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"""Newton's Method."""

# Newton's Method - https://en.wikipedia.org/wiki/Newton%27s_method

from typing import Callable

RealFunc = Callable[[float], float] # type alias for a real -> real function

# function is the f(x) and derivative is the f'(x)

def newton(

function: RealFunc,

derivative: RealFunc,

starting_int: int,

) -> float:

"""

>>> newton(lambda x: x ** 3 - 2 * x - 5, lambda x: 3 * x ** 2 - 2, 3)

2.0945514815423474

>>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -2)

1.0

>>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -4)

1.0000000000000102

>>> import math

>>> newton(math.sin, math.cos, 1)

0.0

>>> newton(math.sin, math.cos, 2)

3.141592653589793

>>> newton(math.cos, lambda x: -math.sin(x), 2)

1.5707963267948966

>>> newton(math.cos, lambda x: -math.sin(x), 0)

Traceback (most recent call last):

...

ZeroDivisionError: Could not find root

"""

prev_guess = float(starting_int)

while True:

try:

next_guess = prev_guess - function(prev_guess) / derivative(prev_guess)

except ZeroDivisionError:

raise ZeroDivisionError("Could not find root") from None

if abs(prev_guess - next_guess) < 10 ** -5:

return next_guess

prev_guess = next_guess

def f(x: float) -> float:

return (x ** 3) - (2 * x) - 5

def f1(x: float) -> float:

return 3 * (x ** 2) - 2

if __name__ == "__main__":

print(newton(f, f1, 3))

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