



"""simple docstring"""
from collections.abc import Callable

__a           =							Callable[[float], float]  # type alias for a real -> real function
def   A_    (     _lowercase,   _lowercase,   _lowercase,   ):







						'''simple docstring'''


						snake_case_					:str		=     float(_lowercase      )
						while True:
												try:
																		snake_case_					:int		=     prev_guess - function(_lowercase      ) / derivative(_lowercase      )
												except ZeroDivisionError:
																		raise ZeroDivisionError("""Could not find root"""      ) from None
												if abs(prev_guess - next_guess      ) < 10**-5:
																		return next_guess
												snake_case_					:str		=     next_guess
def   A_    (     _lowercase      ):







						'''simple docstring'''


						return (x**3) - (2 * x) - 5
def   A_    (     _lowercase      ):







						'''simple docstring'''


						return 3 * (x**2) - 2


if __name__ == "__main__":
						print(newton(f, fa, 3))






'''simple docstring'''







import numpy as np






def 						_lowercase						(      lowercase_    ):



		'''simple docstring'''







		_snake_case			,       _snake_case      			=			a.shape
		_snake_case      			=			min(lowercase_		,				lowercase_    )
		_snake_case      			=			np.eye(lowercase_    )
		_snake_case      			=			a.copy()

		for k in range(t - 1    ):
				# select a column of modified matrix A':
				_snake_case      			=			r[k:, [k]]
				# construct first basis vector
				_snake_case      			=			np.zeros_like(lowercase_    )
				_snake_case      			=			1.0
				# determine scaling factor
				_snake_case      			=			np.linalg.norm(lowercase_    )
				# construct vector v for Householder reflection
				_snake_case      			=			x + np.sign(x[0]    ) * alpha * ea
				v /= np.linalg.norm(lowercase_    )

				# construct the Householder matrix
				_snake_case      			=			np.eye(m - k    ) - 2.0 * v @ v.T
				# pad with ones and zeros as necessary
				_snake_case      			=			np.block([[np.eye(lowercase_    ), np.zeros((k, m - k)    )], [np.zeros((m - k, k)    ), q_k]]    )

				_snake_case      			=			q @ q_k.T
				_snake_case      			=			q_k @ r

		return q, r


if __name__ == "__main__":
						import doctest

						doctest.testmod()





