




import numpy as np
from numpy import floataa
from numpy.typing import NDArray




def        __SCREAMING_SNAKE_CASE (	lowercase_       ,  lowercase_   )							->						NDArray[floataa]:

    """simple docstring"""


    __UpperCamelCase		,   __UpperCamelCase			=    np.shape(lowercase_   )

    __UpperCamelCase			=    np.zeros((rows, 1)       ,  dtype=lowercase_   )
    for row in reversed(range(lowercase_   )   ):
        __UpperCamelCase			=    0
        for col in range(row + 1       ,  lowercase_   ):
            total += coefficients[row, col] * x[col]

        __UpperCamelCase			=    (vector[row] - total) / coefficients[row, row]

    return x




def        __SCREAMING_SNAKE_CASE (	lowercase_       ,  lowercase_   )							->						NDArray[floataa]:

    """simple docstring"""


    __UpperCamelCase		,   __UpperCamelCase			=    np.shape(lowercase_   )
    if rows != columns:
        return np.array(()       ,  dtype=lowercase_   )

    # augmented matrix
    __UpperCamelCase			=    np.concatenate((coefficients, vector)       ,  axis=1   )
    __UpperCamelCase			=    augmented_mat.astype('''float64'''   )

    # scale the matrix leaving it triangular
    for row in range(rows - 1   ):
        __UpperCamelCase			=    augmented_mat[row, row]
        for col in range(row + 1       ,  lowercase_   ):
            __UpperCamelCase			=    augmented_mat[col, row] / pivot
            augmented_mat[col, :] -= factor * augmented_mat[row, :]

    __UpperCamelCase			=    retroactive_resolution(
        augmented_mat[:, 0:columns]       ,  augmented_mat[:, columns : columns + 1]   )

    return x


if __name__ == "__main__":
 import doctest

 doctest.testmod()












class 			_lowerCamelCase    :

      """simple docstring"""

      def __init__(  self       :			Optional[Any]			,							snake_case       :			int					):
          __UpperCamelCase			=    data
          __UpperCamelCase			=    42
          __UpperCamelCase			=    42







def        __SCREAMING_SNAKE_CASE (	lowercase_   )							->						None:

    """simple docstring"""


    __UpperCamelCase			=    0
    __UpperCamelCase			=    x




def        __SCREAMING_SNAKE_CASE (	lowercase_       ,  lowercase_   )							->						None:

    """simple docstring"""


    __UpperCamelCase		,   __UpperCamelCase			=    find_set(lowercase_   ), find_set(lowercase_   )
    if x == y:
        return

    elif x.rank > y.rank:
        __UpperCamelCase			=    x
    else:
        __UpperCamelCase			=    y
        if x.rank == y.rank:
            y.rank += 1




def        __SCREAMING_SNAKE_CASE (	lowercase_   )							->						Node:

    """simple docstring"""


    if x != x.parent:
        __UpperCamelCase			=    find_set(x.parent   )
    return x.parent




def        __SCREAMING_SNAKE_CASE (	lowercase_   )							->						set:

    """simple docstring"""


    __UpperCamelCase			=    ({0, 1, 2}, {3, 4, 5})
    for s in sets:
        if node.data in s:
            return s
    __UpperCamelCase			=    F"{node.data} is not in {sets}"
    raise ValueError(lowercase_   )




def        __SCREAMING_SNAKE_CASE (	)							->						None:

    """simple docstring"""


    __UpperCamelCase			=    [Node(lowercase_   ) for i in range(6   )]
    for v in vertex:
        make_set(lowercase_   )

    union_set(vertex[0]       ,  vertex[1]   )
    union_set(vertex[1]       ,  vertex[2]   )
    union_set(vertex[3]       ,  vertex[4]   )
    union_set(vertex[3]       ,  vertex[5]   )

    for nodea in vertex:
        for nodea in vertex:
            if find_python_set(lowercase_   ).isdisjoint(find_python_set(lowercase_   )   ):
                assert find_set(lowercase_   ) != find_set(lowercase_   )
            else:
                assert find_set(lowercase_   ) == find_set(lowercase_   )


if __name__ == "__main__":
 test_disjoint_set()






