






'''simple docstring'''


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






def   lowerCAmelCase__			(					SCREAMING_SNAKE_CASE__       ,		SCREAMING_SNAKE_CASE__		):
 __a			,    __a    :	str									=						np.shape(SCREAMING_SNAKE_CASE__		)

 __a    :	NDArray[floataa]									=						np.zeros((rows, 1)       ,		dtype=SCREAMING_SNAKE_CASE__		)
 for row in reversed(range(SCREAMING_SNAKE_CASE__		)		):
  __a    :	Optional[Any]									=						0
  for col in range(row + 1       ,		SCREAMING_SNAKE_CASE__		):
   total += coefficients[row, col] * x[col]

  __a    :	List[Any]									=						(vector[row] - total) / coefficients[row, row]

 return x






def   lowerCAmelCase__			(					SCREAMING_SNAKE_CASE__       ,		SCREAMING_SNAKE_CASE__		):
 __a			,    __a    :	List[str]									=						np.shape(SCREAMING_SNAKE_CASE__		)
 if rows != columns:
  return np.array(()       ,		dtype=SCREAMING_SNAKE_CASE__		)

 # augmented matrix
 __a    :	NDArray[floataa]									=						np.concatenate((coefficients, vector)       ,		axis=1		)
 __a    :	Union[str, Any]									=						augmented_mat.astype('float64'		)

 # scale the matrix leaving it triangular
 for row in range(rows - 1		):
  __a    :	Any									=						augmented_mat[row, row]
  for col in range(row + 1       ,		SCREAMING_SNAKE_CASE__		):
   __a    :	Optional[int]									=						augmented_mat[col, row] / pivot
   augmented_mat[col, :] -= factor * augmented_mat[row, :]

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

 return x


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()




