






'''simple docstring'''


import numpy as np






def   lowerCAmelCase__			(					SCREAMING_SNAKE_CASE__		):
 __a    :	str									=						a.shape
 __a    :	Union[str, Any]									=						min(SCREAMING_SNAKE_CASE__       ,		SCREAMING_SNAKE_CASE__		)
 __a    :	List[Any]									=						np.eye(SCREAMING_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    :	str									=						np.zeros_like(SCREAMING_SNAKE_CASE__		)
  __a    :	List[str]									=						1.0
  # determine scaling factor
  __a    :	int									=						np.linalg.norm(SCREAMING_SNAKE_CASE__		)
  # construct vector v for Householder reflection
  __a    :	Optional[int]									=						x + np.sign(x[0]		) * alpha * ea
  v /= np.linalg.norm(SCREAMING_SNAKE_CASE__		)

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

  __a    :	int									=						q @ q_k.T
  __a    :	Dict									=						q_k @ r

 return q, r


if __name__ == "__main__":
  import doctest

  doctest.testmod()












'''simple docstring'''


import math
from timeit import timeit






def   lowerCAmelCase__			(					SCREAMING_SNAKE_CASE__		):
 __a    :	Any									=						0
 __a    :	List[Any]									=						abs(SCREAMING_SNAKE_CASE__		)
 while True:
  __a    :	int									=						n // 10
  digits += 1
  if n == 0:
   break
 return digits






def   lowerCAmelCase__			(					SCREAMING_SNAKE_CASE__		):
 return 1 if n == 0 else math.floor(math.log(abs(SCREAMING_SNAKE_CASE__		)       ,		10		) + 1		)






def   lowerCAmelCase__			(					SCREAMING_SNAKE_CASE__		):
 return len(str(abs(SCREAMING_SNAKE_CASE__		)		)		)






def   lowerCAmelCase__			(					):
 from collections.abc import Callable

 def benchmark_a_function(SCREAMING_SNAKE_CASE__       ,		SCREAMING_SNAKE_CASE__		) -> None:
  __a    :	Tuple									=						f'''{func.__name__}({value})'''
  __a    :	List[Any]									=						timeit(f'''__main__.{call}'''       ,		setup='import __main__'		)
  print(f'''{call}: {func(SCREAMING_SNAKE_CASE__		)} -- {timing} seconds'''		)

 for value in (262144, 1125899906842624, 1267650600228229401496703205376):
  for func in (num_digits, num_digits_fast, num_digits_faster):
   benchmark_a_function(SCREAMING_SNAKE_CASE__       ,		SCREAMING_SNAKE_CASE__		)
  print()


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  benchmark()




