$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
CppAD Speed: Second Derivative of a Polynomial

Specifications

Implementation
# include <cppad/cppad.hpp>

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

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

size_t i;      // temporary index
size_t m = 1;  // number of dependent variables
size_t n = 1;  // number of independent variables

// choose the polynomial coefficients

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

// forward mode first and second differentials
dz[0]  = 1.;
ddz[0] = 0.;

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

// declare independent variables
Independent(Z);

// AD computation of the function value

// 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
Z[0] = z[0];

// declare independent variables
Independent(Z);

// AD computation of the function value

// 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

// 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;
}