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lines 7-63 of file: example/general/bender_quad.cpp
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{xrst_begin bender_quad.cpp app}
{xrst_spell
   argmin
}

BenderQuad: Example and Test
############################

Define
:math:`F : \B{R} \times \B{R} \rightarrow \B{R}` by

.. math::

   F(x, y)
   =
   \frac{1}{2} \sum_{i=1}^N [ y * \sin ( x * t_i ) - z_i ]^2

where :math:`z \in \B{R}^N` is a fixed vector.
It follows that

.. math::
   :nowrap:

   \begin{eqnarray}
   \partial_y F(x, y)
   & = &
   \sum_{i=1}^N [ y * \sin ( x * t_i ) - z_i ] \sin( x * t_i )
   \\
   \partial_y \partial_y F(x, y)
   & = &
   \sum_{i=1}^N \sin ( x t_i )^2
   \end{eqnarray}

Furthermore if we define :math:`Y(x)`
as the argmin of :math:`F(x, y)` with respect to :math:`y`,

.. math::
   :nowrap:

   \begin{eqnarray}
   Y(x)
   & = &
   y - [ \partial_y \partial_y F(x, y) ]^{-1} \partial_y F[x,  y]
   \\
   & = &
   \left.
      \sum_{i=1}^N z_i \sin ( x t_i )
         \right/
            \sum_{i=1}^N z_i \sin ( x * t_i )^2
   \end{eqnarray}

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

{xrst_end bender_quad.cpp}