Fast, transparent, calculations of first and second-order derivatives (automatic differentiation)

## 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

>>> from ad import AD

>>> 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
array([ADF(1.0), ADF(1.41421356237), ADF(1.73205080757)], dtype=object)

## 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 (administrative privileges may be required)

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

3. If setuptools is installed, run easy_install --upgrade 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: Initial release. Nearly full differentiation support for all math module functions.