opt_val_hes: Example and Test

opt_val_hes.cpp

Headings

@(@\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}} }@)@ opt_val_hes: Example and Test Fix @(@ z \in \B{R}^\ell @)@ and define the functions @(@ S_k : \B{R} \times \B{R} \rightarrow \B{R}^\ell @)@ by and @(@ F : \B{R} \times \B{R} \rightarrow \B{R} @)@ by @[@ \begin{array}{rcl} S_k (x, y) & = & \frac{1}{2} [ y * \sin ( x * t_k ) - z_k ]^2 \\ F(x, y) & = & \sum_{k=0}^{\ell-1} S_k (x, y) \end{array} @]@ It follows that @[@ \begin{array}{rcl} \partial_y F(x, y) & = & \sum_{k=0}^{\ell-1} [ y * \sin ( x * t_k ) - z_k ] \sin( x * t_k ) \\ \partial_y \partial_y F(x, y) & = & \sum_{k=0}^{\ell-1} \sin ( x t_k )^2 \end{array} @]@ Furthermore if we define @(@ Y(x) @)@ as solving the equation @(@ \partial F[ x, Y(x) ] = 0 @)@ we have @[@ \begin{array}{rcl} 0 & = & \sum_{k=0}^{\ell-1} [ Y(x) * \sin ( x * t_k ) - z_k ] \sin( x * t_k ) \\ Y(x) \sum_{k=0}^{\ell-1} \sin ( x * t_k )^2 - \sum_{k=0}^{\ell-1} \sin ( x * t_k ) z_k \\ Y(x) & = & \frac{ \sum_{k=0}^{\ell-1} \sin( x * t_k ) z_k }{ \sum_{k=0}^{\ell-1} \sin ( x * t_k )^2 } \end{array} @]@



# include <limits>
# include <cppad/cppad.hpp>

namespace {
     using CppAD::AD;
     typedef CPPAD_TESTVECTOR(double)       BaseVector;
     typedef CPPAD_TESTVECTOR(AD<double>) ADVector;

     class Fun {
     private:
          const BaseVector t_;    // measurement times
          const BaseVector z_;    // measurement values
     public:
          typedef ADVector ad_vector;
          // constructor
          Fun(const BaseVector &t, const BaseVector &z)
          : t_(t) , z_(z)
          {     assert( t.size() == z.size() ); }
          // ell
          size_t ell(void) const
          {     return t_.size(); }
          // Fun.s
          AD<double> s(size_t k, const ad_vector& x, const ad_vector& y) const
          {
               AD<double> residual = y[0] * sin( x[0] * t_[k] ) - z_[k];
               AD<double> s_k      = .5 * residual * residual;

               return s_k;
          }
          // Fun.sy
          ad_vector sy(size_t k, const ad_vector& x, const ad_vector& y) const
          {     assert( y.size() == 1);
               ad_vector sy_k(1);

               AD<double> residual = y[0] * sin( x[0] * t_[k] ) - z_[k];
               sy_k[0] = residual * sin( x[0] * t_[k] );

               return sy_k;
          }
     };
     // Used to test calculation of Hessian of V
     AD<double> V(const ADVector& x, const BaseVector& t, const BaseVector& z)
     {     // compute Y(x)
          AD<double> numerator = 0.;
          AD<double> denominator = 0.;
          size_t k;
          for(k = 0; k < size_t(t.size()); k++)
          {     numerator   += sin( x[0] * t[k] ) * z[k];
               denominator += sin( x[0] * t[k] ) * sin( x[0] * t[k] );
          }
          AD<double> y = numerator / denominator;

          // V(x) = F[x, Y(x)]
          AD<double> sum = 0;
          for(k = 0; k < size_t(t.size()); k++)
          {     AD<double> residual = y * sin( x[0] * t[k] ) - z[k];
               sum += .5 * residual * residual;
          }
          return sum;
     }
}

bool opt_val_hes(void)
{     bool ok = true;
     using CppAD::AD;
     using CppAD::NearEqual;

     // temporary indices
     size_t j, k;

     // x space vector
     size_t n = 1;
     BaseVector x(n);
     x[0] = 2. * 3.141592653;

     // y space vector
     size_t m = 1;
     BaseVector y(m);
     y[0] = 1.;

     // t and z vectors
     size_t ell = 10;
     BaseVector t(ell);
     BaseVector z(ell);
     for(k = 0; k < ell; k++)
     {     t[k] = double(k) / double(ell);       // time of measurement
          z[k] = y[0] * sin( x[0] * t[k] );     // data without noise
     }

     // construct the function object
     Fun fun(t, z);

     // evaluate the Jacobian and Hessian
     BaseVector jac(n), hes(n * n);
# ifndef NDEBUG
     int signdet =
# endif
     CppAD::opt_val_hes(x, y, fun, jac, hes);

     // we know that F_yy is positive definate for this case
     assert( signdet == 1 );

     // create ADFun object g corresponding to V(x)
     ADVector a_x(n), a_v(1);
     for(j = 0; j < n; j++)
          a_x[j] = x[j];
     Independent(a_x);
     a_v[0] = V(a_x, t, z);
     CppAD::ADFun<double> g(a_x, a_v);

     // accuracy for checks
     double eps = 10. * CppAD::numeric_limits<double>::epsilon();

     // check Jacobian
     BaseVector check_jac = g.Jacobian(x);
     for(j = 0; j < n; j++)
          ok &= NearEqual(jac[j], check_jac[j], eps, eps);

     // check Hessian
     BaseVector check_hes = g.Hessian(x, 0);
     for(j = 0; j < n*n; j++)
          ok &= NearEqual(hes[j], check_hes[j], eps, eps);

     return ok;
}

Input File: example/general/opt_val_hes.cpp