Personal tools

Functional differentiation

From HaskellWiki

(Difference between revisions)
Jump to: navigation, search
Current revision (08:32, 19 December 2010) (edit) (undo)
m (→Code)
 
(8 intermediate revisions not shown.)
Line 1: Line 1:
 +
== 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 ==
Line 6: Line 28:
* [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]]

Current revision

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 \frac{f(x+h)-f(x)}{h} where h is close to zero. (or at best the square root of the machine precision \varepsilon.
  • 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

D f (x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}

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 D.

In functional analysis D 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.

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.

4 Code

Retrieved from "http://www.haskell.org/haskellwiki/Functional_differentiation"

Category: Mathematics