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@(@\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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation .
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>
# 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 definate 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;
}
Input File: example/general/opt_val_hes.cpp