





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



def 							lowerCAmelCase_				(    A_  ,A_):
    UpperCamelCase__:					Any												=      np.shape(A_)

    UpperCamelCase__:					NDArray[floataa]												=      np.zeros((rows, 1)  ,dtype=A_)
    for row in reversed(range(A_)):
        UpperCamelCase__:					Dict												=      0
        for col in range(row + 1  ,A_):
            total += coefficients[row, col] * x[col]

        UpperCamelCase__:					Dict												=      (vector[row] - total) / coefficients[row, row]

    return x



def 							lowerCAmelCase_				(    A_  ,A_):
    UpperCamelCase__:					Tuple												=      np.shape(A_)
    if rows != columns:
        return np.array(()  ,dtype=A_)

    # augmented matrix
    UpperCamelCase__:					NDArray[floataa]												=      np.concatenate((coefficients, vector)  ,axis=1)
    UpperCamelCase__:					str												=      augmented_mat.astype("float64")

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

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

    return x


if __name__ == "__main__":
   import doctest

   doctest.testmod()








from .chunk_utils import chunk_layer
from .data_transforms import make_atomaa_masks
from .feats import atomaa_to_atomaa, frames_and_literature_positions_to_atomaa_pos, torsion_angles_to_frames
from .loss import compute_predicted_aligned_error, compute_tm
from .protein import Protein as OFProtein
from .protein import to_pdb
from .rigid_utils import Rigid, Rotation
from .tensor_utils import dict_multimap, flatten_final_dims, permute_final_dims

