/* $Id: checkpoint.cpp 1662 2010-03-11 18:31:11Z bradbell $ */
/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-10 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the 
                    Common Public License Version 1.0.

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 checkpoint.cpp$$
$spell
	Taylor
$$

$section Checkpoint and Function Composition: Example and Test$$

$index checkpoint, example$$
$index example, checkpoint$$
$index test, checkpoint$$

$head Purpose$$
Break a large computation into pieces and only store values at the 
interface of the pieces.
In actual applications, there may be many functions, but 
for this example there are only two.
The functions 
$latex F : \R^2 \rightarrow \R^2$$ 
and
$latex G : \R^2 \rightarrow \R^2$$ 
defined by
$latex \[
	F(x) = \left( \begin{array}{c} x_0 x_1   \\ x_1 - x_0 \end{array} \right) 
	\; , \;
	G(y) = \left( \begin{array}{c} y_0 - y_1 \\ y_1  y_0   \end{array} \right) 
\] $$

$head Processing Steps$$
We apply reverse mode to compute the derivative of
$latex H : \R^2 \rightarrow \R$$
is defined by
$latex \[
\begin{array}{rcl}
	H(x) 
	& = & G_0 [ F(x) ] + G_1 [ F(x)  ] 
	\\
	& = & x_0 x_1 - ( x_1 - x_0 ) + x_0 x_1 ( x_1 - x_0 ) 
	\\
	& = & x_0 x_1 ( 1 - x_0 + x_1 ) - x_1 + x_0
\end{array}
\] $$ 
Given the zero and first order Taylor coefficients 
$latex x^{(0)} $$ and $latex x^{(1)}$$,
we use $latex X(t)$$, $latex Y(t)$$ and $latex Z(t)$$
for the corresponding functions; i.e.,
$latex \[
\begin{array}{rcl}
	X(t) & = & x^{(0)} + x^{(1)} t 
	\\
	Y(t) & = & F[X(t)] = y^{(0)} + y^{(1)} t  + O(t^2)
	\\
	Z(t) & = & G \{ F [ X(t) ] \} = z^{(0)} + z^{(1)} t  + O(t^2)
	\\
	h^{(0)} & = & z^{(0)}_0 + z^{(0)}_1
	\\
	h^{(1)} & = & z^{(1)}_0 + z^{(1)}_1
\end{array}
\] $$
Here are the processing steps:

$list number$$
Use forward mode on $latex F(x)$$ to compute 
$latex y^{(0)}$$ and $latex y^{(1)}$$.
$lnext
Free some, or all, of the memory corresponding to $latex F(x)$$.
$lnext
Use forward mode on $latex G(y)$$ to compute 
$latex z^{(0)}$$ and $latex z^{(1)}$$ 
$lnext
Use reverse mode on $latex G(y)$$ to compute the derivative of 
$latex h^{(1)}$$ with respect to
$latex y^{(0)}$$ and $latex y^{(1)}$$.
$lnext
Free all the memory corresponding to $latex G(y)$$.
$lnext
Use reverse mode on $latex F(x)$$ to compute the derivative of
$latex h^{(1)}$$ with respect to
$latex x^{(0)}$$ and $latex x^{(1)}$$.
$lend
This uses the following relations:
$latex \[
\begin{array}{rcl}
	\partial_{x(0)} h^{(1)} [ x^{(0)} , x^{(1)} ]
	& = &
	\partial_{y(0)} h^{(1)} [ y^{(0)} , y^{(1)} ]
	\partial_{x(0)} y^{(0)} [ x^{(0)} , x^{(1)} ]
	\\
	& + &
	\partial_{y(1)} h^{(1)} [ y^{(0)} , y^{(1)} ]
	\partial_{x(0)} y^{(1)} [ x^{(0)} , x^{(1)} ]
	\\
	\partial_{x(1)} h^{(1)} [ x^{(0)} , x^{(1)} ]
	& = &
	\partial_{y(0)} h^{(1)} [ y^{(0)} , y^{(1)} ]
	\partial_{x(1)} y^{(0)} [ x^{(0)} , x^{(1)} ]
	\\
	& + &
	\partial_{y(1)} h^{(1)} [ y^{(0)} , y^{(1)} ]
	\partial_{x(1)} y^{(1)} [ x^{(0)} , x^{(1)} ]
\end{array}
\] $$
where $latex \partial_{x(0)}$$ denotes the partial with respect
to $latex x^{(0)}$$.

$code
$verbatim%example/checkpoint.cpp%0%// BEGIN PROGRAM%// END PROGRAM%1%$$
$$


$end
*/
// BEGIN PROGRAM

# include <cppad/cppad.hpp>

namespace {
	template <class Vector>
	Vector F(const Vector& x)
	{	Vector y(2);
		y[0] = x[0] * x[1];
		y[1] = x[1] - x[0];
		return y;
	}
	template <class Vector>
	Vector G(const Vector& y)
	{	Vector z(2);
		z[0] = y[0] - y[1];
		z[1] = y[1] * y[0];
		return z;
	}
}

namespace {
	bool checkpoint_case(bool free_all)
	{	bool ok = true;
     	double eps = 10. * std::numeric_limits<double>::epsilon();

		using CppAD::AD;
		using CppAD::NearEqual;
		CppAD::ADFun<double> f, g, empty;

		// specify the Taylor coefficients for X(t)
		size_t n    = 2;
		CPPAD_TEST_VECTOR<double> x0(n), x1(n);
		x0[0] = 1.; x0[1] = 2.;
		x1[0] = 3.; x1[1] = 4.;

		// record the function F(x)
		CPPAD_TEST_VECTOR< AD<double> > X(n), Y(n);
		size_t i;
		for(i = 0; i < n; i++)
			X[i] = x0[i];
		CppAD::Independent(X);
		Y = F(X);
		f.Dependent(X, Y);

		// a fucntion object with an almost empty operation sequence
		CppAD::Independent(X);
		empty.Dependent(X, X);

		// compute the Taylor coefficients for Y(t)
		CPPAD_TEST_VECTOR<double> y0(n), y1(n);
		y0 = f.Forward(0, x0);
		y1 = f.Forward(1, x1);
		if( free_all )
			f = empty;
		else	
		{	// free all the Taylor coefficients stored in f
			f.capacity_taylor(0);
		}

		// record the function G(x)
		CPPAD_TEST_VECTOR< AD<double> > Z(n);
		CppAD::Independent(Y);
		Z = G(Y);
		g.Dependent(Y, Z);

		// compute the Taylor coefficients for Z(t)
		CPPAD_TEST_VECTOR<double> z0(n), z1(n);
		z0 = g.Forward(0, y0);
		z1 = g.Forward(1, y1);

		// check zero order Taylor coefficient for h^0 = z_0^0 + z_1^0
		double check = x0[0] * x0[1] * (1. - x0[0] + x0[1]) - x0[1] + x0[0];
		double h0    = z0[0] + z0[1];
		ok          &= NearEqual(h0, check, eps, eps);

		// check first order Taylor coefficient h^1
		check     = x0[0] * x0[1] * (- x1[0] + x1[1]) - x1[1] + x1[0];
		check    += x1[0] * x0[1] * (1. - x0[0] + x0[1]);
		check    += x0[0] * x1[1] * (1. - x0[0] + x0[1]);
		double h1 = z1[0] + z1[1];
		ok       &= NearEqual(h1, check, eps, eps);
	
		// compute the derivative with respect to y^0 and y^0 of h^1
		size_t p = 2;
		CPPAD_TEST_VECTOR<double> w(n*p), dw(n*p);
		w[0*p+0] = 0.; // coefficient for z_0^0
		w[0*p+1] = 1.; // coefficient for z_0^1
		w[1*p+0] = 0.; // coefficient for z_1^0
		w[1*p+1] = 1.; // coefficient for z_1^1
		dw       = g.Reverse(p, w);

		// We are done using g, so we can free its memory.
		g = empty;
		// We need to use f next. 
		if( free_all )
		{	// we must again record the operation sequence for F(x)
			CppAD::Independent(X);
			Y = F(X);
			f.Dependent(X, Y);
		}
		// now recompute the Taylor coefficients corresponding to F(x)
		// (we already know the result; i.e., y0 and y1).
		f.Forward(0, x0);
		f.Forward(1, x1);

		// compute the derivative with respect to x^0 and x^0 of
		//	h^1 = z_0^1 + z_1^1
		CPPAD_TEST_VECTOR<double> dv(n*p);
		dv   = f.Reverse(p, dw); 

		// check partial of h^1 w.r.t x^0_0
		check  = x0[1] * (- x1[0] + x1[1]);
		check -= x1[0] * x0[1];
		check += x1[1] * (1. - x0[0] + x0[1]) - x0[0] * x1[1];
		ok    &= NearEqual(dv[0*p+0], check, eps, eps);

		// check partial of h^1 w.r.t x^0_1
		check  = x0[0] * (- x1[0] + x1[1]);
		check += x1[0] * (1. - x0[0] + x0[1]) + x1[0] * x0[1]; 
		check += x0[0] * x1[1];
		ok    &= NearEqual(dv[1*p+0], check, eps, eps);

		// check partial of h^1 w.r.t x^1_0
		check  = 1. - x0[0] * x0[1];
		check += x0[1] * (1. - x0[0] + x0[1]);
		ok    &= NearEqual(dv[0*p+1], check, eps, eps);

		// check partial of h^1 w.r.t x^1_1
		check  = x0[0] * x0[1] - 1.;
		check += x0[0] * (1. - x0[0] + x0[1]);
		ok    &= NearEqual(dv[1*p+1], check, eps, eps);

		return ok;
	}
}
bool checkpoint(void)
{	bool ok = true;
	ok     &= checkpoint_case(true);
	ok     &= checkpoint_case(false);
	return ok;
}

// END PROGRAM
