Derivative Towers and how to use the vector-space package (haskell)

Question

I am working with Haskell for quite a while now, but I am far from being an expert. But I see that the functional approach to programming suits me the best.

So far I am working on a project to calculate some serious stuff, like currents and potentials radiated from a given structure.

I followed the blog written by Conal Elliott (here is some more Linear Maps) which is very nice and fundamental.

Unfortunately, I am lacking a simple example :)

To be more precise, I have a curve

f:[0,1] in R -> R³
t -> a*e_y + 2*t*e_z

which is a simple straight line at (0,a,2*t). When I want to calculate the derivative of f, e.g. for the length of the curve, I know the mathematical result, which is quite simple (0,0,2), but how do I accomplish this in Haskell, especially with the vector-space package?

I really want to use this library because of its functionality, it is exactly the approach I would have take too (but I am not that far ahead on the Haskell road)

What I have so far is this:

{-# LANGUAGE Rank2Types, TypeOperators, FlexibleContexts, TypeFamilies #-}
{-# OPTIONS_GHC -Wall #-}
import Numeric.GSL.Integration
import Data.VectorSpace
import Data.Basis
import Data.Cross
import Data.Derivative
import Data.LinearMap

type Vec3 s = Three s

prec :: Double
prec = 1E-9

f1 :: (Floating s, VectorSpace s, Scalar s ~ s) => s -> s
f1 = id

c1 :: Double -> Vec3 Double
c1  = \t -> linearCombo [((v 0 0 1),f1 t),(( v 0 1 0),2)]

derivC :: Double -> Vec3 (Double :> Double)
derivC t = c1 (pureD t)

It is the the actual implementation of the pureD function, so far nothing that I have tried works to get this snippet to compile. I get the following error:

tests.hs:26:12:
   Couldn't match expected type `Double :> Double'
               with actual type `Double'
   Expected type: Vec3 (Double :> Double)
     Actual type: Vec3 Double
   In the return type of a call of `c1'
   In the expression: c1 (pureD t)
Failed, modules loaded: none.

There is also a graphics library which uses vector-space and there is even an example on a torus, where pureD is used. I tried to deduce the example but I don't see how I can map it to my problem.

Any help would be greatly appreciated.

Thanks in advance

PS: I cannot post all the links I'd like to, but am willing to provide

Answer 1

That's an interesting library.. Thanks for sharing. Although I don't understand the concept of the library yet, how about this code:

{-# LANGUAGE Rank2Types, TypeOperators, FlexibleContexts, TypeFamilies #-}
module Main where

import Data.LinearMap
import Data.Maclaurin

diff :: (Double :~> (Double,Double,Double) ) -> (Double :~> (Double,Double,Double))
diff f = \x ->  (atBasis (derivative  (f x)) ())

eval :: (Double :~> (Double,Double,Double)) -> Double -> (Double,Double,Double)
eval f x = powVal (f x)        

f :: Double :~> (Double,Double,Double)
f x = tripleD (pureD 0,pureD 1,(2*idD) x)


*Main> map (eval f) [0,0.2 .. 1]
[(0.0,1.0,0.0),(0.0,1.0,0.4),(0.0,1.0,0.8),(0.0,1.0,1.2000000000000002),
 (0.0,1.0,1.6000000000000003),(0.0,1.0,2.0000000000000004)]

*Main> map (eval (diff f)) [0,0.2 .. 1]
[(0.0,0.0,2.0),(0.0,0.0,2.0),(0.0,0.0,2.0),(0.0,0.0,2.0),(0.0,0.0,2.0), 
 (0.0,0.0,2.0)]

*Main> map (eval (diff $ diff f)) [0,0.2 .. 1]
 [(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0)]

Try also g x = tripleD (pureD 0,idD x,(idD*idD) x) (which seem to represent the curve (0,x,x^2)).