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

hessian.cpp

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Hessian: Example and Test

# include <cppad/cppad.hpp>
namespace { // ---------------------------------------------------------
// define the template function HessianCases<Vector> in empty namespace
template <class 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;
}