lines 6-376 of file: example/atomic_three/reverse.cpp

{xrst_begin atomic_three_reverse.cpp}

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

Purpose
*******
This example demonstrates reverse 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 {          // isolate items below to this file
using CppAD::vector; // abbreviate as vector
//
class atomic_reverse : public CppAD::atomic_three<double> {
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Constructor
***********
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public:
   atomic_reverse(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 first order forward mode
      bool ok = order_up <= 1;
      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 one 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];
      }
      return ok;
   }
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reverse
*******
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{xrst_code cpp} */
   // reverse mode routine called by CppAD
   bool reverse(
      const vector<double>&               parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      size_t                              order_up    ,
      const vector<double>&               taylor_x    ,
      const vector<double>&               taylor_y    ,
      vector<double>&                     partial_x   ,
      const vector<double>&               partial_y   ) override
   {
      size_t q1 = order_up + 1;
      size_t n = taylor_x.size() / q1;
# ifndef NDEBUG
      size_t m = taylor_y.size() / q1;
# endif
      assert( n == 3 );
      assert( m == 2 );

      // this example only implements up to second order reverse mode
      bool ok = q1 <= 2;
      if( ! ok )
         return ok;
      //
      // initalize summation as zero
      for(size_t j = 0; j < n; j++)
         for(size_t k = 0; k < q1; k++)
            partial_x[j * q1 + k] = 0.0;
      //
      if( q1 == 2 )
      {  // --------------------------------------------------------------
         // Second order reverse first compute partials of first order
         // We use the notation pg_ij^k for partial of F_i^1 w.r.t. x_j^k
         //
         // y_0^1    = 2 * x_2^0 * x_2^1
         // pg_02^0  = 2 * x_2^1
         // pg_02^1  = 2 * x_2^0
         //
         // y_1^1    = x_1^0 * x_0^1 + x_0^0 * x_1^1
         // pg_10^0  = x_1^1
         // pg_11^0  = x_0^1
         // pg_10^1  = x_1^0
         // pg_11^1  = x_0^0
         //
         // px_0^0 += py_0^1 * pg_00^0 + py_1^1 * pg_10^0
         //        += py_1^1 * x_1^1
         partial_x[0*q1+0] += partial_y[1*q1+1] * taylor_x[1*q1+1];
         //
         // px_0^1 += py_0^1 * pg_00^1 + py_1^1 * pg_10^1
         //        += py_1^1 * x_1^0
         partial_x[0*q1+1] += partial_y[1*q1+1] * taylor_x[1*q1+0];
         //
         // px_1^0 += py_0^1 * pg_01^0 + py_1^1 * pg_11^0
         //        += py_1^1 * x_0^1
         partial_x[1*q1+0] += partial_y[1*q1+1] * taylor_x[0*q1+1];
         //
         // px_1^1 += py_0^1 * pg_01^1 + py_1^1 * pg_11^1
         //        += py_1^1 * x_0^0
         partial_x[1*q1+1] += partial_y[1*q1+1] * taylor_x[0*q1+0];
         //
         // px_2^0 += py_0^1 * pg_02^0 + py_1^1 * pg_12^0
         //        += py_0^1 * 2 * x_2^1
         partial_x[2*q1+0] += partial_y[0*q1+1] * 2.0 * taylor_x[2*q1+1];
         //
         // px_2^1 += py_0^1 * pg_02^1 + py_1^1 * pg_12^1
         //        += py_0^1 * 2 * x_2^0
         partial_x[2*q1+1] += partial_y[0*q1+1] * 2.0 * taylor_x[2*q1+0];
      }
      // --------------------------------------------------------------
      // First order reverse computes partials of zero order coefficients
      // We use the notation pg_ij for partial of F_i^0 w.r.t. x_j^0
      //
      // y_0^0 = x_2^0 * x_2^0
      // pg_00 = 0,     pg_01 = 0,  pg_02 = 2 * x_2^0
      //
      // y_1^0 = x_0^0 * x_1^0
      // pg_10 = x_1^0, pg_11 = x_0^0,  pg_12 = 0
      //
      // px_0^0 += py_0^0 * pg_00 + py_1^0 * pg_10
      //        += py_1^0 * x_1^0
      partial_x[0*q1+0] += partial_y[1*q1+0] * taylor_x[1*q1+0];
      //
      // px_1^0 += py_1^0 * pg_01 + py_1^0 * pg_11
      //        += py_1^0 * x_0^0
      partial_x[1*q1+0] += partial_y[1*q1+0] * taylor_x[0*q1+0];
      //
      // px_2^0 += py_1^0 * pg_02 + py_1^0 * pg_12
      //        += py_0^0 * 2.0 * x_2^0
      partial_x[2*q1+0] += partial_y[0*q1+0] * 2.0 * taylor_x[2*q1+0];
      // --------------------------------------------------------------
      return ok;
   }
};
}  // End empty namespace
/* {xrst_code}
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Use Atomic Function
*******************
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{xrst_code cpp} */
bool reverse(void)
{  bool ok = true;
   using CppAD::AD;
   using CppAD::NearEqual;
   double eps = 10. * CppAD::numeric_limits<double>::epsilon();
   //
   // Create the atomic_reverse object corresponding to g(x)
   atomic_reverse afun("atomic_reverse");
   //
   // 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 reverse
   //
   // value of Jacobian of f
   double check_jac[] = {
      0.0, 0.0, 2.0 * u_2,
      u_1, u_0,       0.0
   };
   vector<double> w(m), dw(n);
   //
   // check derivative of f_0 (x)
   for(size_t i = 0; i < m; i++)
   {  w[i]   = 1.0;
      w[1-i] = 0.0;
      dw = f.Reverse(1, w);
      for(size_t j = 0; j < n; j++)
      {  // compute partial in j-th component direction
         ok &= NearEqual(dw[j], check_jac[i * n + j], eps, eps);
      }
   }
   // --------------------------------------------------------------------
   // second order reverse
   //
   // value of Hessian of f_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 f_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> u1(n), dw2( 2 * n );
   for(size_t j = 0; j < n; j++)
   {  for(size_t j1 = 0; j1 < n; j1++)
         u1[j1] = 0.0;
      u1[j] = 1.0;
      // first order forward
      f.Forward(1, u1);
      w[0] = 1.0;
      w[1] = 0.0;
      dw2  = f.Reverse(2, w);
      for(size_t i = 0; i < n; i++)
         ok &= NearEqual(dw2[i * 2 + 1], check_hes_0[i * n + j], eps, eps);
      w[0] = 0.0;
      w[1] = 1.0;
      dw2  = f.Reverse(2, w);
      for(size_t i = 0; i < n; i++)
         ok &= NearEqual(dw2[i * 2 + 1], check_hes_1[i * n + j], eps, eps);
   }
   // --------------------------------------------------------------------
   return ok;
}
/* {xrst_code}
{xrst_spell_on}

{xrst_end atomic_three_reverse.cpp}