sqrt-newton

Calculate square roots using the Newton-Raphson method.

For univariate differentiable functions between real numbers, the Newton-Raphson method is an iterative root finding method that works by repeatedly finding the intersection between the tangent of a function at the current guess, and the f(x) == 0 line.

Having our current guess n0, we can calculate n1 = n0 - f(n0)/diff(f)(n0).

That means that we can easily devise calculating a square root using the following method:

1. Take equation x == sqrt(n)
2. Square both sides x*x == n (this implies that our algorithm only works for a non-negative n)
3. Subtract n from both sides x*x - n == 0. This is our f(x)
4. Get derivative 2*x
5. Get recurrence relation x1 = x0 - (x0*x0 - n)/(2*x0)
6. Iterate until satisfied with precision