lines 6-74 of file: example/atomic_four/mat_mul/rev_depend.cpp

{xrst_begin atomic_four_mat_mul_rev_depend.cpp}
{xrst_spell
   multiplications
}

Atomic Matrix Multiply Reverse Dependency: Example and Test
###########################################################

Purpose
*******
This example uses the atomic matrix multiply
``rev_depend`` function to reduce the number of variables in
the recording of :math:`g(u)`.

f(u)
****

.. math::

   f(u) =
   \left( \begin{array}{cc}
   2 u_0 & 2 u_1  \\
   2 u_2 & 2 u_3  \\
   \end{array} \right)
   \left( \begin{array}{cc}
   2 u_4 & 2 u_5  \\
   2 u_6 & 2 u_7
   \end{array} \right)
   =
   \left( \begin{array}{cc}
   4( u_0 u_4 + u_1 u_6 )  & 4( u_0 u_5 + u_1 u_7 )  \\
   4( u_2 u_4 + u_3 u_6 )  & 4( u_2 u_5 + u_3 u_7 )  \\
   \end{array} \right)

.. math::

   f_{0,0} (u)
   =
   4 ( u_0 u_4 + u_1 u_6 )

Forward Analysis
****************
Forward dependency analysis determines that there
is a new variable for each of the 8 multiplications by 2.0.
It also determines, using :ref:`for_type<atomic_four_mat_mul_for_type.hpp-name>`
that each of the 4 elements in the matrix product result is a new variable.

Reverse Analysis
****************
Reverse analysis detect that only 1 of the 4 elements
in the matrix product is used.
In addition it determines,
using :ref:`rev_depend<atomic_four_mat_mul_rev_depend.hpp-name>` ,
that only 4 of the 8 multiplications by 2.0 are used.

size_var
********
The difference in :ref:`fun_property@size_var`
is the difference between only using forward dependency and using both; i.e.,
(8 - 4) + (4 - 1) = 7.

Source
******
{xrst_literal
   // BEGIN C++
   // END C++
}

{xrst_end atomic_four_mat_mul_rev_depend.cpp}