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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