lines 7-65 of file: example/general/opt_val_hes.cpp

{xrst_begin opt_val_hes.cpp app}

opt_val_hes: Example and Test
#############################

Fix :math:`z \in \B{R}^\ell` and define the functions
:math:`S_k : \B{R} \times \B{R} \rightarrow \B{R}^\ell` by and
:math:`F : \B{R} \times \B{R} \rightarrow \B{R}` by

.. math::
   :nowrap:

   \begin{eqnarray}
   S_k (x, y) & = & \frac{1}{2} [ y * \sin ( x * t_k ) - z_k ]^2
   \\
   F(x, y)    & = &  \sum_{k=0}^{\ell-1} S_k (x, y)
   \end{eqnarray}

It follows that

.. math::
   :nowrap:

   \begin{eqnarray}
   \partial_y F(x, y)
   & = &
   \sum_{k=0}^{\ell-1} [ y * \sin ( x * t_k ) - z_k ] \sin( x * t_k )
   \\
   \partial_y \partial_y F(x, y)
   & = &
   \sum_{k=0}^{\ell-1} \sin ( x t_k )^2
   \end{eqnarray}

Furthermore if we define :math:`Y(x)`
as solving the equation :math:`\partial F[ x, Y(x) ] = 0` we have

.. math::
   :nowrap:

   \begin{eqnarray}
   0 & = &
   \sum_{k=0}^{\ell-1} [ Y(x) * \sin ( x * t_k ) - z_k ] \sin( x * t_k )
   \\
   Y(x) \sum_{k=0}^{\ell-1} \sin ( x * t_k )^2
   - \sum_{k=0}^{\ell-1} \sin ( x * t_k ) z_k
   \\
   Y(x) & = & \frac{
      \sum_{k=0}^{\ell-1} \sin( x * t_k ) z_k
      }{
      \sum_{k=0}^{\ell-1} \sin ( x * t_k )^2
   }
   \end{eqnarray}

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

{xrst_end opt_val_hes.cpp}