'''simple docstring'''


from dataclasses import dataclass
from typing import List, Optional, Union

import numpy as np
import PIL
from PIL import Image

from ...utils import is_torch_available, is_transformers_available


if is_transformers_available() and is_torch_available():
 from .image_encoder import PaintByExampleImageEncoder
 from .pipeline_paint_by_example import PaintByExamplePipeline











import numpy as np
def 				UpperCAmelCase__		(					_SCREAMING_SNAKE_CASE :						Tuple					)->Union[str, Any]:
      _lowerCAmelCase      , _lowerCAmelCase			      =	a.shape
      _lowerCAmelCase			      =	min(_SCREAMING_SNAKE_CASE  ,					_SCREAMING_SNAKE_CASE					)
      _lowerCAmelCase			      =	np.eye(_SCREAMING_SNAKE_CASE					)
      _lowerCAmelCase			      =	a.copy()

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

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

            _lowerCAmelCase			      =	q @ q_k.T
            _lowerCAmelCase			      =	q_k @ r

      return q, r


if __name__ == "__main__":
    import doctest

    doctest.testmod()