-- Exercise 4.2
-- George C. Privon
-- 2023-12-22
type R = Double
type Derivative = (R -> R) -> R -> R

-- a function to take the numerical derivative, approximated for some dt
derivative :: R -> Derivative
derivative dt x t = (x (t + dt/2) - x (t - dt/2)) / dt

-- the function we want to take the derivative of
myfunc :: R -> R
myfunc x = x**3

-- exact derivative of our function of interest
dmyfunc :: R -> R
dmyfunc x = 3 * x**2

-- take the derivative with dt=1
dOne :: R -> R
dOne x = derivative 1 myfunc x

-- compare the exact derivative with the numerical derivative using dt=1
abserr :: R -> R
abserr x = abs (dmyfunc x - dOne x)

-- compare the exact derivative with the numerical derivative, for a fixed
-- value, varying the dt
abserrdt :: R -> R
abserrdt dt = abs (dmyfunc 0.1 - derivative dt myfunc 0.1)


-- Solution discussion
-- a) The error introduced by the numerical derivative is a constant value of
--    0.25.
-- b) The error in terms of `dt` (a in the variables of the example) follows
--    0.25*dt**2
-- c) When x=4, a fractional error of 1% can be achieved with a
--    dt=sqrt(0.01*48/0.25)~1.3856406460551018.
--    When x=0.1, the same fractional error can be achieved with:
--    dt=sqrt(0.01*0.03/0.25)~3.4641016151377546e-2