


'''simple docstring'''




from typing import Any

import numpy as np





def 				a_							(    lowercase     )      ->bool:






				'''simple docstring'''

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

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

				# Get eigenvalues and eignevectors for a symmetric matrix.
				__lowerCamelCase    ,       __lowerCamelCase		:   Dict           =       np.linalg.eigh(lowercase     )

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





def 				a_							(    lowercase     )      ->Any:






				'''simple docstring'''

				__lowerCamelCase		:   Dict           =       np.random.randn(lowercase				,    lowercase     )
				__lowerCamelCase		:   Any           =       np.dot(lowercase				,    random_matrix.T     )
				assert _is_matrix_spd(lowercase     )
				return spd_matrix





def 				a_							(    lowercase				,    lowercase				,    lowercase = 1000				,    lowercase = 1E-8				,    )      ->Any:






				'''simple docstring'''

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

				# Initialize solution guess, residual, search direction.
				__lowerCamelCase		:   Any           =       np.zeros((np.shape(lowercase     )[0], 1)     )
				__lowerCamelCase		:   int           =       np.copy(lowercase     )
				__lowerCamelCase		:   Tuple           =       np.copy(lowercase     )

				# Set initial errors in solution guess and residual.
				__lowerCamelCase		:   Optional[int]           =       1E9
				__lowerCamelCase		:   Optional[Any]           =       1E9
				__lowerCamelCase		:   Dict           =       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		:   Tuple           =       np.dot(lowercase				,    lowercase     )

								# The main algorithm.

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

								# Calculate errors.
								__lowerCamelCase		:   int           =       np.linalg.norm(r - ra     )
								__lowerCamelCase		:   Dict           =       np.linalg.norm(x - xa     )
								__lowerCamelCase		:   int           =       np.maximum(lowercase				,    lowercase     )

								# Update variables.
								__lowerCamelCase		:   int           =       np.copy(lowercase     )
								__lowerCamelCase		:   int           =       np.copy(lowercase     )
								__lowerCamelCase		:   Any           =       np.copy(lowercase     )

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

				return x





def 				a_							(    )      ->None:






				'''simple docstring'''

				__lowerCamelCase		:   Optional[int]           =       3
				__lowerCamelCase		:   Optional[int]           =       _create_spd_matrix(lowercase     )
				__lowerCamelCase		:   Union[str, Any]           =       np.random.randn(lowercase				,    1     )
				__lowerCamelCase		:   Any           =       np.dot(lowercase				,    lowercase     )

				# Numpy solution.
				__lowerCamelCase		:   int           =       np.linalg.solve(lowercase				,    lowercase     )

				# Our implementation.
				__lowerCamelCase		:   Optional[int]           =       conjugate_gradient(lowercase				,    lowercase     )

				# 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'''




import os





def 				a_							(    )      ->int:






				'''simple docstring'''

				__lowerCamelCase		:   Dict           =       os.path.dirname(os.path.realpath(lowercase     )     )
				__lowerCamelCase		:   Any           =       os.path.join(lowercase				,    "triangle.txt"     )

				with open(lowercase     ) as in_file:
								__lowerCamelCase		:   List[str]           =       [[int(lowercase     ) for i in line.split()] for line in in_file]

				while len(lowercase     ) != 1:
								__lowerCamelCase		:   List[str]           =       triangle.pop()
								__lowerCamelCase		:   Union[str, Any]           =       triangle[-1]
								for j in range(len(lowercase     ) - 1     ):
												curr_row[j] += max(last_row[j]				,    last_row[j + 1]     )
				return triangle[0][0]


if __name__ == "__main__":
				print(solution())





