Copyright : 2008-2009, Barak A. Pearlmutter and Jeffrey Mark Siskind License : BSD3

Maintainer : bjorn.buckwalter@gmail.com Stability : experimental Portability: GHC only?

Forward Automatic Differentiation via overloading to perform nonstandard interpretation that replaces original numeric type with corresponding generalized dual number type.

Each invocation of the differentiation function introduces a distinct perturbation, which requires a distinct dual number type. In order to prevent these from being confused, tagging, called branding in the Haskell community, is used. This seems to prevent perturbation confusion, although it would be nice to have an actual proof of this. The technique does require adding invocations of lift at appropriate places when nesting is present.

For more information on perturbation confusion and the solution employed in this library see: http://www.bcl.hamilton.ie/~barak/papers/ifl2005.pdf http://thread.gmane.org/gmane.comp.lang.haskell.cafe/22308/

Installation

To install: cabal install

Or: runhaskell Setup.lhs configure runhaskell Setup.lhs build runhaskell Setup.lhs install

Examples

Define an example function 'f':

import Numeric.FAD f x = 6 - 5 * x + x ^ 2 -- Our example function

Basic usage of the differentiation operator:

y = f 2 -- f(2) = 0 y' = diff f 2 -- First derivative f'(2) = -1 y'' = diff (diff f) 2 -- Second derivative f''(2) = 2

List of derivatives:

ys = take 3 $ diffs f 2 -- [0, -1, 2]

Example optimization method; find a zero using Newton's method:

y_newton1 = zeroNewton f 0 -- converges to first zero at 2.0. y_newton2 = zeroNewton f 10 -- converges to second zero at 3.0.

Credits

Authors: Copyright 2008, Barak A. Pearlmutter barak@cs.nuim.ie & Jeffrey Mark Siskind qobi@purdue.edu

Work started as stripped-down version of higher-order tower code published by Jerzy Karczmarczuk jerzy.karczmarczuk@info.unicaen.fr which used a non-standard standard prelude.

Initial perturbation-confusing code is a modified version of http://cdsmith.wordpress.com/2007/11/29/some-playing-with-derivatives/

Tag trick, called "branding" in the Haskell community, from Bjorn Buckwalter bjorn.buckwalter@gmail.com http://thread.gmane.org/gmane.comp.lang.haskell.cafe/22308/