abs_eval: Example and Test

abs_eval.cpp

@(@\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}} }@)@ 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;
}

Input File: example/abs_normal/abs_eval.cpp