from typing import Any

import numpy as np





def 						__SCREAMING_SNAKE_CASE  (    UpperCamelCase__			)  ->    List[str]:







			'''simple docstring'''

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

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

			# Get eigenvalues and eignevectors for a symmetric matrix.
			UpperCAmelCase		,      UpperCAmelCase       		=    np.linalg.eigh(__UpperCamelCase			)

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





def 						__SCREAMING_SNAKE_CASE  (    UpperCamelCase__			)  ->    Union[str, Any]:







			'''simple docstring'''

			UpperCAmelCase       		=    np.random.randn(__UpperCamelCase      ,      __UpperCamelCase			)
			UpperCAmelCase       		=    np.dot(__UpperCamelCase      ,      random_matrix.T			)
			assert _is_matrix_spd(__UpperCamelCase			)
			return spd_matrix





def 						__SCREAMING_SNAKE_CASE  (    UpperCamelCase__      ,      UpperCamelCase__      ,      UpperCamelCase__ = 1000      ,      UpperCamelCase__ = 1E-8      ,      )  ->    Optional[int]:







			'''simple docstring'''

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

			# Initialize solution guess, residual, search direction.
			UpperCAmelCase       		=    np.zeros((np.shape(__UpperCamelCase			)[0], 1)			)
			UpperCAmelCase       		=    np.copy(__UpperCamelCase			)
			UpperCAmelCase       		=    np.copy(__UpperCamelCase			)

			# Set initial errors in solution guess and residual.
			UpperCAmelCase       		=    1E9
			UpperCAmelCase       		=    1E9
			UpperCAmelCase       		=    1E9

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

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

						# The main algorithm.

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

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

						# Update variables.
						UpperCAmelCase       		=    np.copy(__UpperCamelCase			)
						UpperCAmelCase       		=    np.copy(__UpperCamelCase			)
						UpperCAmelCase       		=    np.copy(__UpperCamelCase			)

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

			return x





def 						__SCREAMING_SNAKE_CASE  (    )  ->    Optional[Any]:







			'''simple docstring'''

			UpperCAmelCase       		=    3
			UpperCAmelCase       		=    _create_spd_matrix(__UpperCamelCase			)
			UpperCAmelCase       		=    np.random.randn(__UpperCamelCase      ,      1			)
			UpperCAmelCase       		=    np.dot(__UpperCamelCase      ,      __UpperCamelCase			)

			# Numpy solution.
			UpperCAmelCase       		=    np.linalg.solve(__UpperCamelCase      ,      __UpperCamelCase			)

			# Our implementation.
			UpperCAmelCase       		=    conjugate_gradient(__UpperCamelCase      ,      __UpperCamelCase			)

			# 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()










from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_sentencepiece_available, is_tokenizers_available


__A    :   Any          =       {}

try:
				if not is_sentencepiece_available():
								raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
				pass
else:
				__A    :   Union[str, Any]          =       ["CpmTokenizer"]

try:
				if not is_tokenizers_available():
								raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
				pass
else:
				__A    :   List[Any]          =       ["CpmTokenizerFast"]


if TYPE_CHECKING:
				try:
								if not is_sentencepiece_available():
												raise OptionalDependencyNotAvailable()
				except OptionalDependencyNotAvailable:
								pass
				else:
								from .tokenization_cpm import CpmTokenizer

				try:
								if not is_tokenizers_available():
												raise OptionalDependencyNotAvailable()
				except OptionalDependencyNotAvailable:
								pass
				else:
								from .tokenization_cpm_fast import CpmTokenizerFast

else:
				import sys

				__A    :   str          =       _LazyModule(__name__, globals()["__file__"], _import_structure, module_spec=__spec__)



