






"""simple docstring"""


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




def 			snake_case  (A_      :int	,			A_      :str      ):







  '''simple docstring'''

  a     :			List[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     :			List[str]				=   0
    for col in range(row + 1	,			SCREAMING_SNAKE_CASE__      ):
      total += coefficients[row, col] * x[col]

    a     :			Optional[Any]				=   (vector[row] - total) / coefficients[row, row]

  return x




def 			snake_case  (A_      :Union[str, Any]	,			A_      :Union[str, Any]      ):







  '''simple docstring'''

  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     :			Optional[int]				=   augmented_mat.astype('float64'      )

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

  a     :			int				=   retroactive_resolution(
      augmented_mat[:, 0:columns]	,			augmented_mat[:, columns : columns + 1]      )

  return x


if __name__ == "__main__":
   import doctest

   doctest.testmod()






'''simple docstring'''
from __future__ import annotations

from collections.abc import Callable




def 					_UpperCamelCase		(    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__ = 100      ,    )   ->				float:


		'''simple docstring'''


		snake_case   :    int			       =		x_start
		snake_case   :    Tuple			       =		fnc(SCREAMING_SNAKE_CASE__       )
		snake_case   :    Dict			       =		0.0

		for _ in range(SCREAMING_SNAKE_CASE__       ):
				# Approximates small segments of curve as linear and solve
				# for trapezoidal area
				snake_case   :    str			       =		(x_end - x_start) / steps + xa
				snake_case   :    str			       =		fnc(SCREAMING_SNAKE_CASE__       )
				area += abs(fxa + fxa       ) * (xa - xa) / 2

				# Increment step
				snake_case   :    Any			       =		xa
				snake_case   :    List[str]			       =		fxa
		return area


if __name__ == "__main__":




						def 					_UpperCamelCase		(    SCREAMING_SNAKE_CASE__       )   ->				str:


								'''simple docstring'''


								return x**3

						print("f(x) = x^3")
						print("The area between the curve, x = -10, x = 10 and the x axis is:")
						lowercase__		    =	1_0
						while i <= 1_0_0_0_0_0:
												lowercase__		    =	trapezoidal_area(f, -5, 5, i)
												print(f"with {i} steps: {area}")
												i *= 1_0


