



from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403






def  UpperCAmelCase__    (    lowercase__						,       lowercase__						,       lowercase__ = "x"						,       lowercase__ = 10**-10						,       lowercase__ = 1						,       ) ->      complex:
						__lowercase					=				symbols(lowercase__   )
						__lowercase					=				lambdify(lowercase__						,       lowercase__   )
						__lowercase					=				lambdify(lowercase__						,       diff(lowercase__						,       lowercase__   )   )

						__lowercase					=				starting_point

						while True:
												if diff_function(lowercase__   ) != 0:
																		__lowercase					=				prev_guess - multiplicity * func(lowercase__   ) / diff_function(
																		    lowercase__   )
												else:
																		raise ZeroDivisionError("""Could not find root"""   ) from None

												# Precision is checked by comparing the difference of consecutive guesses
												if abs(next_guess - prev_guess   ) < precision:
																		return next_guess

												__lowercase					=				next_guess


# Let's Execute
if __name__ == "__main__":
			# Find root of trigonometric function
			# Find value of pi
			print(F"""The root of sin(x) = 0 is {newton_raphson("sin(x)", 2)}""")

			# Find root of polynomial
			# Find fourth Root of 5
			print(F"""The root of x**4 - 5 = 0 is {newton_raphson("x**4 -5", 0.4 +5j)}""")

			# Find value of e
			print(
			    "The root of log(y) - 1 = 0 is ",
			    F"""{newton_raphson("log(y) - 1", 2, variable="y")}""",
			)

			# Exponential Roots
			print(
			    "The root of exp(x) - 1 = 0 is",
			    F"""{newton_raphson("exp(x) - 1", 10, precision=0.005)}""",
			)

			# Find root of cos(x)
			print(F"""The root of cos(x) = 0 is {newton_raphson("cos(x)", 0)}""")










from __future__ import annotations

from collections.abc import Callable

UpperCamelCase__ =						list[list[float | int]]






def  UpperCAmelCase__    (    lowercase__						,       lowercase__   ) ->      Matrix:
						__lowercase					=				len(lowercase__   )
						__lowercase					=				[[0 for _ in range(size + 1   )] for _ in range(lowercase__   )]
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42

						for row in range(lowercase__   ):
												for col in range(lowercase__   ):
																		__lowercase					=				matrix[row][col]

												__lowercase					=				vector[row][0]

						__lowercase					=				0
						__lowercase					=				0
						while row < size and col < size:
												# pivoting
												__lowercase					=				max((abs(augmented[rowa][col]   ), rowa) for rowa in range(lowercase__						,       lowercase__   )   )[
												    1
												]
												if augmented[pivot_row][col] == 0:
																		col += 1
																		continue
												else:
																		__lowercase	,			__lowercase					=				augmented[pivot_row], augmented[row]

												for rowa in range(row + 1						,       lowercase__   ):
																		__lowercase					=				augmented[rowa][col] / augmented[row][col]
																		__lowercase					=				0
																		for cola in range(col + 1						,       size + 1   ):
																								augmented[rowa][cola] -= augmented[row][cola] * ratio

												row += 1
												col += 1

						# back substitution
						for col in range(1						,       lowercase__   ):
												for row in range(lowercase__   ):
																		__lowercase					=				augmented[row][col] / augmented[col][col]
																		for cola in range(lowercase__						,       size + 1   ):
																								augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
						return [
						    [round(augmented[row][size] / augmented[row][row]						,       10   )] for row in range(lowercase__   )
						]






def  UpperCAmelCase__    (    lowercase__   ) ->      Callable[[int], int]:
						__lowercase					=				len(lowercase__   )
						__lowercase					=				[[0 for _ in range(lowercase__   )] for _ in range(lowercase__   )]
						__lowercase					=				[[0] for _ in range(lowercase__   )]
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42

						for x_val, y_val in enumerate(lowercase__   ):
												for col in range(lowercase__   ):
																		__lowercase					=				(x_val + 1) ** (size - col - 1)
												__lowercase					=				y_val

						__lowercase					=				solve(lowercase__						,       lowercase__   )

						def interpolated_func(lowercase__   ) -> int:
												return sum(
												    round(coeffs[x_val][0]   ) * (var ** (size - x_val - 1))
												    for x_val in range(lowercase__   )   )

						return interpolated_func






def  UpperCAmelCase__    (    lowercase__   ) ->      int:
						return (
						    1
						    - variable
						    + variable**2
						    - variable**3
						    + variable**4
						    - variable**5
						    + variable**6
						    - variable**7
						    + variable**8
						    - variable**9
						    + variable**10
						)






def  UpperCAmelCase__    (    lowercase__ = question_function						,       lowercase__ = 10   ) ->      int:
						__lowercase					=				[func(lowercase__   ) for x_val in range(1						,       order + 1   )]

						__lowercase					=				[
						    interpolate(data_points[:max_coeff]   ) for max_coeff in range(1						,       order + 1   )
						]

						__lowercase					=				0
						__lowercase					=				42
						__lowercase					=				42

						for poly in polynomials:
												__lowercase					=				1
												while func(lowercase__   ) == poly(lowercase__   ):
																		x_val += 1

												ret += poly(lowercase__   )

						return ret


if __name__ == "__main__":
			print(F"""{solution() = }""")





