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exp_eps: 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 @)@ and @(@ \varepsilon = .2 @)@, of the function
     exp_eps(
xepsilon)
as defined by the exp_eps.hpp include file.

Exercises
  1. 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 @)@ and @(@ \varepsilon = .2 @)@, of the function
         exp_eps(
    xepsilon)
  2. Create and test a modified version of the routine below that computes partial derivative with respect to @(@ x @)@, at the point @(@ x = .1 @)@ and @(@ \varepsilon = .2 @)@, of the function corresponding to the operation sequence for @(@ x = .5 @)@ and @(@ \varepsilon = .2 @)@. Hint: you could define a vector u with two components and use
         
    f.Forward(0, u)
    to run zero order forward mode at a point different form the point where the operation sequence corresponding to f was recorded.
# include <cppad/cppad.hpp>  // http://www.coin-or.org/CppAD/
# include "exp_eps.hpp"      // our example exponential function approximation
bool exp_eps_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 = 2; // dimension of the domain space
     vector< AD<double> > U(n);
     U[0] = .5;    // value of x for this operation sequence
     U[1] = .2;    // value of e for this operation sequence

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

     // evaluate our exponential approximation
     AD<double> x       = U[0];
     AD<double> epsilon = U[1];
     AD<double> apx = exp_eps(x, epsilon);

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

     // first order forward mode sweep that computes partial w.r.t x
     vector<double> du(n);      // differential in domain space
     vector<double> dy(m);      // differential in range space
     du[0] = 1.;                // x direction in domain space
     du[1] = 0.;
     dy    = f.Forward(1, du);  // partial w.r.t. x
     double check = 1.5;
     ok   &= NearEqual(dy[0], check, 1e-10, 1e-10);

     // first order reverse mode 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_eps(x, epsilon)
     check = 1.5;              // partial w.r.t. x
     ok   &= NearEqual(dw[0], check, 1e-10, 1e-10);
     check = 0.;               // partial w.r.t. epsilon
     ok   &= NearEqual(dw[1], check, 1e-10, 1e-10);

     // second order forward sweep that computes
     // second partial of exp_eps(x, epsilon) w.r.t. x
     vector<double> x2(n);     // second order Taylor coefficients
     vector<double> y2(m);
     x2[0] = 0.;               // evaluate partial w.r.t x
     x2[1] = 0.;
     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_eps(x, epsilon) w.r.t. x
     dw.resize(2 * n);         // space for first and second derivative
     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;
}

Input File: introduction/exp_eps_cppad.cpp