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 fThe 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.