-----------------------------------------------------------
lines 6-91 of file: example/atomic_four/mat_mul/forward.cpp
-----------------------------------------------------------

{xrst_begin atomic_four_mat_mul_forward.cpp}
{xrst_spell
   cccccccc
}

Atomic Matrix Multiply Forward Mode: Example and Test
#####################################################

Purpose
*******
This example demonstrates using forward mode with
the :ref:`atomic_four_mat_mul-name` class.

f(x)
****
For this example, the function :math:`f(x)` is

.. math::

   f(x) =
   \left( \begin{array}{cc}
   x_0 & x_1  \\
   x_2 & x_3  \\
   x_4 & x_5
   \end{array} \right)
   \left( \begin{array}{c}
   x_6  \\
   x_7
   \end{array} \right)
   =
   \left( \begin{array}{c}
   x_0 x_6 + x_1 x_7  \\
   x_2 x_6 + x_3 x_7  \\
   x_4 x_6 + x_5 x_7
   \end{array} \right)

Jacobian of f(x)
****************
The Jacobian of :math:`f(x)` is

.. math::

   f^{(1)} (x) = \left( \begin{array}{cccccccc}
   x_6 & x_7 & 0   & 0    & 0    & 0   & x_0  & x_1 \\
   0   & 0   & x_6 & x_7  & 0    & 0   & x_2  & x_3 \\
   0   & 0   & 0   & 0    & x_6  & x_7 & x_4  & x_5
   \end{array} \right)

g(x)
****
We define the function :math:`g(x) = f_1^{(1)} (x)^\R{T}`; i.e.,

.. math::

   g(x) = ( 0,  0,  x_6,  x_7,  0,  0,  x_2,  x_3 )^\R{T}

Hessian
*******
The Hessian of :math:`f_1(x)` is the Jacobian
of :math:`g(x)`; i.e.,

.. math::

   f_1^{(2)} (x)
   =
   g^{(1)} (x)
   =
   \left( \begin{array}{cccccccc}
      0   & 0   & 0   & 0    & 0    & 0   & 0    & 0   \\
      0   & 0   & 0   & 0    & 0    & 0   & 0    & 0   \\
      0   & 0   & 0   & 0    & 0    & 0   & 1    & 0   \\
      0   & 0   & 0   & 0    & 0    & 0   & 0    & 1   \\
      0   & 0   & 0   & 0    & 0    & 0   & 0    & 0   \\
      0   & 0   & 0   & 0    & 0    & 0   & 0    & 0   \\
      0   & 0   & 1   & 0    & 0    & 0   & 0    & 0   \\
      0   & 0   & 0   & 1    & 0    & 0   & 0    & 0   \\
   \end{array} \right)

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

{xrst_end atomic_four_mat_mul_forward.cpp}