Special Functions (:mod:`scipy.special`)
========================================

.. currentmodule:: scipy.special

The main feature of the :mod:`scipy.special` package is the definition of
numerous special functions of mathematical physics. Available
functions include airy, elliptic, bessel, gamma, beta, hypergeometric,
parabolic cylinder, mathieu, spheroidal wave, struve, and
kelvin. There are also some low-level stats functions that are not
intended for general use as an easier interface to these functions is
provided by the ``stats`` module. Most of these functions can take
array arguments and return array results following the same
broadcasting rules as other math functions in Numerical Python. Many
of these functions also accept complex numbers as input. For a
complete list of the available functions with a one-line description
type ``>>> help(special).`` Each function also has its own
documentation accessible using help.  If you don't see a function you
need, consider writing it and contributing it to the library. You can
write the function in either C, Fortran, or Python. Look in the source
code of the library for examples of each of these kinds of functions.


Bessel functions of real order(:func:`jv`, :func:`jn_zeros`)
------------------------------------------------------------

Bessel functions are a family of solutions to Bessel's differential equation
with real or complex order alpha:

.. math::
   x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0

Among other uses, these functions arise in wave propagation problems, such as
the vibrational modes of a thin drum head.  Here is an example of a circular
drum head anchored at the edge:

.. plot::
   :alt: "This code generates a 3-D representation of the vibrational modes on a drum head viewed at a three-quarter angle. A circular region on the X-Y plane is defined with a Z value of 0 around the edge. Within the circle a single smooth valley exists on the -X side and a smooth peak exists on the +X side. The image resembles a yin-yang at this angle."

   >>> from scipy import special
   >>> import numpy as np
   >>> def drumhead_height(n, k, distance, angle, t):
   ...    kth_zero = special.jn_zeros(n, k)[-1]
   ...    return np.cos(t) * np.cos(n*angle) * special.jn(n, distance*kth_zero)
   >>> theta = np.r_[0:2*np.pi:50j]
   >>> radius = np.r_[0:1:50j]
   >>> x = np.array([r * np.cos(theta) for r in radius])
   >>> y = np.array([r * np.sin(theta) for r in radius])
   >>> z = np.array([drumhead_height(1, 1, r, theta, 0.5) for r in radius])

   >>> import matplotlib.pyplot as plt
   >>> fig = plt.figure()
   >>> ax = fig.add_axes(rect=(0, 0.05, 0.95, 0.95), projection='3d')
   >>> ax.plot_surface(x, y, z, rstride=1, cstride=1, cmap='RdBu_r', vmin=-0.5, vmax=0.5)
   >>> ax.set_xlabel('X')
   >>> ax.set_ylabel('Y')
   >>> ax.set_xticks(np.arange(-1, 1.1, 0.5))
   >>> ax.set_yticks(np.arange(-1, 1.1, 0.5))
   >>> ax.set_zlabel('Z')
   >>> plt.show()

..   :caption: Vibrating drum head using
..             :obj:`scipy.special.jn`


Cython Bindings for Special Functions (:mod:`scipy.special.cython_special`)
---------------------------------------------------------------------------

.. highlight:: cython

SciPy also offers Cython bindings for scalar, typed versions of many
of the functions in special. The following Cython code gives a simple
example of how to use these functions::

  cimport scipy.special.cython_special as csc

  cdef:
      double x = 1
      double complex z = 1 + 1j
      double si, ci, rgam
      double complex cgam

  rgam = csc.gamma(x)
  print(rgam)
  cgam = csc.gamma(z)
  print(cgam)
  csc.sici(x, &si, &ci)
  print(si, ci)

(See the `Cython documentation`_ for help with compiling Cython.) In
the example the function ``csc.gamma`` works essentially like its
ufunc counterpart `gamma`, though it takes C types as arguments
instead of NumPy arrays. Note, in particular, that the function is
overloaded to support real and complex arguments; the correct variant
is selected at compile time. The function ``csc.sici`` works slightly
differently from `sici`; for the ufunc we could write ``ai, bi =
sici(x)``, whereas in the Cython version multiple return values are
passed as pointers. It might help to think of this as analogous to
calling a ufunc with an output array: ``sici(x, out=(si, ci))``.

There are two potential advantages to using the Cython bindings:

- they avoid Python function overhead
- they do not require the Python Global Interpreter Lock (GIL)

The following sections discuss how to use these advantages to
potentially speed up your code, though, of course, one should always
profile the code first to make sure putting in the extra effort will
be worth it.

Avoiding Python Function Overhead
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

For the ufuncs in special, Python function overhead is avoided by
vectorizing, that is, by passing an array to the function. Typically,
this approach works quite well, but sometimes it is more convenient to
call a special function on scalar inputs inside a loop, for example,
when implementing your own ufunc. In this case, the Python function
overhead can become significant. Consider the following example::

  import scipy.special as sc
  cimport scipy.special.cython_special as csc

  def python_tight_loop():
      cdef:
          int n
          double x = 1

      for n in range(100):
          sc.jv(n, x)

  def cython_tight_loop():
      cdef:
          int n
          double x = 1

      for n in range(100):
          csc.jv(n, x)

On one computer ``python_tight_loop`` took about 131 microseconds to
run and ``cython_tight_loop`` took about 18.2 microseconds to
run. Obviously this example is contrived: one could just call
``special.jv(np.arange(100), 1)`` and get results just as fast as in
``cython_tight_loop``. The point is that if Python function overhead
becomes significant in your code, then the Cython bindings might be
useful.

Releasing the GIL
^^^^^^^^^^^^^^^^^

One often needs to evaluate a special function at many points, and
typically the evaluations are trivially parallelizable. Since the
Cython bindings do not require the GIL, it is easy to run them in
parallel using Cython's ``prange`` function. For example, suppose that
we wanted to compute the fundamental solution to the Helmholtz
equation:

.. math::

   \Delta_x G(x, y) + k^2G(x, y) = \delta(x - y),

where :math:`k` is the wavenumber and :math:`\delta` is the Dirac
delta function. It is known that in two dimensions the unique
(radiating) solution is

.. math::

   G(x, y) = \frac{i}{4}H_0^{(1)}(k|x - y|),

where :math:`H_0^{(1)}` is the Hankel function of the first kind,
i.e., the function `hankel1`. The following example shows how we could
compute this function in parallel::

  from libc.math cimport fabs
  cimport cython
  from cython.parallel cimport prange

  import numpy as np
  import scipy.special as sc
  cimport scipy.special.cython_special as csc

  def serial_G(k, x, y):
      return 0.25j*sc.hankel1(0, k*np.abs(x - y))

  @cython.boundscheck(False)
  @cython.wraparound(False)
  cdef void _parallel_G(double k, double[:,:] x, double[:,:] y,
                        double complex[:,:] out) nogil:
      cdef int i, j

      for i in prange(x.shape[0]):
          for j in range(y.shape[0]):
              out[i,j] = 0.25j*csc.hankel1(0, k*fabs(x[i,j] - y[i,j]))

  def parallel_G(k, x, y):
      out = np.empty_like(x, dtype='complex128')
      _parallel_G(k, x, y, out)
      return out

(For help with compiling parallel code in Cython see `here`_.) If the
above Cython code is in a file ``test.pyx``, then we can write an
informal benchmark which compares the parallel and serial versions of
the function::

  import timeit

  import numpy as np

  from test import serial_G, parallel_G

  def main():
      k = 1
      x, y = np.linspace(-100, 100, 1000), np.linspace(-100, 100, 1000)
      x, y = np.meshgrid(x, y)

      def serial():
          serial_G(k, x, y)

      def parallel():
          parallel_G(k, x, y)

      time_serial = timeit.timeit(serial, number=3)
      time_parallel = timeit.timeit(parallel, number=3)
      print("Serial method took {:.3} seconds".format(time_serial))
      print("Parallel method took {:.3} seconds".format(time_parallel))

  if __name__ == "__main__":
      main()

On one quad-core computer the serial method took 1.29 seconds and the
parallel method took 0.29 seconds.


Functions not in :mod:`scipy.special`
-------------------------------------

.. highlight:: python

Some functions are not included in special because they are
straightforward to implement with existing functions in NumPy and
SciPy. To prevent reinventing the wheel, this section provides
implementations of several such functions, which hopefully illustrate
how to handle similar functions. In all examples NumPy is imported as
``np`` and special is imported as ``sc``.

The `binary entropy function`_::

  def binary_entropy(x):
      return -(sc.xlogy(x, x) + sc.xlog1py(1 - x, -x))/np.log(2)

A rectangular step function on [0, 1]::

  def step(x):
      return 0.5*(np.sign(x) + np.sign(1 - x))

Translating and scaling can be used to get an arbitrary step function.

The `ramp function`_::

  def ramp(x):
      return np.maximum(0, x)


.. _Cython documentation: http://docs.cython.org/en/latest/src/reference/compilation.html

.. _here: http://docs.cython.org/en/latest/src/userguide/parallelism.html#compiling

.. _`binary entropy function`: https://en.wikipedia.org/wiki/Binary_entropy_function

.. _`ramp function`: https://en.wikipedia.org/wiki/Ramp_function
