

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







def  UpperCAmelCase_   (_lowerCAmelCase :		NDArray[floataa]     ,			_lowerCAmelCase :		NDArray[floataa]		):
			__UpperCamelCase							,				__UpperCamelCase    :    Optional[Any]								=      np.shape(_lowerCAmelCase		)

			__UpperCamelCase    :    NDArray[floataa]								=      np.zeros((rows, 1)     ,			dtype=_lowerCAmelCase		)
			for row in reversed(range(_lowerCAmelCase		)		):
						__UpperCamelCase    :    str								=      0
						for col in range(row + 1     ,			_lowerCAmelCase		):
									total += coefficients[row, col] * x[col]

						__UpperCamelCase    :    Tuple								=      (vector[row] - total) / coefficients[row, row]

			return x







def  UpperCAmelCase_   (_lowerCAmelCase :		NDArray[floataa]     ,			_lowerCAmelCase :		NDArray[floataa]		):
			__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    :    Union[str, Any]								=      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    :    int								=      augmented_mat[col, row] / pivot
									augmented_mat[col, :] -= factor * augmented_mat[row, :]

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

			return x


if __name__ == "__main__":
					import doctest

					doctest.testmod()


from .balanced_parentheses import balanced_parentheses
from .stack import Stack







def  UpperCAmelCase_   (_lowerCAmelCase :		str		):
			return {"+": 1, "-": 1, "*": 2, "/": 2, "^": 3}.get(_lowerCAmelCase     ,			-1		)







def  UpperCAmelCase_   (_lowerCAmelCase :		str		):
			if not balanced_parentheses(_lowerCAmelCase		):
						raise ValueError("Mismatched parentheses"		)
			__UpperCamelCase    :    Stack[str]								=      Stack()
			__UpperCamelCase    :    int								=      []
			for char in expression_str:
						if char.isalpha() or char.isdigit():
									postfix.append(_lowerCAmelCase		)
						elif char == "(":
									stack.push(_lowerCAmelCase		)
						elif char == ")":
									while not stack.is_empty() and stack.peek() != "(":
												postfix.append(stack.pop()		)
									stack.pop()
						else:
									while not stack.is_empty() and precedence(_lowerCAmelCase		) <= precedence(stack.peek()		):
												postfix.append(stack.pop()		)
									stack.push(_lowerCAmelCase		)
			while not stack.is_empty():
						postfix.append(stack.pop()		)
			return " ".join(_lowerCAmelCase		)


if __name__ == "__main__":
					from doctest import testmod

					testmod()
					lowercase     :       int        =       "a+b*(c^d-e)^(f+g*h)-i"

					print("Infix to Postfix Notation demonstration:\n")
					print("Infix notation: " + expression)
					print("Postfix notation: " + infix_to_postfix(expression))