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@(@\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}} }@)@
CppAD Speed: Second Derivative of a Polynomial

Specifications
See link_poly .

Implementation
# include <cppad/cppad.hpp>
# include <cppad/speed/uniform_01.hpp>

// Note that CppAD uses global_option["memory"] at the main program level
# include <map>
extern std::map<std::string, bool> global_option;

bool link_poly(
     size_t                     size     ,
     size_t                     repeat   ,
     CppAD::vector<double>     &a        ,  // coefficients of polynomial
     CppAD::vector<double>     &z        ,  // polynomial argument value
     CppAD::vector<double>     &ddp      )  // second derivative w.r.t z
{
     // --------------------------------------------------------------------
     // check global options
     const char* valid[] = { "memory", "onetape", "optimize"};
     size_t n_valid = sizeof(valid) / sizeof(valid[0]);
     typedef std::map<std::string, bool>::iterator iterator;
     //
     for(iterator itr=global_option.begin(); itr!=global_option.end(); ++itr)
     {     if( itr->second )
          {     bool ok = false;
               for(size_t i = 0; i < n_valid; i++)
                    ok |= itr->first == valid[i];
               if( ! ok )
                    return false;
          }
     }
     // --------------------------------------------------------------------
     // optimization options: no conditional skips or compare operators
     std::string options="no_compare_op";
     // -----------------------------------------------------
     // setup
     typedef CppAD::AD<double>     ADScalar;
     typedef CppAD::vector<ADScalar> ADVector;

     size_t i;      // temporary index
     size_t m = 1;  // number of dependent variables
     size_t n = 1;  // number of independent variables
     ADVector Z(n); // AD domain space vector
     ADVector P(m); // AD range space vector

     // choose the polynomial coefficients
     CppAD::uniform_01(size, a);

     // AD copy of the polynomial coefficients
     ADVector A(size);
     for(i = 0; i < size; i++)
          A[i] = a[i];

     // forward mode first and second differentials
     CppAD::vector<double> p(1), dp(1), dz(1), ddz(1);
     dz[0]  = 1.;
     ddz[0] = 0.;

     // AD function object
     CppAD::ADFun<double> f;

     // --------------------------------------------------------------------
     if( ! global_option["onetape"] ) while(repeat--)
     {
          // choose an argument value
          CppAD::uniform_01(1, z);
          Z[0] = z[0];

          // declare independent variables
          Independent(Z);

          // AD computation of the function value
          P[0] = CppAD::Poly(0, A, Z[0]);

          // create function object f : A -> detA
          f.Dependent(Z, P);

          if( global_option["optimize"] )
               f.optimize(options);

          // skip comparison operators
          f.compare_change_count(0);

          // pre-allocate memory for three forward mode calculations
          f.capacity_order(3);

          // evaluate the polynomial
          p = f.Forward(0, z);

          // evaluate first order Taylor coefficient
          dp = f.Forward(1, dz);

          // second derivative is twice second order Taylor coef
          ddp     = f.Forward(2, ddz);
          ddp[0] *= 2.;
     }
     else
     {
          // choose an argument value
          CppAD::uniform_01(1, z);
          Z[0] = z[0];

          // declare independent variables
          Independent(Z);

          // AD computation of the function value
          P[0] = CppAD::Poly(0, A, Z[0]);

          // create function object f : A -> detA
          f.Dependent(Z, P);

          if( global_option["optimize"] )
               f.optimize(options);

          // skip comparison operators
          f.compare_change_count(0);

          while(repeat--)
          {     // sufficient memory is allocated by second repetition

               // get the next argument value
               CppAD::uniform_01(1, z);

               // evaluate the polynomial at the new argument value
               p = f.Forward(0, z);

               // evaluate first order Taylor coefficient
               dp = f.Forward(1, dz);

               // second derivative is twice second order Taylor coef
               ddp     = f.Forward(2, ddz);
               ddp[0] *= 2.;
          }
     }
     return true;
}

Input File: speed/cppad/poly.cpp