Functional differentiation
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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 == | ||
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* [http://hackage.haskell.org/package/fad Forward accumulation mode Automatic Differentiation] Hackage package | * [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
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
In functional analysis D is called a (linear) function operator, because it maps functions to functions.
In Haskellderive 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 chain rule.
