exp_2_cppad¶

exp_2: CppAD Forward and Reverse Sweeps¶

Purpose¶

Use CppAD forward and reverse modes to compute the partial derivative with respect to \(x\), at the point \(x = .5\), of the function

exp_2 ( x )

as defined by the exp_2.hpp include file.

Exercises¶

Create and test a modified version of the routine below that computes the same order derivatives with respect to \(x\), at the point \(x = .1\) of the function

exp_2 ( x )
Create a routine called

exp_3 ( x )

that evaluates the function

\[f(x) = 1 + x^2 / 2 + x^3 / 6\]

Test a modified version of the routine below that computes the derivative of \(f(x)\) at the point \(x = .5\).

# include <cppad/cppad.hpp>  // https://www.coin-or.org/CppAD/
# include "exp_2.hpp"        // second order exponential approximation
bool exp_2_cppad(void)
{  bool ok = true;
   using CppAD::AD;
   using CppAD::vector;    // can use any simple vector template class
   using CppAD::NearEqual; // checks if values are nearly equal

   // domain space vector
   size_t n = 1; // dimension of the domain space
   vector< AD<double> > X(n);
   X[0] = .5;    // value of x for this operation sequence

   // declare independent variables and start recording operation sequence
   CppAD::Independent(X);

   // evaluate our exponential approximation
   AD<double> x   = X[0];
   AD<double> apx = exp_2(x);

   // range space vector
   size_t m = 1;  // dimension of the range space
   vector< AD<double> > Y(m);
   Y[0] = apx;    // variable that represents only range space component

   // Create f: X -> Y corresponding to this operation sequence
   // and stop recording. This also executes a zero order forward
   // sweep using values in X for x.
   CppAD::ADFun<double> f(X, Y);

   // first order forward sweep that computes
   // partial of exp_2(x) with respect to x
   vector<double> dx(n);  // differential in domain space
   vector<double> dy(m);  // differential in range space
   dx[0] = 1.;            // direction for partial derivative
   dy    = f.Forward(1, dx);
   double check = 1.5;
   ok   &= NearEqual(dy[0], check, 1e-10, 1e-10);

   // first order reverse sweep that computes the derivative
   vector<double>  w(m);   // weights for components of the range
   vector<double> dw(n);   // derivative of the weighted function
   w[0] = 1.;              // there is only one weight
   dw   = f.Reverse(1, w); // derivative of w[0] * exp_2(x)
   check = 1.5;            // partial of exp_2(x) with respect to x
   ok   &= NearEqual(dw[0], check, 1e-10, 1e-10);

   // second order forward sweep that computes
   // second partial of exp_2(x) with respect to x
   vector<double> x2(n);     // second order Taylor coefficients
   vector<double> y2(m);
   x2[0] = 0.;               // evaluate second partial .w.r.t. x
   y2    = f.Forward(2, x2);
   check = 0.5 * 1.;         // Taylor coef is 1/2 second derivative
   ok   &= NearEqual(y2[0], check, 1e-10, 1e-10);

   // second order reverse sweep that computes
   // derivative of partial of exp_2(x) w.r.t. x
   dw.resize(2 * n);         // space for first and second derivatives
   dw    = f.Reverse(2, w);
   check = 1.;               // result should be second derivative
   ok   &= NearEqual(dw[0*2+1], check, 1e-10, 1e-10);

   return ok;
}