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import math

def fx(x: float, a: float) -> float:

return math.pow(x, 2) - a

def fx_derivative(x: float) -> float:

return 2 * x

def get_initial_point(a: float) -> float:

start = 2.0

while start <= a:

start = math.pow(start, 2)

return start

def square_root_iterative(

a: float, max_iter: int = 9999, tolerance: float = 0.00000000000001

) -> float:

"""

Square root is aproximated using Newtons method.

https://en.wikipedia.org/wiki/Newton%27s_method

>>> all(abs(square_root_iterative(i)-math.sqrt(i)) <= .00000000000001

... for i in range(500))

True

>>> square_root_iterative(-1)

Traceback (most recent call last):

...

ValueError: math domain error

>>> square_root_iterative(4)

2.0

>>> square_root_iterative(3.2)

1.788854381999832

>>> square_root_iterative(140)

11.832159566199232

"""

if a < 0:

raise ValueError("math domain error")

value = get_initial_point(a)

for i in range(max_iter):

prev_value = value

value = value - fx(value, a) / fx_derivative(value)

if abs(prev_value - value) < tolerance:

return value

if __name__ == "__main__":

from doctest import testmod

testmod()

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