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// SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
// SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
// SPDX-FileContributor: 2003-22 Bradley M. Bell
// ----------------------------------------------------------------------------
/*
{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}
*/
// BEGIN C++
# include <limits>
# include <cppad/cppad.hpp>
namespace {
using CppAD::AD;
typedef CPPAD_TESTVECTOR(double) BaseVector;
typedef CPPAD_TESTVECTOR(AD<double>) ADVector;
class Fun {
private:
const BaseVector t_; // measurement times
const BaseVector z_; // measurement values
public:
typedef ADVector ad_vector;
// constructor
Fun(const BaseVector &t, const BaseVector &z)
: t_(t) , z_(z)
{ assert( t.size() == z.size() ); }
// ell
size_t ell(void) const
{ return t_.size(); }
// Fun.s
AD<double> s(size_t k, const ad_vector& x, const ad_vector& y) const
{
AD<double> residual = y[0] * sin( x[0] * t_[k] ) - z_[k];
AD<double> s_k = .5 * residual * residual;
return s_k;
}
// Fun.sy
ad_vector sy(size_t k, const ad_vector& x, const ad_vector& y) const
{ assert( y.size() == 1);
ad_vector sy_k(1);
AD<double> residual = y[0] * sin( x[0] * t_[k] ) - z_[k];
sy_k[0] = residual * sin( x[0] * t_[k] );
return sy_k;
}
};
// Used to test calculation of Hessian of V
AD<double> V(const ADVector& x, const BaseVector& t, const BaseVector& z)
{ // compute Y(x)
AD<double> numerator = 0.;
AD<double> denominator = 0.;
size_t k;
for(k = 0; k < size_t(t.size()); k++)
{ numerator += sin( x[0] * t[k] ) * z[k];
denominator += sin( x[0] * t[k] ) * sin( x[0] * t[k] );
}
AD<double> y = numerator / denominator;
// V(x) = F[x, Y(x)]
AD<double> sum = 0;
for(k = 0; k < size_t(t.size()); k++)
{ AD<double> residual = y * sin( x[0] * t[k] ) - z[k];
sum += .5 * residual * residual;
}
return sum;
}
}
bool opt_val_hes(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
// temporary indices
size_t j, k;
// x space vector
size_t n = 1;
BaseVector x(n);
x[0] = 2. * 3.141592653;
// y space vector
size_t m = 1;
BaseVector y(m);
y[0] = 1.;
// t and z vectors
size_t ell = 10;
BaseVector t(ell);
BaseVector z(ell);
for(k = 0; k < ell; k++)
{ t[k] = double(k) / double(ell); // time of measurement
z[k] = y[0] * sin( x[0] * t[k] ); // data without noise
}
// construct the function object
Fun fun(t, z);
// evaluate the Jacobian and Hessian
BaseVector jac(n), hes(n * n);
# ifndef NDEBUG
int signdet =
# endif
CppAD::opt_val_hes(x, y, fun, jac, hes);
// we know that F_yy is positive definite for this case
assert( signdet == 1 );
// create ADFun object g corresponding to V(x)
ADVector a_x(n), a_v(1);
for(j = 0; j < n; j++)
a_x[j] = x[j];
Independent(a_x);
a_v[0] = V(a_x, t, z);
CppAD::ADFun<double> g(a_x, a_v);
// accuracy for checks
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
// check Jacobian
BaseVector check_jac = g.Jacobian(x);
for(j = 0; j < n; j++)
ok &= NearEqual(jac[j], check_jac[j], eps, eps);
// check Hessian
BaseVector check_hes = g.Hessian(x, 0);
for(j = 0; j < n*n; j++)
ok &= NearEqual(hes[j], check_hes[j], eps, eps);
return ok;
}
// END C++
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