exp_eps: CppAD Forward and Reverse Sweeps
Keywords: exp_eps: cppad forward reverse sweeps purpose exercises

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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 ε = .2 , of the function `
exp_eps(x, epsilon)
` 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 ε = .2 , of the function `
   exp_eps(x, epsilon)
   `
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 ε = .2 , of the function corresponding to the operation sequence for x = .5 and ε = .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_apx/exp_eps_cppad.cpp