/* $Id: forward_order.cpp 3214 2014-03-18 20:50:38Z bradbell $ */
/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-14 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 forward_order.cpp$$
$spell
	Cpp
$$

$section Forward Mode: Example and Test of Multiple Orders$$
$index forward, multiple orders$$
$index multiple, forward orders$$
$index order, multiple forward$$

$code
$verbatim%example/forward_order.cpp%0%// BEGIN C++%// END C++%1%$$
$$

$end
*/
// BEGIN C++
# include <limits>
# include <cppad/cppad.hpp>
bool forward_order(void)
{	bool ok = true;
	using CppAD::AD;
	using CppAD::NearEqual;
	double eps = 10. * std::numeric_limits<double>::epsilon();

	// domain space vector
	size_t n = 2;
	CPPAD_TESTVECTOR(AD<double>) ax(n);
	ax[0] = 0.; 
	ax[1] = 1.;

	// declare independent variables and starting recording
	CppAD::Independent(ax);

	// range space vector
	size_t m = 1;
	CPPAD_TESTVECTOR(AD<double>) ay(m);
	ay[0] = ax[0] * ax[0] * ax[1];

	// create f: x -> y and stop tape recording
	CppAD::ADFun<double> f(ax, ay);

	// initially, the variable values during taping are stored in f
	ok &= f.size_order() == 1;

	// Compute three forward orders at one
	size_t q = 2, q1 = q+1;
	CPPAD_TESTVECTOR(double) xq(n*q1), yq;
	xq[q1*0 + 0] = 3.;    xq[q1*1 + 0] = 4.; // x^0 (order zero)
	xq[q1*0 + 1] = 1.;    xq[q1*1 + 1] = 0.; // x^1 (order one)
	xq[q1*0 + 2] = 0.;    xq[q1*1 + 2] = 0.; // x^2 (order two)
	// X(t) =   x^0 + x^1 * t + x^2 * t^2
	//      = [ 3 + t, 4 ]   
	yq  = f.Forward(q, xq);
	ok &= yq.size() == m*q1;
	// Y(t) = F[X(t)] 
	//      = (3 + t) * (3 + t) * 4
	//      = y^0 + y^1 * t + y^2 * t^2 + o(t^3)
	//
	// check y^0 (order zero)
	CPPAD_TESTVECTOR(double) x0(n);
	x0[0] = xq[q1*0 + 0];
	x0[1] = xq[q1*1 + 0];
	ok  &= NearEqual(yq[q1*0 + 0] , x0[0]*x0[0]*x0[1], eps, eps);
	//
	// check y^1 (order one)
	ok  &= NearEqual(yq[q1*0 + 1] , 2.*x0[0]*x0[1], eps, eps);
	//
	// check y^2 (order two)
	double F_00 = 2. * yq[q1*0 + 2]; // second partial F w.r.t. x_0, x_0
	ok   &= NearEqual(F_00, 2.*x0[1], eps, eps);

	// check number of orders per variable
	ok   &= f.size_order() == 3;

	return ok;
}
// END C++