// $Id: exp_eps_cppad.cpp 3785 2016-02-08 12:53:06Z bradbell $
/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-15 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the
                    GNU General Public License Version 3.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */
/*
$begin exp_eps_cppad$$
$spell
	Taylor
	dy
	coef
	resize
	cppad.hpp
	cmath
	fabs
	bool
	exp_eps_cppad
	du
	dv
	dw
	endl
	hpp
	http
	org
	std
	www
	CppAD
	apx
$$

$section exp_eps: CppAD Forward and Reverse Sweeps$$.

$head Purpose$$
Use CppAD forward and reverse modes to compute the
partial derivative with respect to $latex x$$,
at the point $latex x = .5$$ and $latex \varepsilon = .2$$,
of the function
$codei%
	exp_eps(%x%, %epsilon%)
%$$
as defined by the $cref exp_eps.hpp$$ include file.

$head Exercises$$
$list number$$
Create and test a modified version of the routine below that computes
the same order derivatives with respect to $latex x$$,
at the point $latex x = .1$$ and $latex \varepsilon = .2$$,
of the function
$codei%
	exp_eps(%x%, %epsilon%)
%$$
$lnext
Create and test a modified version of the routine below that computes
partial derivative with respect to $latex x$$,
at the point $latex x = .1$$ and $latex \varepsilon = .2$$,
of the function corresponding to the operation sequence
for $latex x = .5$$ and $latex \varepsilon = .2$$.
Hint: you could define a vector u with two components and use
$codei%
	%f%.Forward(0, %u%)
%$$
to run zero order forward mode at a point different
form the point where the operation sequence corresponding to
$icode f$$ was recorded.
$lend
$codep */
# 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;
}
/* $$
$end
*/
