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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: Example and Test
 # include <cppad/cppad.hpp> namespace { // --------------------------------------------------------- // define the template function HessianCases<Vector> in empty namespace template <typename Vector> bool HessianCases() { bool ok = true; using CppAD::AD; using CppAD::NearEqual; double eps99 = 99.0 * std::numeric_limits<double>::epsilon(); using CppAD::exp; using CppAD::sin; using CppAD::cos; // domain space vector size_t n = 2; CPPAD_TESTVECTOR(AD<double>) X(n); X[0] = 1.; X[1] = 2.; // declare independent variables and starting recording CppAD::Independent(X); // a calculation between the domain and range values AD<double> Square = X[0] * X[0]; // range space vector size_t m = 3; CPPAD_TESTVECTOR(AD<double>) Y(m); Y[0] = Square * exp( X[1] ); Y[1] = Square * sin( X[1] ); Y[2] = Square * cos( X[1] ); // create f: X -> Y and stop tape recording CppAD::ADFun<double> f(X, Y); // new value for the independent variable vector Vector x(n); x[0] = 2.; x[1] = 1.; // second derivative of y[1] Vector hes( n * n ); hes = f.Hessian(x, 1); /* F_1 = x[0] * x[0] * sin(x[1]) F_1^{(1)} = [ 2 * x[0] * sin(x[1]) , x[0] * x[0] * cos(x[1]) ] F_1^{(2)} = [ 2 * sin(x[1]) , 2 * x[0] * cos(x[1]) ] [ 2 * x[0] * cos(x[1]) , - x[0] * x[0] * sin(x[1]) ] */ ok &= NearEqual( 2.*sin(x[1]), hes[0*n+0], eps99, eps99); ok &= NearEqual( 2.*x[0]*cos(x[1]), hes[0*n+1], eps99, eps99); ok &= NearEqual( 2.*x[0]*cos(x[1]), hes[1*n+0], eps99, eps99); ok &= NearEqual( - x[0]*x[0]*sin(x[1]), hes[1*n+1], eps99, eps99); return ok; } } // End empty namespace # include <vector> # include <valarray> bool Hessian(void) { bool ok = true; // Run with Vector equal to three different cases // all of which are Simple Vectors with elements of type double. ok &= HessianCases< CppAD::vector <double> >(); ok &= HessianCases< std::vector <double> >(); ok &= HessianCases< std::valarray <double> >(); return ok; } 
Input File: example/general/hessian.cpp