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fad: Forward Automatic Differentiation.

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Forward Automatic Differentiation via overloading to perform nonstandard interpretation that replaces original numeric type with corresponding generalized dual number type. Existential type "branding" is used to prevent perturbation confusion. **Note: In general we recommend using the ad package maintained by Edward Kmett instead of this package.**

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Versions [RSS] 1.0, 1.1.0.1
Dependencies base (<5) [details]
License BSD-3-Clause
Copyright Barak A. Pearlmutter and Jeffrey Mark Siskind 2008-2009
Author Barak A. Pearlmutter and Jeffrey Mark Siskind
Maintainer bjorn.buckwalter@gmail.com
Category Math
Home page http://github.com/bjornbm/fad
Uploaded by BjornBuckwalter at 2012-12-22T17:26:31Z
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Reverse Dependencies 1 direct, 0 indirect [details]
Downloads 2085 total (11 in the last 30 days)
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Readme for fad-1.1.0.1

[back to package description] 
   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/>