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

forward.cpp

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Forward Mode: Example and Test

# include <limits>
# include <cppad/cppad.hpp>
namespace { // --------------------------------------------------------
// define the template function ForwardCases<Vector> in empty namespace
template <class Vector>
bool ForwardCases(void)
{  bool ok = true;
   using CppAD::AD;
   using CppAD::NearEqual;
   double eps = 10. * std::numeric_limits<double>::epsilon();

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

   // declare independent variables and starting recording
   CppAD::Independent(ax);

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

   // create f: x -> y and stop tape recording
   CppAD::ADFun<double> f(ax, ay);

   // initially, the variable values during taping are stored in f
   ok &= f.size_order() == 1;

   // zero order forward mode using notation in forward_zero
   // use the template parameter Vector for the vector type
   Vector x0(n), y0(m);
   x0[0] = 3.;
   x0[1] = 4.;
   y0    = f.Forward(0, x0);
   ok  &= NearEqual(y0[0] , x0[0]*x0[0]*x0[1], eps, eps);
   ok  &= f.size_order() == 1;

   // first order forward mode using notation in forward_one
   // X(t)           = x0 + x1 * t
   // Y(t) = F[X(t)] = y0 + y1 * t + o(t)
   Vector x1(n), y1(m);
   x1[0] = 1.;
   x1[1] = 0.;
   y1    = f.Forward(1, x1); // partial F w.r.t. x_0
   ok   &= NearEqual(y1[0] , 2.*x0[0]*x0[1], eps, eps);
   ok   &= f.size_order() == 2;

   // second order forward mode using notation in forward_order
   // X(t) =           x0 + x1 * t + x2 * t^2
   // Y(t) = F[X(t)] = y0 + y1 * t + y2 * t^2 + o(t^3)
   Vector x2(n), y2(m);
   x2[0]      = 0.;
   x2[1]      = 0.;
   y2         = f.Forward(2, x2);
   double F_00 = 2. * y2[0]; // second partial F w.r.t. x_0, x_0
   ok         &= NearEqual(F_00, 2.*x0[1], eps, eps);
   ok         &= f.size_order() == 3;

   return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool Forward(void)
{  bool ok = true;
   // Run with Vector equal to three different cases
   // all of which are Simple Vectors with elements of type double.
   ok &= ForwardCases< CppAD::vector  <double> >();
   ok &= ForwardCases< std::vector    <double> >();
   ok &= ForwardCases< std::valarray  <double> >();
   return ok;
}