ad
Fast, transparent first- and second-order automatic differentiation
``ad`` Package Documentation
============================
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.
**All base numeric types are supported** (``int``, ``float``, ``complex``,
etc.). This package is designed so that the underlying numeric types will
interact with each other *as they normally do* when performing any
calculations. Thus, this package acts more like a "wrapper" that simply helps
keep track of derivatives while **maintaining the original functionality** of
the numeric calculations.
From the Wikipedia entry on `Automatic differentiation`_ (AD):
"AD exploits the fact that every computer program, no matter how
complicated, executes a sequence of elementary arithmetic operations
(addition, subtraction, multiplication, division, etc.) and elementary
functions (exp, log, sin, cos, etc.). By applying the chain rule
repeatedly to these operations, derivatives of arbitrary order can be
computed automatically, and accurate to working precision."
See the `package documentation`_ for details and examples.
.. image:: https://travis-ci.org/tisimst/ad.png?branch=master
Basic examples
--------------
Let's start with the main import that all numbers use to track derivatives::
>>> from ad import adnumber
Creating AD objects (either a scalar or an N-dimensional array is acceptable)::
>>> x = adnumber(2.0)
>>> x
ad(2.0)
>>> y = adnumber([1, 2, 3])
>>> y
[ad(1), ad(2), ad(3)]
>>> z = adnumber(3, tag='z') # tags can help track variables
>>> z
ad(3, z)
Now for some math::
>>> square = x**2
>>> square
ad(4.0)
>>> sum_value = sum(y)
>>> sum_value
ad(6)
>>> w = x*z**2
>>> w
ad(18.0)
Using more advanced math functions like those in the standard `math`_
and `cmath`_ modules::
>>> from ad.admath import * # sin, cos, log, exp, sqrt, etc.
>>> sin(1 + x**2)
ad(-0.9589242746631385)
Calculating derivatives (evaluated at the given input values)::
>>> square.d(x) # get the first derivative wrt x
4.0
>>> square.d2(x) # get the second derivative wrt x
2.0
>>> z.d(x) # returns zero if the derivative doesn't exist
0.0
>>> w.d2c(x, z) # second cross-derivatives, order doesn't matter
6.0
>>> w.d2c(z, z) # equivalent to "w.d2(z)"
4.0
>>> w.d() # a dict of all relevant derivatives shown if no input
{ad(2.0): 9.0, ad(3, z): 12.0}
Some convenience functions (useful in optimization)::
>>> w.gradient([x, z]) # show the gradient in the order given
[9.0, 12.0]
>>> w.hessian([x, z])
[[0.0, 6.0], [6.0, 4.0]]
>>> sum_value.gradient(y) # works well with input arrays
[1.0, 1.0, 1.0]
# multiple dependents, multiple independents, first derivatives
>>> from ad import jacobian
>>> jacobian([w, square], [x, z])
[[9.0, 12.0], [4.0, 0.0]]
Working with `NumPy`_ arrays (many functions should work out-of-the-box)::
>>> import numpy as np
>>> arr = np.array([1, 2, 3])
>>> a = adnumber(arr)
>>> a.sum()
ad(6)
>>> a.max()
ad(3)
>>> a.mean()
ad(2.0)
>>> a.var() # array variance
ad(0.6666666666666666)
>>> print sqrt(a) # vectorized operations supported with ad operators
[ad(1.0) ad(1.4142135623730951) ad(1.7320508075688772)]
Interfacing with `scipy.optimize`_
----------------------------------
To make it easier to work with the `scipy.optimize`_ module, there's a
**convenient way to wrap functions** that will generate appropriate gradient
and hessian functions::
>>> from ad import gh # the gradient and hessian function generator
>>> def objective(x):
... return (x[0] - 10.0)**2 + (x[1] + 5.0)**2
>>> grad, hess = gh(objective) # now gradient and hessian are automatic!
>>> from scipy.optimize import minimize
>>> x0 = np.array([24, 17])
>>> bnds = ((0, None), (0, None))
>>> method = 'L-BFGS-B'
>>> res = minimize(objective, x0, method=method, jac=grad, bounds=bnds,
... options={'ftol': 1e-8, 'disp': False})
>>> res.x # optimal parameter values
array([ 10., 0.])
>>> res.fun # optimal objective
25.0
>>> res.jac # gradient at optimum
array([ 7.10542736e-15, 1.00000000e+01])
Main Features
-------------
- **Transparent calculations with derivatives: no or little
modification of existing code** is needed, including when using
the `Numpy`_ module.
- **Almost all mathematical operations** are supported, including
functions from the standard math_ module (sin, cos, exp, erf,
etc.) and cmath_ module (phase, polar, etc.) with additional convenience
trigonometric, hyperbolic, and logarithmic functions (csc, acoth, ln, etc.).
Comparison operators follow the **same rules as the underlying numeric
types**.
- **Real and complex** arithmetic handled seamlessly. Treat objects as you
normally would using the `math`_ and `cmath`_ functions, but with their new
``admath`` counterparts.
- **Automatic gradient and hessian function generator** for optimization
studies using `scipy.optimize`_ routines with ``gh(your_func_here)``.
- **Compatible Linear Algebra Routines** in the ``ad.linalg`` submodule,
similar to those found in NumPy's ``linalg`` submodule, that are not
dependent on LAPACK. There are currently:
a. Decompositions
1. ``chol``: Cholesky Decomposition
2. ``lu``: LU Decomposition
3. ``qr``: QR Decomposition
b. Solving equations and inverting matrices
1. ``solve``: General solver for linear systems of equations
2. ``lstsq``: Least-squares solver for linear systems of equations
3. ``inv``: Solve for the (multiplicative) inverse of a matrix
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 install --upgrade ad`` from the
command-line.
5. Download the *bleeding-edge* version on GitHub_
Python 3
--------
Download the file below, unzip it to any directory, and run::
$ python setup.py install
or::
$ python3 setup.py install
If bugs continue to pop up, please email the author.
Contact
-------
Please send **feature requests, bug reports, or feedback** to
`Abraham Lee`_.
Acknowledgements
----------------
The author expresses his thanks to :
- `Eric O. LEBIGOT (EOL)`_, author of the `uncertainties`_ package, for providing
code insight and inspiration
- Stephen Marks, professor at Pomona College, for useful feedback concerning
the interface with optimization routines in ``scipy.optimize``.
.. _NumPy: http://numpy.scipy.org/
.. _math: http://docs.python.org/library/math.html
.. _cmath: http://docs.python.org/library/cmath.html
.. _Automatic differentiation: http://en.wikipedia.org/wiki/Automatic_differentiation
.. _Eric O. LEBIGOT (EOL): http://www.linkedin.com/pub/eric-lebigot/22/293/277
.. _uncertainties: http://pypi.python.org/pypi/uncertainties
.. _scipy.optimize: http://docs.scipy.org/doc/scipy/reference/optimize.html
.. _Abraham Lee: mailto:tisimst@gmail.com
.. _package documentation: http://pythonhosted.org/ad
.. _GitHub: https://github.com/tisimst/ad
Abraham Lee
dd50815c28163fed87c5b94a222f7cc00d1acadd
1.2.2