Functional differentiation

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.

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

$Df(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

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 $D(f,x)$.

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.

Code

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