You can use some numeric solver like Runge-Kutta
-- define 4th order Runge-Kutta map (RK4)
rk4 :: Floating a => (a -> a) -> a -> a -> a
rk4 f h x = x + (1/6) * (k1 + 2*k2 + 2*k3 + k4)
where k1 = h * f (x)
k2 = h * f (x + 0.5*k1)
k3 = h * f (x + 0.5*k2)
k4 = h * f (x + k3)
in that case function signature is Floating
but you can use RealFloat
instead (you can use runge-kutta in complex).
Complete example:
Prelude> import Data.Complex
Prelude Data.Complex> let rk4 f h x = x + (1/6) * (k1 + 2*k2 + 2*k3 + k4) where {k1 = h * f(x);k2 = h * f (x + 0.5*k1);k3 = h * f (x + 0.5*k2);k4 = h * f (x + k3)}
Prelude Data.Complex> let f z = 2 * z
Prelude Data.Complex> rk4 f (0.1 :+ 0.2) (0.3 :+ 1.2)
(-0.2334199999999999) :+ 1.4925599999999999
Prelude Data.Complex>
On the other hand, @leftaroundabout suggest extend that behavior to VectorSpace
(great! of course! :D )