.. algorithm::

.. summary::

.. relatedalgorithms::

.. properties::

Description
-----------

This algorithm is used to calculate the asymmetry for a Muon workspace.
It first groups the input workspace according to the spectra numbers
provided as *ForwardSpectra* and *BackwardSpectra*. If these properties
are not supplied, the algorithm assumes that the first spectrum in the
workspace is the forward group and the second one is the backward
group. It then calculates the asymmetry, :math:`A`, as:

.. math:: A = \frac{F-\alpha B}{F+\alpha B},

where :math:`F` is the front spectra, :math:`B` is the back spectra
and :math:`\alpha` is the balance parameter [1]_.

The error in :math:`A`, :math:`\sigma_A` is calculated using standard error propagation. First the errors in :math:`F-\alpha B` and :math:`F+\alpha B` are calculated
by adding the errors in :math:`F` and :math:`B` in quadrature (:math:`\alpha` is assumed to have no error);

.. math:: \sigma_{F + \alpha B} = \sigma_{F - \alpha B} = \sqrt{ \sigma_F^2 + \alpha^2 \sigma_B^2  },

Then the error in the asymmetry is given by;

.. math:: \sigma_A = A \sqrt{ \left( \frac{\sigma_{F + \alpha B} }{F + \alpha B} \right)^2 + \left( \frac{\sigma_{F- \alpha B}}{ F - \alpha B} \right)^2    },

using the fact that the errors on counts :math:`F,B` can be taken to be Poisson errors (:math:`\sigma_{F} = \sqrt{F}`, :math:`\sigma_{B} = \sqrt{B}`), this can be simplified to

.. math:: \sigma_A = \frac{\sqrt{ F + \alpha^2 B} \sqrt{1 + A^2} }{F + \alpha B}.

If any bins have :math:`F=B=0`, then the result is :math:`A=0.0, \sigma_A =1.0`.

The output workspace contains one set of data for the time of flight:
the asymmetry and the asymmetry errors.

.. [1] See :ref:`algm-AlphaCalc`

Usage
-----

**Example - Calculating asymmetry:**

.. testcode:: ExSimple

   y = [1,2,3] + [3,1,12]
   x = [1,2,3,4] * 2
   e = [1,1,1] * 2
   input = CreateWorkspace(x, y, e, NSpec=2)

   asymmetry = AsymmetryCalc(input, Alpha=0.5)

   print('Asymmetry: {}'.format(asymmetry.readY(0)))
   print('Errors: {}'.format(asymmetry.readE(0)))

Output:

.. testoutput:: ExSimple

   Asymmetry: [-0.2         0.6        -0.33333333]
   Errors: [0.5396295  0.69971423 0.28688766]

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