# Functional differentiation

### From HaskellWiki

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

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* [http://vandreev.wordpress.com/2006/12/04/non-standard-analysis-and-automatic-differentiation/ Non-standard analysis, automatic differentiation, Haskell, and other stories.] | * [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://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]] | [[Category:Mathematics]] |

## Latest revision as of 08:32, 19 December 2010

## Contents |

## [edit] 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.

## [edit] 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*).

## [edit] 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.

## [edit] 4 Code

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