



'''simple docstring'''
import numpy as np
from numpy import floataa
from numpy.typing import NDArray


def   lowercase_	(   lowerCAmelCase__	:  NDArray[floataa]			,							lowerCAmelCase__	:  NDArray[floataa] ):

			"""simple docstring"""





			__UpperCAmelCase				,  __UpperCAmelCase   :      int										=     np.shape(lowerCAmelCase__ )

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

						__UpperCAmelCase   :      str										=     (vector[row] - total) / coefficients[row, row]

			return x


def   lowercase_	(   lowerCAmelCase__	:  NDArray[floataa]			,							lowerCAmelCase__	:  NDArray[floataa] ):

			"""simple docstring"""





			__UpperCAmelCase				,  __UpperCAmelCase   :      Optional[Any]										=     np.shape(lowerCAmelCase__ )
			if rows != columns:
						return np.array(()			,							dtype=lowerCAmelCase__ )

			# augmented matrix
			__UpperCAmelCase   :      NDArray[floataa]										=     np.concatenate((coefficients, vector)			,							axis=1 )
			__UpperCAmelCase   :      Tuple										=     augmented_mat.astype("""float64""" )

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

			__UpperCAmelCase   :      Union[str, Any]										=     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 typing import Any



class _A  :





		def __init__(     self ,       __UpperCAmelCase ,       __UpperCAmelCase				)       ->     None:  # noqa: A002

					'''simple docstring'''
					__UpperCAmelCase   :      Tuple										=     item
					__UpperCAmelCase   :      str										=     next



class _A  :





		def __init__(     self				)       ->     None:

					'''simple docstring'''
					__UpperCAmelCase   :      Node | None										=     None
					__UpperCAmelCase   :      Optional[int]										=     0





		def 							__A       (     self ,       __UpperCAmelCase				)       ->     None:

					'''simple docstring'''
					__UpperCAmelCase   :      Optional[Any]										=     Node(__UpperCAmelCase ,       self.head				)
					self.size += 1





		def 							__A       (     self				)       ->     Any:

					'''simple docstring'''
					# Switched 'self.is_empty()' to 'self.head is None'
					# because mypy was considering the possibility that 'self.head'
					# can be None in below else part and giving error
					if self.head is None:
								return None
					else:
								__UpperCAmelCase   :      str										=     self.head.item
								__UpperCAmelCase   :      Tuple										=     self.head.next
								self.size -= 1
								return item





		def 							__A       (     self				)       ->     bool:

					'''simple docstring'''
					return self.head is None





		def __str__(     self				)       ->     str:

					'''simple docstring'''
					if self.is_empty():
								return ""
					else:
								__UpperCAmelCase   :      Union[str, Any]										=     self.head
								__UpperCAmelCase   :      List[Any]										=     """"""
								__UpperCAmelCase   :      list[str]										=     []
								while iterate:
											item_list.append(str(iterate.item				)				)
											__UpperCAmelCase   :      Any										=     iterate.next

								__UpperCAmelCase   :      Dict										=     """ --> """.join(__UpperCAmelCase				)

								return item_str





		def __len__(     self				)       ->     int:

					'''simple docstring'''
					return self.size




