





"""simple docstring"""



import numpy as np







def 	__lowercase  (     snake_case_						:						List[Any]		)     ->Optional[Any]:



    '''simple docstring'''







    __A				, __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"""



from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_tf_available,
    is_tokenizers_available,
    is_torch_available,
)


a_				=    {
    """configuration_layoutlm""": ["""LAYOUTLM_PRETRAINED_CONFIG_ARCHIVE_MAP""", """LayoutLMConfig""", """LayoutLMOnnxConfig"""],
    """tokenization_layoutlm""": ["""LayoutLMTokenizer"""],
}

try:
       if not is_tokenizers_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       a_				=    ["""LayoutLMTokenizerFast"""]

try:
       if not is_torch_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       a_				=    [
           """LAYOUTLM_PRETRAINED_MODEL_ARCHIVE_LIST""",
           """LayoutLMForMaskedLM""",
           """LayoutLMForSequenceClassification""",
           """LayoutLMForTokenClassification""",
           """LayoutLMForQuestionAnswering""",
           """LayoutLMModel""",
           """LayoutLMPreTrainedModel""",
       ]

try:
       if not is_tf_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       a_				=    [
           """TF_LAYOUTLM_PRETRAINED_MODEL_ARCHIVE_LIST""",
           """TFLayoutLMForMaskedLM""",
           """TFLayoutLMForSequenceClassification""",
           """TFLayoutLMForTokenClassification""",
           """TFLayoutLMForQuestionAnswering""",
           """TFLayoutLMMainLayer""",
           """TFLayoutLMModel""",
           """TFLayoutLMPreTrainedModel""",
       ]


if TYPE_CHECKING:
       from .configuration_layoutlm import LAYOUTLM_PRETRAINED_CONFIG_ARCHIVE_MAP, LayoutLMConfig, LayoutLMOnnxConfig
       from .tokenization_layoutlm import LayoutLMTokenizer

       try:
              if not is_tokenizers_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .tokenization_layoutlm_fast import LayoutLMTokenizerFast

       try:
              if not is_torch_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .modeling_layoutlm import (
                  LAYOUTLM_PRETRAINED_MODEL_ARCHIVE_LIST,
                  LayoutLMForMaskedLM,
                  LayoutLMForQuestionAnswering,
                  LayoutLMForSequenceClassification,
                  LayoutLMForTokenClassification,
                  LayoutLMModel,
                  LayoutLMPreTrainedModel,
              )
       try:
              if not is_tf_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .modeling_tf_layoutlm import (
                  TF_LAYOUTLM_PRETRAINED_MODEL_ARCHIVE_LIST,
                  TFLayoutLMForMaskedLM,
                  TFLayoutLMForQuestionAnswering,
                  TFLayoutLMForSequenceClassification,
                  TFLayoutLMForTokenClassification,
                  TFLayoutLMMainLayer,
                  TFLayoutLMModel,
                  TFLayoutLMPreTrainedModel,
              )

else:
       import sys

       a_				=    _LazyModule(__name__, globals()["""__file__"""], _import_structure, module_spec=__spec__)




