$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
opt_val_hes: Example and Test
Fix $z \in \B{R}^\ell$ and define the functions $S_k : \B{R} \times \B{R} \rightarrow \B{R}^\ell$ by and $F : \B{R} \times \B{R} \rightarrow \B{R}$ by $$\begin{array}{rcl} S_k (x, y) & = & \frac{1}{2} [ y * \sin ( x * t_k ) - z_k ]^2 \\ F(x, y) & = & \sum_{k=0}^{\ell-1} S_k (x, y) \end{array}$$ It follows that $$\begin{array}{rcl} \partial_y F(x, y) & = & \sum_{k=0}^{\ell-1} [ y * \sin ( x * t_k ) - z_k ] \sin( x * t_k ) \\ \partial_y \partial_y F(x, y) & = & \sum_{k=0}^{\ell-1} \sin ( x t_k )^2 \end{array}$$ Furthermore if we define $Y(x)$ as solving the equation $\partial F[ x, Y(x) ] = 0$ we have $$\begin{array}{rcl} 0 & = & \sum_{k=0}^{\ell-1} [ Y(x) * \sin ( x * t_k ) - z_k ] \sin( x * t_k ) \\ Y(x) \sum_{k=0}^{\ell-1} \sin ( x * t_k )^2 - \sum_{k=0}^{\ell-1} \sin ( x * t_k ) z_k \\ Y(x) & = & \frac{ \sum_{k=0}^{\ell-1} \sin( x * t_k ) z_k }{ \sum_{k=0}^{\ell-1} \sin ( x * t_k )^2 } \end{array}$$

# include <limits>

namespace {

class Fun {
private:
const BaseVector t_;    // measurement times
const BaseVector z_;    // measurement values
public:
// 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> residual = y[0] * sin( x[0] * t_[k] ) - z_[k];
AD<double> s_k      = .5 * residual * residual;

return s_k;
}
// Fun.sy
{     assert( y.size() == 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
{     // compute Y(x)
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)]
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;

// 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

// we know that F_yy is positive definate for this case
assert( signdet == 1 );

// create ADFun object g corresponding to V(x)
for(j = 0; j < n; j++)
a_x[j] = x[j];
Independent(a_x);
a_v[0] = V(a_x, t, z);

// 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;
}

Input File: example/general/opt_val_hes.cpp