abs_eval.cpp¶

abs_eval: Example and Test¶

Purpose¶

The function \(f : \B{R}^3 \rightarrow \B{R}\) defined by

\[f( x_0, x_1, x_2 ) = | x_0 + x_1 | + | x_1 + x_2 |\]

is affine, except for its absolute value terms. For this case, the abs_normal approximation should be equal to the function itself.

Source¶

# include <cppad/cppad.hpp>
# include "abs_eval.hpp"

namespace {
   CPPAD_TESTVECTOR(double) join(
      const CPPAD_TESTVECTOR(double)& x ,
      const CPPAD_TESTVECTOR(double)& u )
   {  size_t n = x.size();
      size_t s = u.size();
      CPPAD_TESTVECTOR(double) xu(n + s);
      for(size_t j = 0; j < n; j++)
         xu[j] = x[j];
      for(size_t j = 0; j < s; j++)
         xu[n + j] = u[j];
      return xu;
   }
}
bool abs_eval(void)
{  bool ok = true;
   //
   using CppAD::AD;
   using CppAD::ADFun;
   //
   typedef CPPAD_TESTVECTOR(double)       d_vector;
   typedef CPPAD_TESTVECTOR( AD<double> ) ad_vector;
   //
   double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
   //
   size_t n = 3; // size of x
   size_t m = 1; // size of y
   size_t s = 2; // number of absolute value terms
   //
   // record the function f(x)
   ad_vector ad_x(n), ad_y(m);
   for(size_t j = 0; j < n; j++)
      ad_x[j] = double(j + 1);
   Independent( ad_x );
   // for this example, we ensure first absolute value is | x_0 + x_1 |
   AD<double> ad_0 = abs( ad_x[0] + ad_x[1] );
   // and second absolute value is | x_1 + x_2 |
   AD<double> ad_1 = abs( ad_x[1] + ad_x[2] );
   ad_y[0]         = ad_0 + ad_1;
   ADFun<double> f(ad_x, ad_y);

   // create its abs_normal representation in g, a
   ADFun<double> g, a;
   f.abs_normal_fun(g, a);

   // check dimension of domain and range space for g
   ok &= g.Domain() == n + s;
   ok &= g.Range()  == m + s;

   // check dimension of domain and range space for a
   ok &= a.Domain() == n;
   ok &= a.Range()  == s;

   // --------------------------------------------------------------------
   // Choose a point x_hat
   d_vector x_hat(n);
   for(size_t j = 0; j < n; j++)
      x_hat[j] = double(j - 1);

   // value of a_hat = a(x_hat)
   d_vector a_hat = a.Forward(0, x_hat);

   // (x_hat, a_hat)
   d_vector xu_hat = join(x_hat, a_hat);

   // value of g[ x_hat, a_hat ]
   d_vector g_hat = g.Forward(0, xu_hat);

   // Jacobian of g[ x_hat, a_hat ]
   d_vector g_jac = g.Jacobian(xu_hat);

   // value of delta_x
   d_vector delta_x(n);
   delta_x[0] =  1.0;
   delta_x[1] = -2.0;
   delta_x[2] = +2.0;

   // value of x
   d_vector x(n);
   for(size_t j = 0; j < n; j++)
      x[j] = x_hat[j] + delta_x[j];

   // value of f(x)
   d_vector y = f.Forward(0, x);

   // value of g_tilde
   d_vector g_tilde = CppAD::abs_eval(n, m, s, g_hat, g_jac, delta_x);

   // should be equal because f is affine, except for abs terms
   ok &= CppAD::NearEqual(y[0], g_tilde[0], eps99, eps99);

   return ok;
}