCamelCase_							:	Optional[int]							     =							np.linalg.norm(r - ra    )
		lowerCamelCase_							:	List[Any]							     =							np.linalg.norm(x - xa    )
		lowerCamelCase_							:	Dict							     =							np.maximum(A		,     A    )

		# Update variables.
		lowerCamelCase_							:	List[Any]							     =							np.copy(A    )
		lowerCamelCase_							:	Dict							     =							np.copy(A    )
		lowerCamelCase_							:	List[str]							     =							np.copy(A    )

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

	return x






def       snake_case      (							)							->							None:




	'''simple docstring'''


	lowerCamelCase_							:	Optional[int]							     =							3
	lowerCamelCase_							:	List[Any]							     =							_create_spd_matrix(A    )
	lowerCamelCase_							:	List[Any]							     =							np.random.randn(A		,     1    )
	lowerCamelCase_							:	Tuple							     =							np.dot(A		,     A    )

	# Numpy solution.
	lowerCamelCase_							:	Optional[Any]							     =							np.linalg.solve(A		,     A    )

	# Our implementation.
	lowerCamelCase_							:	List[Any]							     =							conjugate_gradient(A		,     A    )

	# 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()






# this script builds a small sample spm file tests/fixtures/test_sentencepiece_no_bos.model, with features needed by pegasus

# 1. pip install sentencepiece
#
# 2. wget https://raw.githubusercontent.com/google/sentencepiece/master/data/botchan.txt

# 3. build
import sentencepiece as spm

# pegasus:
# 1. no bos
# 2. eos_id is 1
# 3. unk_id is 2
# build a sample spm file accordingly
spm.SentencePieceTrainer.train(
    '--input=botchan.txt --model_prefix=test_sentencepiece_no_bos --bos_id=-1 --unk_id=2  --eos_id=1  --vocab_size=1000'
)

# 4. now update the fixture
# mv test_sentencepiece_no_bos.model ../../tests/fixtures/
