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lines 6-110 of file: example/atomic_four/mat_mul/sparsity.cpp
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{xrst_begin atomic_four_mat_mul_sparsity.cpp}
{xrst_spell
   cccccccc
   cccccccccc
   rvec
}

Atomic Matrix Multiply Sparsity Patterns: Example and Test
##########################################################

Purpose
*******
This example demonstrates computing sparsity patterns with
the :ref:`atomic_four_mat_mul-name` class.

f(x)
****
For a matrix :math:`A` we define the function :math:`\R{rvec} ( A )`
to be the elements of :math:`A` in row major order.
For this example, the function :math:`f(x)` is

.. math::

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

.. math::

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

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

.. math::

   f^{(1)} (x) = \left( \begin{array}{cccccccc}
   % 0   1     2     3      4      5     6      7
   x_4 & x_6 & 0   & 0    & x_0  & 0   & x_1  & 0   \\ % 0
   x_5 & x_7 & 0   & 0    & 0    & x_0 & 0    & x_1 \\ % 1
   0   & 0   & x_4 & x_6  & x_2  & 0   & x_3  & 0   \\ % 2
   0   & 0   & x_5 & x_7  & 0    & x_2 & 0    & x_3 \\ % 3
   \end{array} \right)

Hessian
*******
The function :math:`f_2 (x)` is

.. math::

   f_2 (x) = x_2 x_4 + x_3 x_6

The Hessian of :math:`f_2(x)` is

.. math::

   f_2^{(2)} (x)
   =
   \left( \begin{array}{cccccccccc}
            & 0    & 1    & 2    & 3    & 4    & 5    & 6    & 7    \\
            & -    & -    & -    & -    & -    & -    & -    & -    \\
      0 \; |  & 0    & 0    & 0    & 0    & 0    & 0    & 0    & 0    \\
      1 \; |  & 0    & 0    & 0    & 0    & 0    & 0    & 0    & 0    \\
      2 \; |  & 0    & 0    & 0    & 0    & 1    & 0    & 0    & 0    \\
      3 \; |  & 0    & 0    & 0    & 0    & 0    & 0    & 1    & 0    \\
      4 \; |  & 0    & 0    & 1    & 0    & 0    & 0    & 0    & 0    \\
      5 \; |  & 0    & 0    & 0    & 0    & 0    & 0    & 0    & 0    \\
      6 \; |  & 0    & 0    & 0    & 1    & 0    & 0    & 0    & 0    \\
      7 \; |  & 0    & 0    & 0    & 0    & 0    & 0    & 0    & 0    \\
   \end{array} \right)

where the first row is the column index,
and the first column is the row index,
for the corresponding matrix entries above.

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

{xrst_end atomic_four_mat_mul_sparsity.cpp}