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Newton’s Method using Automatic Differentiation

Newton’s Method is a numerical algorithm for finding roots of real-valued functions. It requires us to know the derivative of the function, which we use Futhark’s support for automatic differentiation to obtain.

def newton (tol: f64) (f: f64 -> f64) (x0: f64) =
  let iteration (_, x, i) =
    let (y, dy) = jvp2 f x 1
    let x' = x - y / dy
    in (f64.abs (x - x') < tol, x', i+1)
  let (_, x, steps) = iterate_until (.0) iteration (false, x0, 0)
  in (x, steps)

The jvp2 function is like jvp (see reverse-mode automatic differentiation), but also returns the normal result of the function.

To use newton, we first define an appropriate objective function, in this case a Cubic equation with the roots 1, pi, and 42:

def f x : f64 = (x - 1) * (x - f64.pi) * (x - 42)

def f_roots = newton f64.epsilon f

The number of steps needed to find a root depends on the initial guess of x0:

> f_roots 0.0
(1.0f64, 7i32)
> f_roots 4.0
(3.141592653589793f64, 6i32)
> f_roots 43.0
(42.0f64, 5i32)
> f_roots 1000000000.0
(42.0f64, 49i32)

Thanks to Gusten Isfeldt for this example.