





"""simple docstring"""



import numpy as np







def 	__lowercase  (     snake_case_						:						List[Any]		):



    '''simple docstring'''







    __A					:		Optional[int]						=  a.shape
    __A					:		Optional[int]						=  min(snake_case_       ,snake_case_		)
    __A					:		List[str]						=  np.eye(snake_case_		)
    __A					:		Union[str, Any]						=  a.copy()

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

        # construct the Householder matrix
        __A					:		List[str]						=  np.eye(m - k		) - 2.0 * v @ v.T
        # pad with ones and zeros as necessary
        __A					:		List[Any]						=  np.block([[np.eye(snake_case_		), np.zeros((k, m - k)		)], [np.zeros((m - k, k)		), q_k]]		)

        __A					:		int						=  q @ q_k.T
        __A					:		Tuple						=  q_k @ r

    return q, r


if __name__ == "__main__":
       import doctest

       doctest.testmod()











"""simple docstring"""



def 	__lowercase  (     snake_case_						:						list		)     ->list:



    '''simple docstring'''







    for insert_index, insert_value in enumerate(collection[1:]		):
        __A					:		Any						=  insert_index
        while insert_index >= 0 and insert_value < collection[insert_index]:
            __A					:		List[Any]						=  collection[insert_index]
            insert_index -= 1
        if insert_index != temp_index:
            __A					:		str						=  insert_value
    return collection


if __name__ == "__main__":
       from doctest import testmod

       testmod()

       a_				=    input("""Enter numbers separated by a comma:\n""").strip()
       a_				=    [int(item) for item in user_input.split(""",""")]
       print(f'''{insertion_sort(unsorted) = }''')




