View source for Functional differentiation

== Introduction ==

Functional differentiation means computing or approximating the derivative of a function.
There are several ways to do this:
* Approximate the derivative <math>f'(x)</math> by <math>\frac{f(x+h)-f(x)}{h}</math> where <math>h</math> is close to zero. (or at best the square root of the machine precision <math>\varepsilon</math>.
* Compute the derivative of <math>f</math> [[Typeful_symbolic_differentiation|symbolically]]. This approach is particularly interesting for Haskell.

== Functional analysis ==

If you want to explain the terms [[Higher order function]] and [[Currying]] to mathematicians, this is certainly a good example.
The mathematician writes
: <math> D f (x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}</math>
and the Haskell programmer writes
<haskell>
derive :: (Fractional a) => a -> (a -> a) -> (a -> a)
derive h f x = (f (x+h) - f x) / h    .
</haskell>
Haskell's <hask>derive h</hask> approximates the mathematician's <math> D </math>.
In functional analysis <math> D </math> is called a (linear) function operator, because it maps functions to functions.
In Haskell <hask>derive h</hask> is called a higher order function for the same reason.
<math> D </math> is in curried form. If it would be uncurried, you would write <math> D(f,x) </math>.

== Blog Posts ==

There have been several blog posts on this recently.  I think we should gather the information together and make a nice wiki article on it here.  For now, here are links to articles on the topic.

* [http://augustss.blogspot.com/2007/04/overloading-haskell-numbers-part-2.html Overloading Haskell numbers, part 2, Forward Automatic Differentiation.]
* [http://vandreev.wordpress.com/2006/12/04/non-standard-analysis-and-automatic-differentiation/ Non-standard analysis, automatic differentiation, Haskell, and other stories.]
* [http://sigfpe.blogspot.com/2005/07/automatic-differentiation.html Automatic Differentiation]
* [http://cdsmith.wordpress.com/2007/11/29/some-playing-with-derivatives/ Some Playing with Derivatives]
* [http://conal.net/blog/posts/paper-beautiful-differentiation/ Beautiful differentiation by Conal Elliott.] The paper itself and link to video of ICFP talk on the subject are available from his [http://conal.net/papers/beautiful-differentiation/ site].

== Code ==

* [http://hackage.haskell.org/package/fad Forward accumulation mode Automatic Differentiation] Hackage package
* [http://hackage.haskell.org/package/vector-space Vector-space package], including derivatives as linear transformations satisfying chain rule.

[[Category:Mathematics]]