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Fast, transparent, calculations of first and second-order automatic differentiation package

Overview
========
The ``ad`` package allows you to **easily** and **transparently** perform
**first and second-order automatic differentiation**. Advanced math
involving trigonometric, logarithmic, hyperbolic, etc. functions can also
be evaluated directly using the ``admath`` sub-module.

`Automatic differentiation`_ is different from numerical and symbolic
differentiation in that it uses prior knowledge of how derivatives
are calculated, `but that's the part you don't need to worry about
while using this package`. They are then transmitted through subsequent
calculations (using the generalized `chain rule`_).

Basic examples
==============
::

>>> x

>>> square = x**2
>>> square
>>> square.d(x) # get the first derivative wrt x
4.0
>>> square.d2(x) # get the second derivative wrt x
2.0

>>> from ad.admath import * # sin, cos, log, exp, sqrt, etc.
>>> sin(1 + x**2)

>>> print (2*x + 1000).d() # no inputs shows dict of all derivatives

>>> y = AD(3, tag='y') # tags are useful for tracking original variables
>>> y
y(3.0)
>>> y.d(x) # returns zero if the derivative doesn't exist
0.0

>>> z = x*y**2
>>> z
[9.0, 12.0]
>>> z.d2c(x, y) # second cross-derivatives, order doesn't matter -> (x,y) or (y,x)
6.0
>>> z.hessian([x, y])
[[0.0, 6.0], [6.0, 4.0]]

>>> import numpy as np # most numpy functions work out of the box
>>> arr = np.array(AD([1, 2, 3])) # multiple input support
>>> arr.sum()
>>> arr.max()
>>> arr.mean()
>>> arr.var() # array variance
>>> sqrt(arr) # vectorized operations supported with ad operators

Main Features
=============

- **Transparent calculations with derivatives: no or little
modification of existing code** is needed, including when using
the `Numpy`_ module. The only function (that I have tested, and I
certainly haven't tested most of them) that doesn't work right
out of the box is ``numpy.std`` since it internally calls its
builtin ``sqrt`` function. Two alternatives exist to work around
this: 1) use ``**0.5`` or 2) using the ``admath.sqrt``
operator.
- **Almost all mathematical operations** are supported, including
functions from the standard `math`_ module (sin, cos, exp, erf,
etc.) with additional convenience trigonometric, hyperbolic,
and logarithmic functions (csc, acoth, ln, etc.). Comparison
operators follow the same rules as ``float`` types.
- Nearly all derivative calculations are performed **analytically**
(only the ``gamma`` and ``lgamma`` functions use a high-accuracy
finite difference formula).

Installation
============

You have several easy, convenient options to install the ``ad`` package

1. Download the package files below, unzip to any directory, and run
``python setup.py install`` from the command-line
2. Simply copy the unzipped ``ad-XYZ`` directory to any other location
that python can find it and rename it ``ad``
from the command-line
4. If ``pip`` is installed, run ``pip --upgrade ad`` from the command-line

Contact
=======

Please send **feature requests, bug reports, or feedback** to
`Abraham Lee`_.

Version History
===============

Main changes:

- 1.0.1: Smashed some vectorization bugs
- 1.0: Initial release. Nearly full differentiation support for all
`math`_ module functions.

.. _NumPy: http://numpy.scipy.org/
.. _math: http://docs.python.org/library/math.html
.. _Automatic differentiation: http://en.wikipedia.org/wiki/Automatic_differentiation
.. _chain rule: http://en.wikipedia.org/wiki/Chain_rule
.. _Abraham Lee: mailto:tisimst@gmail.com
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