





"""simple docstring"""



import numpy as np







def 						__lowercase	(  snake_case_    :    np.ndarray							,snake_case_    :    np.ndarray							,snake_case_    :    float = 1e-12							,snake_case_    :    int = 100							,)  ->tuple[float, np.ndarray]:



    '''simple docstring'''







    assert np.shape(snake_case_		)[0] == np.shape(snake_case_		)[1]
    # Ensure proper dimensionality.
    assert np.shape(snake_case_		)[0] == np.shape(snake_case_		)[0]
    # Ensure inputs are either both complex or both real
    assert np.iscomplexobj(snake_case_		) == np.iscomplexobj(snake_case_		)
    __A       :		Tuple							=	np.iscomplexobj(snake_case_		)
    if is_complex:
        # Ensure complex input_matrix is Hermitian
        assert np.array_equal(snake_case_							,input_matrix.conj().T		)

    # Set convergence to False. Will define convergence when we exceed max_iterations
    # or when we have small changes from one iteration to next.

    __A       :		Optional[Any]							=	False
    __A       :		Tuple							=	0
    __A       :		Union[str, Any]							=	0
    __A       :		str							=	1e12

    while not convergence:
        # Multiple matrix by the vector.
        __A       :		Dict							=	np.dot(snake_case_							,snake_case_		)
        # Normalize the resulting output vector.
        __A       :		Union[str, Any]							=	w / np.linalg.norm(snake_case_		)
        # Find rayleigh quotient
        # (faster than usual b/c we know vector is normalized already)
        __A       :		Optional[Any]							=	vector.conj().T if is_complex else vector.T
        __A       :		Optional[Any]							=	np.dot(snake_case_							,np.dot(snake_case_							,snake_case_		)		)

        # Check convergence.
        __A       :		Any							=	np.abs(lambda_ - lambda_previous		) / lambda_
        iterations += 1

        if error <= error_tol or iterations >= max_iterations:
            __A       :		str							=	True

        __A       :		Tuple							=	lambda_

    if is_complex:
        __A       :		Tuple							=	np.real(lambda_		)

    return lambda_, vector







def 						__lowercase	(  )  ->None:



    '''simple docstring'''







    __A       :		str							=	np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]		)
    __A       :		Optional[Any]							=	np.array([41, 4, 20]		)
    __A       :		Optional[Any]							=	real_input_matrix.astype(np.complexaaa		)
    __A       :		List[Any]							=	np.triu(1j * complex_input_matrix							,1		)
    complex_input_matrix += imag_matrix
    complex_input_matrix += -1 * imag_matrix.T
    __A       :		Optional[Any]							=	np.array([41, 4, 20]		).astype(np.complexaaa		)

    for problem_type in ["real", "complex"]:
        if problem_type == "real":
            __A       :		Dict							=	real_input_matrix
            __A       :		Optional[int]							=	real_vector
        elif problem_type == "complex":
            __A       :		Dict							=	complex_input_matrix
            __A       :		Optional[Any]							=	complex_vector

        # Our implementation.
        __A					,			__A       :		str							=	power_iteration(snake_case_							,snake_case_		)

        # Numpy implementation.

        # Get eigenvalues and eigenvectors using built-in numpy
        # eigh (eigh used for symmetric or hermetian matrices).
        __A					,			__A       :		str							=	np.linalg.eigh(snake_case_		)
        # Last eigenvalue is the maximum one.
        __A       :		List[str]							=	eigen_values[-1]
        # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
        __A       :		Optional[Any]							=	eigen_vectors[:, -1]

        # Check our implementation and numpy gives close answers.
        assert np.abs(eigen_value - eigen_value_max		) <= 1e-6
        # Take absolute values element wise of each eigenvector.
        # as they are only unique to a minus sign.
        assert np.linalg.norm(np.abs(snake_case_		) - np.abs(snake_case_		)		) <= 1e-6


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       test_power_iteration()











"""simple docstring"""



def 						__lowercase	(  snake_case_    :    int = 2000000		)  ->int:



    '''simple docstring'''







    __A       :		List[Any]							=	[0 for i in range(n + 1		)]
    __A       :		str							=	1
    __A       :		Dict							=	1

    for i in range(2							,int(n**0.5		) + 1		):
        if primality_list[i] == 0:
            for j in range(i * i							,n + 1							,snake_case_		):
                __A       :		List[Any]							=	1
    __A       :		Union[str, Any]							=	0
    for i in range(snake_case_		):
        if primality_list[i] == 0:
            sum_of_primes += i
    return sum_of_primes


if __name__ == "__main__":
       print(f'''{solution() = }''')




