




'''simple docstring'''




import numpy as np




def     snake_case__     (			lowerCamelCase__					: List[str]      ) -> Any:
					A_			,A_			:  List[str]			=				a.shape
					A_			:  List[str]			=				min(lowerCamelCase__      ,		lowerCamelCase__      )
					A_			:  Union[str, Any]			=				np.eye(lowerCamelCase__      )
					A_			:  int			=				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_			:  List[str]			=				np.zeros_like(lowerCamelCase__      )
										A_			:  Union[str, Any]			=				1.0
										# determine scaling factor
										A_			:  str			=				np.linalg.norm(lowerCamelCase__      )
										# construct vector v for Householder reflection
										A_			:  List[str]			=				x + np.sign(x[0]      ) * alpha * ea
										v /= np.linalg.norm(lowerCamelCase__      )

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

										A_			:  Optional[int]			=				q @ q_k.T
										A_			:  List[str]			=				q_k @ r

					return q, r


if __name__ == "__main__":
							import doctest

							doctest.testmod()









'''simple docstring'''




import abc
import copy
import dataclasses
from dataclasses import dataclass
from typing import ClassVar, Dict, Type, TypeVar

from ..features import Features


snake_case__ 							=    TypeVar("""T""", bound="""TaskTemplate""")






@dataclass(frozen=a__  )
class      UpperCamelCase_  (abc.ABC  ):







						"""simple docstring"""


						_lowerCAmelCase         =       42
						_lowerCAmelCase         =       42
						_lowerCAmelCase         =       42







						def    _a     (  self   :							T				,			_lowerCamelCase   :							Features     ):





											"""simple docstring"""







											return copy.deepcopy(self     )







						@property
						def    _a     (  self   :							List[str]     ):





											"""simple docstring"""







											return Features(**self.input_schema				,			**self.label_schema     )







						@property
						@abc.abstractmethod
						def    _a     (  self   :							int     ):





											"""simple docstring"""







											raise NotImplementedError







						@classmethod
						def    _a     (  cls   :							Type[T]				,			_lowerCamelCase   :							dict     ):





											"""simple docstring"""







											A_			:  Dict			=				{f.name for f in dataclasses.fields(cls     )}
											return cls(**{k: v for k, v in template_dict.items() if k in field_names}     )



