\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

for_two.cpp

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Subset of Second Order Partials: Example and Test

# include <cppad/cppad.hpp>
namespace { // -----------------------------------------------------
// define the template function in empty namespace
// bool ForTwoCases<BaseVector, SizeVector_t>(void)
template <class BaseVector, class SizeVector_t>
bool ForTwoCases()
{  bool ok = true;
   using CppAD::AD;
   using CppAD::NearEqual;
   double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
   using CppAD::exp;
   using CppAD::sin;
   using CppAD::cos;

   // domain space vector
   size_t n = 2;
   CPPAD_TESTVECTOR(AD<double>)  X(n);
   X[0] = 1.;
   X[1] = 2.;

   // declare independent variables and starting recording
   CppAD::Independent(X);

   // a calculation between the domain and range values
   AD<double> Square = X[0] * X[0];

   // range space vector
   size_t m = 3;
   CPPAD_TESTVECTOR(AD<double>)  Y(m);
   Y[0] = Square * exp( X[1] );
   Y[1] = Square * sin( X[1] );
   Y[2] = Square * cos( X[1] );

   // create f: X -> Y and stop tape recording
   CppAD::ADFun<double> f(X, Y);

   // new value for the independent variable vector
   BaseVector x(n);
   x[0] = 2.;
   x[1] = 1.;

   // set j and k to compute specific second partials of y
   size_t p = 2;
   SizeVector_t j(p);
   SizeVector_t k(p);
   j[0] = 0; k[0] = 0; // for second partial w.r.t. x[0] and x[0]
   j[1] = 0; k[1] = 1; // for second partial w.r.t x[0] and x[1]

   // compute the second partials
   BaseVector ddy(m * p);
   ddy = f.ForTwo(x, j, k);
   /*
   partial of y w.r.t x[0] is
   [ 2 * x[0] * exp(x[1]) ]
   [ 2 * x[0] * sin(x[1]) ]
   [ 2 * x[0] * cos(x[1]) ]
   */
   // second partial of y w.r.t x[0] and x[1]
   ok &=  NearEqual( 2.*exp(x[1]), ddy[0*p+0], eps99, eps99);
   ok &=  NearEqual( 2.*sin(x[1]), ddy[1*p+0], eps99, eps99);
   ok &=  NearEqual( 2.*cos(x[1]), ddy[2*p+0], eps99, eps99);

   // second partial of F w.r.t x[0] and x[1]
   ok &=  NearEqual( 2.*x[0]*exp(x[1]), ddy[0*p+1], eps99, eps99);
   ok &=  NearEqual( 2.*x[0]*cos(x[1]), ddy[1*p+1], eps99, eps99);
   ok &=  NearEqual(-2.*x[0]*sin(x[1]), ddy[2*p+1], eps99, eps99);

   return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool ForTwo(void)
{  bool ok = true;
      // Run with BaseVector equal to three different cases
      // all of which are Simple Vectors with elements of type double.
   ok &= ForTwoCases< CppAD::vector <double>, std::vector<size_t> >();
   ok &= ForTwoCases< std::vector   <double>, std::vector<size_t> >();
   ok &= ForTwoCases< std::valarray <double>, std::vector<size_t> >();

      // Run with SizeVector_t equal to two other cases
      // which are Simple Vectors with elements of type size_t.
   ok &= ForTwoCases< std::vector <double>, CppAD::vector<size_t> >();
   ok &= ForTwoCases< std::vector <double>, std::valarray<size_t> >();

   return ok;
}