[Haskell-cafe] Effective function representation.

[Henning Thielemann]

On Sat, 22 Jan 2011, Grigory Sarnitskiy wrote:

> I need to deal with functions of type (Double -> Double). I need Fourier 
> transform, integration, + - * / operations, value of a function in a 
> point, probably composition.

Incidentally I am currently working on a function representation based on 
Gaussians. It supports exact Fourier transform, convolution, but not 
division and composition.

http://code.haskell.org/numericprelude/src/MathObj/Gaussian/Polynomial.hs
http://users.informatik.uni-halle.de/~thielema/Research/fourieralgebra.pdf

> Is there some libraries which allow me to work with functions as if 
> their type was (Double -> Double), but indeed it was something faster?

No, you have to use a special type for the particular function 
representation and use an 'apply' function, that converts your 
representation to a real function:

   apply :: FunctionRepresentation a b -> a -> b

You can also define an infix operator

   ($$) :: FunctionRepresentation a b -> a -> b

and then write

   f $$ x

where you would have written (f x) if 'f' would be a function and not a 
function representation.