exp_2: CppAD Forward and Reverse Sweeps
Keywords: exp_2: cppad forward reverse sweeps purpose exercises

| Prev | Next |  |  |  |  |  | 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

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 of the function `
   exp_2(x)
   `
2. 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>  // http://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;
    }

`

Input File: introduction/exp_apx/exp_2_cppad.cpp