# Functional differentiation

### From HaskellWiki

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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> | ||

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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>. | ||

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

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

## Contents |

## 1 Introduction

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

- Approximate the derivative
*f*'(*x*) by where*h*is close to zero. (or at best the square root of the machine precision . - Compute the derivative of
*f*symbolically. This approach is particularly interesting for Haskell.

## 2 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 .

derive h

*D*.

In functional analysis *D* is called a (linear) function operator, because it maps functions to functions.

derive h

*D* is in curried form. If it would be uncurried, you would write *D*(*f*,*x*).

## 3 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.

## 4 Code

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