Functional differentiation

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1 Introduction
2 Functional analysis
3 Blog Posts
4 Code

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.

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.

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

Category: Mathematics

@@ Line 34: / Line 34: @@
 * [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 product rule.
+* [http://hackage.haskell.org/package/vector-space Vector-space package], including derivatives as linear transformations satisfying chain rule.
 [[Category:Mathematics]]

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