lines 6-321 of file: example/atomic_three/forward.cpp

{xrst_begin atomic_three_forward.cpp}

Atomic Functions and Forward Mode: Example and Test
###################################################

Purpose
*******
This example demonstrates forward mode derivative calculation
using an :ref:`atomic_three-name` function.

Function
********
For this example, the atomic function
:math:`g : \B{R}^3 \rightarrow \B{R}^2` is defined by

.. math::

   g(x) = \left( \begin{array}{c}
      x_2 * x_2 \\
      x_0 * x_1
   \end{array} \right)

Jacobian
********
The corresponding Jacobian is

.. math::

   g^{(1)} (x) = \left( \begin{array}{ccc}
     0  &   0 & 2 x_2 \\
   x_1  & x_0 & 0
   \end{array} \right)

Hessian
*******
The Hessians of the component functions are

.. math::

   g_0^{(2)} ( x ) = \left( \begin{array}{ccc}
      0 & 0 & 0  \\
      0 & 0 & 0  \\
      0 & 0 & 2
   \end{array} \right)
   \W{,}
   g_1^{(2)} ( x ) = \left( \begin{array}{ccc}
      0 & 1 & 0 \\
      1 & 0 & 0 \\
      0 & 0 & 0
   \end{array} \right)

Start Class Definition
**********************
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# include <cppad/cppad.hpp>
namespace {          // begin empty namespace
using CppAD::vector; // abbreviate CppAD::vector using vector
//
class atomic_forward : public CppAD::atomic_three<double> {
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Constructor
***********
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public:
   atomic_forward(const std::string& name) :
   CppAD::atomic_three<double>(name)
   { }
private:
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for_type
********
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   // calculate type_y
   bool for_type(
      const vector<double>&               parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      vector<CppAD::ad_type_enum>&        type_y      ) override
   {  assert( parameter_x.size() == type_x.size() );
      bool ok = type_x.size() == 3; // n
      ok     &= type_y.size() == 2; // m
      if( ! ok )
         return false;
      type_y[0] = type_x[2];
      type_y[1] = std::max(type_x[0], type_x[1]);
      return true;
   }
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forward
*******
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{xrst_code cpp} */
   // forward mode routine called by CppAD
   bool forward(
      const vector<double>&              parameter_x  ,
      const vector<CppAD::ad_type_enum>& type_x       ,
      size_t                             need_y       ,
      size_t                             order_low    ,
      size_t                             order_up     ,
      const vector<double>&              taylor_x     ,
      vector<double>&                    taylor_y     ) override
   {
      size_t q1 = order_up + 1;
# ifndef NDEBUG
      size_t n = taylor_x.size() / q1;
      size_t m = taylor_y.size() / q1;
# endif
      assert( n == 3 );
      assert( m == 2 );
      assert( order_low <= order_up );

      // this example only implements up to second order forward mode
      bool ok = order_up <=  2;
      if( ! ok )
         return ok;

      // ------------------------------------------------------------------
      // Zero forward mode.
      // This case must always be implemented
      // g(x) = [ x_2 * x_2 ]
      //        [ x_0 * x_1 ]
      // y^0  = f( x^0 )
      if( order_low <= 0 )
      {  // y_0^0 = x_2^0 * x_2^0
         taylor_y[0*q1+0] = taylor_x[2*q1+0] * taylor_x[2*q1+0];
         // y_1^0 = x_0^0 * x_1^0
         taylor_y[1*q1+0] = taylor_x[0*q1+0] * taylor_x[1*q1+0];
      }
      if( order_up <=  0 )
         return ok;
      // ------------------------------------------------------------------
      // First order forward mode.
      // This case is needed if first order forward mode is used.
      // g'(x) = [   0,   0, 2 * x_2 ]
      //         [ x_1, x_0,       0 ]
      // y^1 =  f'(x^0) * x^1
      if( order_low <= 1 )
      {  // y_0^1 = 2 * x_2^0 * x_2^1
         taylor_y[0*q1+1] = 2.0 * taylor_x[2*q1+0] * taylor_x[2*q1+1];
         // y_1^1 = x_1^0 * x_0^1 + x_0^0 * x_1^1
         taylor_y[1*q1+1]  = taylor_x[1*q1+0] * taylor_x[0*q1+1];
         taylor_y[1*q1+1] += taylor_x[0*q1+0] * taylor_x[1*q1+1];
      }
      if( order_up <=  1 )
         return ok;
      // ------------------------------------------------------------------
      // Second order forward mode.
      // This case is neede if second order forwrd mode is used.
      // g'(x) = [   0,   0, 2 x_2 ]
      //         [ x_1, x_0,     0 ]
      //
      //            [ 0 , 0 , 0 ]                  [ 0 , 1 , 0 ]
      // g_0''(x) = [ 0 , 0 , 0 ]  g_1^{(2)} (x) = [ 1 , 0 , 0 ]
      //            [ 0 , 0 , 2 ]                  [ 0 , 0 , 0 ]
      //
      //  y_0^2 = x^1 * g_0''( x^0 ) x^1 / 2! + g_0'( x^0 ) x^2
      //        = ( x_2^1 * 2.0 * x_2^1 ) / 2!
      //        + 2.0 * x_2^0 * x_2^2
      taylor_y[0*q1+2]  = taylor_x[2*q1+1] * taylor_x[2*q1+1];
      taylor_y[0*q1+2] += 2.0 * taylor_x[2*q1+0] * taylor_x[2*q1+2];
      //
      //  y_1^2 = x^1 * g_1''( x^0 ) x^1 / 2! + g_1'( x^0 ) x^2
      //        = ( x_1^1 * x_0^1 + x_0^1 * x_1^1) / 2
      //        + x_1^0 * x_0^2 + x_0^0 + x_1^2
      taylor_y[1*q1+2]  = taylor_x[1*q1+1] * taylor_x[0*q1+1];
      taylor_y[1*q1+2] += taylor_x[1*q1+0] * taylor_x[0*q1+2];
      taylor_y[1*q1+2] += taylor_x[0*q1+0] * taylor_x[1*q1+2];
      // ------------------------------------------------------------------
      return ok;
   }
};
}  // End empty namespace
/* {xrst_code}
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Use Atomic Function
*******************
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{xrst_code cpp} */
bool forward(void)
{  bool ok = true;
   using CppAD::AD;
   using CppAD::NearEqual;
   double eps = 10. * CppAD::numeric_limits<double>::epsilon();
   //
   // Create the atomic_forward object corresponding to g(x)
   atomic_forward afun("atomic_forward");
   //
   // Create the function f(u) = g(u) for this example.
   //
   // domain space vector
   size_t n  = 3;
   double u_0 = 1.00;
   double u_1 = 2.00;
   double u_2 = 3.00;
   vector< AD<double> > au(n);
   au[0] = u_0;
   au[1] = u_1;
   au[2] = u_2;

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

   // range space vector
   size_t m = 2;
   vector< AD<double> > ay(m);

   // call atomic function
   vector< AD<double> > ax = au;
   afun(ax, ay);

   // create f: u -> y and stop tape recording
   CppAD::ADFun<double> f;
   f.Dependent (au, ay);  // y = f(u)
   //
   // check function value
   double check = u_2 * u_2;
   ok &= NearEqual( Value(ay[0]) , check,  eps, eps);
   check = u_0 * u_1;
   ok &= NearEqual( Value(ay[1]) , check,  eps, eps);

   // --------------------------------------------------------------------
   // zero order forward
   //
   vector<double> u0(n), y0(m);
   u0[0] = u_0;
   u0[1] = u_1;
   u0[2] = u_2;
   y0   = f.Forward(0, u0);
   check = u_2 * u_2;
   ok &= NearEqual(y0[0] , check,  eps, eps);
   check = u_0 * u_1;
   ok &= NearEqual(y0[1] , check,  eps, eps);
   // --------------------------------------------------------------------
   // first order forward
   //
   // value of Jacobian of f
   double check_jac[] = {
      0.0, 0.0, 2.0 * u_2,
      u_1, u_0,       0.0
   };
   vector<double> u1(n), y1(m);
   // check first order forward mode
   for(size_t j = 0; j < n; j++)
      u1[j] = 0.0;
   for(size_t j = 0; j < n; j++)
   {  // compute partial in j-th component direction
      u1[j] = 1.0;
      y1    = f.Forward(1, u1);
      u1[j] = 0.0;
      // check this direction
      for(size_t i = 0; i < m; i++)
         ok &= NearEqual(y1[i], check_jac[i * n + j], eps, eps);
   }
   // --------------------------------------------------------------------
   // second order forward
   //
   // value of Hessian of g_0
   double check_hes_0[] = {
      0.0, 0.0, 0.0,
      0.0, 0.0, 0.0,
      0.0, 0.0, 2.0
   };
   //
   // value of Hessian of g_1
   double check_hes_1[] = {
      0.0, 1.0, 0.0,
      1.0, 0.0, 0.0,
      0.0, 0.0, 0.0
   };
   vector<double> u2(n), y2(m);
   for(size_t j = 0; j < n; j++)
      u2[j] = 0.0;
   // compute diagonal elements of the Hessian
   for(size_t j = 0; j < n; j++)
   {  // first order forward in j-th direction
      u1[j] = 1.0;
      f.Forward(1, u1);
      y2 = f.Forward(2, u2);
      // check this element of Hessian diagonal
      ok &= NearEqual(y2[0], check_hes_0[j * n + j] / 2.0, eps, eps);
      ok &= NearEqual(y2[1], check_hes_1[j * n + j] / 2.0, eps, eps);
      //
      for(size_t k = 0; k < n; k++) if( k != j )
      {  u1[k] = 1.0;
         f.Forward(1, u1);
         y2 = f.Forward(2, u2);
         //
         // y2 = (H_jj + H_kk + H_jk + H_kj) / 2.0
         // y2 = (H_jj + H_kk) / 2.0 + H_jk
         //
         double H_jj = check_hes_0[j * n + j];
         double H_kk = check_hes_0[k * n + k];
         double H_jk = y2[0] - (H_kk + H_jj) / 2.0;
         ok &= NearEqual(H_jk, check_hes_0[j * n + k], eps, eps);
         //
         H_jj = check_hes_1[j * n + j];
         H_kk = check_hes_1[k * n + k];
         H_jk = y2[1] - (H_kk + H_jj) / 2.0;
         ok &= NearEqual(H_jk, check_hes_1[j * n + k], eps, eps);
         //
         u1[k] = 0.0;
      }
      u1[j] = 0.0;
   }
   // --------------------------------------------------------------------
   return ok;
}
/* {xrst_code}
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{xrst_end atomic_three_forward.cpp}