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Functional differentiation

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== Introduction ==
 
== Introduction ==
   
Functional differentiation means computing or approximating the deriviative of a function.
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Functional differentiation means computing or approximating the derivative of a function.
 
There are several ways to do this:
 
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>.
 
* 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.
 
* Compute the derivative of <math>f</math> [[Typeful_symbolic_differentiation|symbolically]]. This approach is particularly interesting for Haskell.
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== Functional analysis ==
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If you want to explain the terms [[Higher order function]] and [[Currying]] to mathematicians, this is certainly a good example.
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The mathematician writes
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: <math> D f (x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}</math>
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and the Haskell programmer writes
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<haskell>
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derive :: a -> (a -> a) -> (a -> a)
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derive h f x = (f (x+h) - f x) / h .
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</haskell>
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Haskell's <hask>derive h</hask> approximates the mathematician's <math> D </math>.
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In functional analysis <math> D </math> is called a (linear) function operator, because it maps functions to functions.
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In Haskell <hask>derive h</hask> is called a higher order function for the same reason.
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<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 ==

Revision as of 19:31, 10 October 2007

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 :: 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.