



'''simple docstring'''



from typing import Any

import numpy as np



def 	__lowerCAmelCase       (				lowerCamelCase :    np.ndarray       ):






	'''simple docstring'''




	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.
	__lowerCAmelCase				,							__lowerCAmelCase       				=	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 :    int       ):






	'''simple docstring'''




	__lowerCAmelCase       				=	np.random.randn(lowerCamelCase					,		lowerCamelCase       )
	__lowerCAmelCase       				=	np.dot(lowerCamelCase					,		random_matrix.T       )
	assert _is_matrix_spd(lowerCamelCase       )
	return spd_matrix



def 	__lowerCAmelCase       (				lowerCamelCase :    np.ndarray					,		lowerCamelCase :    np.ndarray					,		lowerCamelCase :    int = 10_00					,		lowerCamelCase :    float = 1e-8					,		):






	'''simple docstring'''




	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.
	__lowerCAmelCase       				=	np.zeros((np.shape(lowerCamelCase       )[0], 1)       )
	__lowerCAmelCase       				=	np.copy(lowerCamelCase       )
	__lowerCAmelCase       				=	np.copy(lowerCamelCase       )

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

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

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

		# The main algorithm.

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

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

		# Update variables.
		__lowerCAmelCase       				=	np.copy(lowerCamelCase       )
		__lowerCAmelCase       				=	np.copy(lowerCamelCase       )
		__lowerCAmelCase       				=	np.copy(lowerCamelCase       )

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

	return x



def 	__lowerCAmelCase       (				):






	'''simple docstring'''




	__lowerCAmelCase       				=	3
	__lowerCAmelCase       				=	_create_spd_matrix(lowerCamelCase       )
	__lowerCAmelCase       				=	np.random.randn(lowerCamelCase					,		1       )
	__lowerCAmelCase       				=	np.dot(lowerCamelCase					,		lowerCamelCase       )

	# Numpy solution.
	__lowerCAmelCase       				=	np.linalg.solve(lowerCamelCase					,		lowerCamelCase       )

	# Our implementation.
	__lowerCAmelCase       				=	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'''



def 	__lowerCAmelCase       (				lowerCamelCase :    list[list[int]]					,		lowerCamelCase :    list[list[int]]       ):






	'''simple docstring'''




	__lowerCAmelCase       				=	[]
	__lowerCAmelCase       				=	len(lowerCamelCase       )
	for i in range(lowerCamelCase       ):
		__lowerCAmelCase       				=	[]
		for j in range(lowerCamelCase       ):
			__lowerCAmelCase       				=	0
			for k in range(lowerCamelCase       ):
				__lowerCAmelCase       				=	val + matrix_a[i][k] * matrix_b[k][j]
			list_a.append(lowerCamelCase       )
		matrix_c.append(lowerCamelCase       )
	return matrix_c



def 	__lowerCAmelCase       (				lowerCamelCase :    int       ):






	'''simple docstring'''




	return [[int(row == column       ) for column in range(lowerCamelCase       )] for row in range(lowerCamelCase       )]



def 	__lowerCAmelCase       (				lowerCamelCase :    int       ):






	'''simple docstring'''




	if n <= 1:
		return n
	__lowerCAmelCase       				=	identity(2       )
	__lowerCAmelCase       				=	[[1, 1], [1, 0]]
	__lowerCAmelCase       				=	n - 1
	while n > 0:
		if n % 2 == 1:
			__lowerCAmelCase       				=	multiply(lowerCamelCase					,		lowerCamelCase       )
		__lowerCAmelCase       				=	multiply(lowerCamelCase					,		lowerCamelCase       )
		__lowerCAmelCase       				=	int(n / 2       )
	return res_matrix[0][0]



def 	__lowerCAmelCase       (				lowerCamelCase :    int       ):






	'''simple docstring'''




	if n <= 1:
		return n
	__lowerCAmelCase       				=	0
	__lowerCAmelCase       				=	1
	for _ in range(2					,		n + 1       ):
		__lowerCAmelCase				,							__lowerCAmelCase       				=	fiba, fiba + fiba
	return fiba



def 	__lowerCAmelCase       (				):






	'''simple docstring'''




	for ordinal in "0th 1st 2nd 3rd 10th 100th 1000th".split():
		__lowerCAmelCase       				=	int("".join(c for c in ordinal if c in "0123456789"       )       )  # 1000th --> 1000
		print(
		    f'''{ordinal} fibonacci number using matrix exponentiation is '''
		    f'''{nth_fibonacci_matrix(lowerCamelCase       )} and using bruteforce is '''
		    f'''{nth_fibonacci_bruteforce(lowerCamelCase       )}\n'''       )
	# from timeit import timeit
	# print(timeit("nth_fibonacci_matrix(1000000)",
	#              "from main import nth_fibonacci_matrix", number=5))
	# print(timeit("nth_fibonacci_bruteforce(1000000)",
	#              "from main import nth_fibonacci_bruteforce", number=5))
	# 2.3342058970001744
	# 57.256506615000035


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	main()