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

reverse_two.cpp

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Second Order Reverse ModeExample and Test

# include <cppad/cppad.hpp>
namespace { // ----------------------------------------------------------
// define the template function reverse_two_cases<Vector> in empty namespace
template <class Vector>
bool reverse_two_cases(void)
{  bool ok = true;
   using CppAD::AD;
   using CppAD::NearEqual;
   double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

   // domain space vector
   size_t n = 2;
   CPPAD_TESTVECTOR(AD<double>) X(n);
   X[0] = 0.;
   X[1] = 1.;

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

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

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

   // use zero order forward mode to evaluate y at x = (3, 4)
   // use the template parameter Vector for the vector type
   Vector x(n), y(m);
   x[0]  = 3.;
   x[1]  = 4.;
   y     = f.Forward(0, x);
   ok    &= NearEqual(y[0] , x[0]*x[0]*x[1], eps99, eps99);

   // use first order forward mode in x[0] direction
   // (all second order partials below involve x[0])
   Vector dx(n), dy(m);
   dx[0] = 1.;
   dx[1] = 1.;
   dy    = f.Forward(1, dx);
   double check = 2.*x[0]*x[1]*dx[0] + x[0]*x[0]*dx[1];
   ok   &= NearEqual(dy[0], check, eps99, eps99);

   // use second order reverse mode to evalaute second partials of y[0]
   // with respect to (x[0], x[0]) and with respect to (x[0], x[1])
   Vector w(m), dw( n * 2 );
   w[0]  = 1.;
   dw    = f.Reverse(2, w);

   // check derivative of f
   ok   &= NearEqual(dw[0*2+0] , 2.*x[0]*x[1], eps99, eps99);
   ok   &= NearEqual(dw[1*2+0] ,    x[0]*x[0], eps99, eps99);

   // check derivative of f^{(1)} (x) * dx
   check = 2.*x[1]*dx[1] + 2.*x[0]*dx[1];
   ok   &= NearEqual(dw[0*2+1] , check, eps99, eps99);
   check = 2.*x[0]*dx[1];
   ok   &= NearEqual(dw[1*2+1] , check, eps99, eps99);

   return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool reverse_two(void)
{  bool ok = true;
   ok &= reverse_two_cases< CppAD::vector  <double> >();
   ok &= reverse_two_cases< std::vector    <double> >();
   ok &= reverse_two_cases< std::valarray  <double> >();
   return ok;
}