Fermi-Dirac Integrals (FDINT)

FDINT is a free, open-source python package that provides fast, double precision (64-bit floating point) approximations to the Fermi-Dirac integrals of integer and half integer order, based on the work by Prof. Fukushima [1].

[1]	T. Fukushima, "Precise and fast computation of Fermi-Dirac integral of integer and half integer order by piecewise minimax rational approximation," Applied Mathematics and Computation, vol. 259, pp. 708-729, May 2015.

The source code and documentation (coming soon) are graciously hosted by GitHub.

Installation

In order to use FDINT, you must having a working Python distribution installed. Python 3 support has not yet been tested, so Python 2.7 is suggested. In order to achieve the highest performance, a Fortran 90 compiler, such as gfortran, is required.

From PyPi

This is the easiest method. Install from PyPi by running the following command:

pip install fdint

From Github

First, you will need to install the following prerequisite package:

- Numpy_

Additional functionality is provided by the following optional packages:

- Matplotlib_

Once these are installed, download the latest release .zip or .tar.gz source package from the github page, extract its contents, and run python setup.py install from within the extracted directory.

Testing

Once installed, you can test the package by running the following command:

python -m fdint.tests

If you have Matplotlib installed, you can also plot a sample of the available functions by running the following command:

python -m fdint

Tutorial

First, start up an interactive python shell from the command line:

$ python

Next, import everything from the fdint package:

>>> from fdint import *

Now you can access the Fermi-Dirac integral and derivative convenience functions, fdk and dfdk:

>>> fdk(k=0.5,phi=-10)
4.0233994366893939e-05
>>> fdk(0.5,-10)
4.0233994366893939e-05
>>> fdk(k=0.5,phi=5)
7.837976057293096
>>> fdk(k=0.5,phi=50)
235.81861512588432
>>> dfdk(k=0.5,phi=-10,d=1) # first derivative
4.0233348580568672e-05
>>> dfdk(k=0.5,phi=5,d=1)
2.1916282173557855
>>> dfdk(k=0.5,phi=50,d=1)
7.0699026455055112
>>> dfdk(k=0.5,phi=50,d=2) # second derivative
0.07074571454521902

You can also pass in lists or arrays as phi:

>>> fdk(k=0.5,phi=[-10,0,10])
array([  4.02339944e-05,   6.78093895e-01,   2.13444715e+01])

If you request an order or derivative that is not implemented, a NotImplementedError is raised:

>>> fdk(22,0)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "fdint/fdint.py", line 68, in fdk
    raise NotImplementedError()
NotImplementedError
>>> dfdk(k=0.5,phi=50,d=10)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "fdint/fdint.py", line 99, in dfdk
    raise NotImplementedError()
NotImplementedError

If you prefer to call the low-level functions directly to avoid overhead, you can access them from the fd module:

>>> fd.fd1h(-10) # k=1/2
4.023399436689394e-05
>>> fd.vfd1h([-10,0,10]) # k=1/2
array([  4.02339944e-05,   6.78093895e-01,   2.13444715e+01])

Benchmarks

For single values, calling the function for a specific order is ~7x faster than calling fdk:

$ python -m timeit -s "from fdint import fdk" "fdk(0.5, 10)"
1000000 loops, best of 3: 1.1 usec per loop
$ python -m timeit -s "from fdint.fd import fd1h" "fd1h(10)"
10000000 loops, best of 3: 0.153 usec per loop

However, even for a fairly small array of 1000, most of the advantage is lost:

$ python -m timeit -s "from fdint import fdk; import numpy; x=numpy.linspace(-100,100,1000)" "fdk(0.5, x)"
100000 loops, best of 3: 13.8 usec per loop
$ python -m timeit -s "from fdint.fd import vfd1h; import numpy; x=numpy.linspace(-100,100,1000)" "vfd1h(x)"
100000 loops, best of 3: 12.9 usec per loop

Overall, the performance is excellent. Note that the call time is within a factor of 2 of numpy.exp.

$ python -m timeit -s "import numpy; from numpy import exp; x=numpy.linspace(-100,100,1000)" "exp(x)" 100000 loops, best of 3: 7.49 usec per loop

Documentation

The documentation (coming soon) is graciously hosted by GitHub.