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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 .
Hessian Times Direction: Example and Test
// Example and test of computing the Hessian times a direction; i.e.,
// given F : R^n -> R and a direction dx in R^n, we compute F''(x) * dx

# include <cppad/cppad.hpp>

namespace { // put this function in the empty namespace
    // F(x) = |x|^2 = x[0]^2 + ... + x[n-1]^2
    template <class Type>
    Type F(CPPAD_TESTVECTOR(Type) &x)
    {   Type sum = 0;
        size_t i = x.size();
        while(i--)
            sum += x[i] * x[i];
        return sum;
    }
}

bool HesTimesDir(void)
{   bool ok = true;                   // initialize test result
    size_t j;                         // a domain variable variable

    using CppAD::AD;
    using CppAD::NearEqual;
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

    // domain space vector
    size_t n = 5;
    CPPAD_TESTVECTOR(AD<double>)  X(n);
    for(j = 0; j < n; j++)
        X[j] = AD<double>(j);

    // declare independent variables and start recording
    CppAD::Independent(X);

    // range space vector
    size_t m = 1;
    CPPAD_TESTVECTOR(AD<double>) Y(m);
    Y[0] = F(X);

    // create f : X -> Y and stop recording
    CppAD::ADFun<double> f(X, Y);

    // choose a direction dx and compute dy(x) = F'(x) * dx
    CPPAD_TESTVECTOR(double) dx(n);
    CPPAD_TESTVECTOR(double) dy(m);
    for(j = 0; j < n; j++)
        dx[j] = double(n - j);
    dy = f.Forward(1, dx);

    // compute ddw = F''(x) * dx
    CPPAD_TESTVECTOR(double) w(m);
    CPPAD_TESTVECTOR(double) ddw(2 * n);
    w[0] = 1.;
    ddw  = f.Reverse(2, w);

    // F(x)        = x[0]^2 + x[1]^2 + ... + x[n-1]^2
    // F''(x)      = 2 * Identity_Matrix
    // F''(x) * dx = 2 * dx
    for(j = 0; j < n; j++)
        ok &= NearEqual(ddw[j * 2 + 1], 2.*dx[j], eps99, eps99);

    return ok;
}
Input File: example/general/hes_times_dir.cpp