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$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
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