# Difference between revisions of "Functional differentiation"

(→Code) |
(use correct type signature) |
||

Line 13: | Line 13: | ||

and the Haskell programmer writes |
and the Haskell programmer writes |
||

<haskell> |
<haskell> |
||

− | derive :: a -> (a -> a) -> (a -> a) |
+ | derive :: (Fractional a) => a -> (a -> a) -> (a -> a) |

derive h f x = (f (x+h) - f x) / h . |
derive h f x = (f (x+h) - f x) / h . |
||

</haskell> |
</haskell> |
||

Line 20: | Line 20: | ||

In Haskell <hask>derive h</hask> is called a higher order function for the same reason. |
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>. |
<math> D </math> is in curried form. If it would be uncurried, you would write <math> D(f,x) </math>. |
||

− | |||

== Blog Posts == |
== Blog Posts == |

## Revision as of 22:34, 8 December 2010

## Introduction

Functional differentiation means computing or approximating the derivative of a function. There are several ways to do this:

- Approximate the derivative by where is close to zero. (or at best the square root of the machine precision .
- Compute the derivative of 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

and the Haskell programmer writes

```
derive :: (Fractional a) => a -> (a -> a) -> (a -> a)
derive h f x = (f (x+h) - f x) / h .
```

Haskell's `derive h`

approximates the mathematician's .
In functional analysis is called a (linear) function operator, because it maps functions to functions.
In Haskell `derive h`

is called a higher order function for the same reason.
is in curried form. If it would be uncurried, you would write .

## 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.

- Overloading Haskell numbers, part 2, Forward Automatic Differentiation.
- Non-standard analysis, automatic differentiation, Haskell, and other stories.
- Automatic Differentiation
- Some Playing with Derivatives
- Beautiful differentiation by Conal Elliott. The paper itself and link to video of ICFP talk on the subject are available from his site.

## Code

- Forward accumulation mode Automatic Differentiation Hackage package
- Vector-space package, including derivatives as linear transformations satisfying product rule.