




'''simple docstring'''

from __future__ import annotations

import numpy as np


def 						__magic_name__       (      __UpperCAmelCase      )						->   tuple[np.ndarray, np.ndarray]:

					'''simple docstring'''

					snake_case_     ,snake_case_		     =							np.shape(__UpperCAmelCase      )
					if rows != columns:
										snake_case_		     =							(
										    '''\'table\' has to be of square shaped array but got a '''
										    F"{rows}x{columns} array:\n{table}"
										)
										raise ValueError(__UpperCAmelCase      )

					snake_case_		     =							np.zeros((rows, columns)      )
					snake_case_		     =							np.zeros((rows, columns)      )
					for i in range(__UpperCAmelCase      ):
										for j in range(__UpperCAmelCase      ):
															snake_case_		     =							sum(lower[i][k] * upper[k][j] for k in range(__UpperCAmelCase      )      )
															if upper[j][j] == 0:
																				raise ArithmeticError('''No LU decomposition exists'''      )
															snake_case_		     =							(table[i][j] - total) / upper[j][j]
										snake_case_		     =							1
										for j in range(__UpperCAmelCase, __UpperCAmelCase      ):
															snake_case_		     =							sum(lower[i][k] * upper[k][j] for k in range(__UpperCAmelCase      )      )
															snake_case_		     =							table[i][j] - total
					return lower, upper


if __name__ == "__main__":
							import doctest

							doctest.testmod()







'''simple docstring'''

from numpy import exp, pi, sqrt


def 						__magic_name__       (      __UpperCAmelCase, __UpperCAmelCase = 0.0, __UpperCAmelCase = 1.0      )						->   int:

					'''simple docstring'''

					return 1 / sqrt(2 * pi * sigma**2      ) * exp(-((x - mu) ** 2) / (2 * sigma**2)      )


if __name__ == "__main__":
							import doctest

							doctest.testmod()

