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Automatic Differentiation

Automatic Differentiation roughly means that a numerical value is equipped with a derivative part, which is updated accordingly on every function application. Let the number $x 0$ be equipped with the derivative $x 1$ : $\langle x_0,x_1 \rangle$ . For example the sinus is defined as:

$\sin\langle x_0,x_1 \rangle = \langle \sin x_0, x_1\cdot\cos x_0\rangle$

You see, that's just estimating errors as in physics. However, it becomes more interesting for vector functions.

Implementations:

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1 Power Series

You may count arithmetic with power series also as Automatic Differentiation, since this means just working with all derivatives simultaneously.

Implementation with Haskell 98 type classes: http://darcs.haskell.org/htam/src/PowerSeries/Taylor.hs

With advanced type classes in Numeric Prelude: http://hackage.haskell.org/packages/archive/numeric-prelude/0.0.5/doc/html/MathObj-PowerSeries.html

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2 See also

Functional differentiation
Chris Smith in Haskell-cafe on Hit a wall with the type system

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