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Adolc Speed: Second Derivative of a Polynomial
Specifications
See link_poly .
Implementation

    // suppress conversion warnings before other includes
    # include <cppad/wno_conversion.hpp>
    //
    # include <vector>
    # include <adolc/adolc.h>

    # include <cppad/speed/uniform_01.hpp>
    # include <cppad/utility/poly.hpp>
    # include <cppad/utility/vector.hpp>
    # include <cppad/utility/thread_alloc.hpp>
    # include "adolc_alloc_mat.hpp"

    // list of possible options
    # 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
    {
         if( global_option["atomic"] )
              return false;
         if( global_option["memory"] || global_option["optimize"] )
              return false;
         // -----------------------------------------------------
         // setup
         size_t i;
         int tag  = 0;  // tape identifier
         int keep = 0;  // do not keep forward mode results in buffer
         int m    = 1;  // number of dependent variables
         int n    = 1;  // number of independent variables
         int d    = 2;  // highest derivative degree
         double f;      // function value

         // set up for thread_alloc memory allocator (fast and checks for leaks)
         using CppAD::thread_alloc; // the allocator
         size_t capacity;           // capacity of an allocation

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

         // AD copy of the polynomial coefficients
         std::vector<adouble> A(size);
         for(i = 0; i < size; i++)
              A[i] = a[i];

         // domain and range space AD values
         adouble Z, P;

         // allocate arguments to hos_forward
         double* x0 = thread_alloc::create_array<double>(size_t(n), capacity);
         double* y0 = thread_alloc::create_array<double>(size_t(m), capacity);
         double** x = adolc_alloc_mat(size_t(n), size_t(d));
         double** y = adolc_alloc_mat(size_t(m), size_t(d));

         // Taylor coefficient for argument
         x[0][0] = 1.;  // first order
         x[0][1] = 0.;  // second order

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

              // declare independent variables
              trace_on(tag, keep);
              Z <<= z[0];

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

              // create function object f : Z -> P
              P >>= f;
              trace_off();

              // set the argument value
              x0[0] = z[0];

              // evaluate the polynomial at the new argument value
              hos_forward(tag, m, n, d, keep, x0, x, y0, y);

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

              // declare independent variables
              trace_on(tag, keep);
              Z <<= z[0];

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

              // create function object f : Z -> P
              P >>= f;
              trace_off();

              while(repeat--)
              {     // get the next argument value
                   CppAD::uniform_01(1, z);
                   x0[0] = z[0];

                   // evaluate the polynomial at the new argument value
                   hos_forward(tag, m, n, d, keep, x0, x, y0, y);

                   // second derivative is twice second order Taylor coef
                   ddp[0] = 2. * y[0][1];
              }
         }
         // ------------------------------------------------------
         // tear down
         adolc_free_mat(x);
         adolc_free_mat(y);
         thread_alloc::delete_array(x0);
         thread_alloc::delete_array(y0);

         return true;
    }

Input File: speed/adolc/poly.cpp