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<center><b><big><big>Using AD to Compute Atomic Function Derivatives</big></big></b></center>
<br/>
<b><big><a name="Deprecated" id="Deprecated">Deprecated</a></big></b>
<br/>
This example has been deprecated because it is easier to use the 
<a href="checkpoint.xml" target="_top"><span style='white-space: nowrap'>checkpoint</span></a>
 class instead.

<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
Consider the case where an inner function is used repeatedly in the 
definition of an outer function.
In this case, it may reduce the number of variables
<a href="seq_property.xml#size_var" target="_top"><span style='white-space: nowrap'>size_var</span></a>
,
and hence the required memory.

<code><font color="blue">
<pre style='display:inline'> 
# include &lt;cppad/cppad.hpp&gt;

namespace { // Begin empty namespace 
	using CppAD::AD;
	using CppAD::ADFun;
	using CppAD::vector;

	// ----------------------------------------------------------------------
	// ODE for [t, t^2 / 2 ] in form required by Runge45
	class Fun {
	public:
		void Ode(
			const <a href="ad.xml" target="_top">AD</a>&lt;double&gt;           &amp;t, 
			const vector&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt; &amp;z, 
			vector&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt;       &amp;f)
		{	assert( z.size() == 2 );
			assert( f.size() == 2 );
			f[0] =  1.0;
			f[1] =  z[0];
		}
	};

	// ----------------------------------------------------------------------
	// Create function that takes on Runge45 step for the ODE above
	<a href="funconstruct.xml" target="_top">ADFun</a>&lt;double&gt;* r_ptr_;
	void create_r(void)
	{	size_t n = 3, m = 2;
		vector&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt; x(n), zi(m), y(m), e(m);
		// The value of x does not matter because the operation sequence
		// does not depend on x. 
		x[0]  = 0.0;  // initial value z_0 (t) at t = ti
		x[1]  = 0.0;  // initial value z_1 (t) at t = ti
		x[2]  = 0.1;  // final time for this integration
		CppAD::<a href="independent.xml" target="_top">Independent</a>(x);
		zi[0]         = x[0];  // z_0 (t) at t = ti
		zi[1]         = x[1];  // z_1 (t) at t = ti
		<a href="ad.xml" target="_top">AD</a>&lt;double&gt; ti = 0.0;   // t does not appear in ODE so does not matter
		<a href="ad.xml" target="_top">AD</a>&lt;double&gt; tf = x[2];  // final time
		size_t M      = 3;     // number of Runge45 steps to take
		Fun F;
		y             = CppAD::Runge45(F, M, ti, tf, zi, e);
		r_ptr_        = new <a href="funconstruct.xml" target="_top">ADFun</a>&lt;double&gt;(x, y);
	}
	void destroy_r(void)
	{	delete r_ptr_; 
		r_ptr_ = CPPAD_NULL;
	}

	// ----------------------------------------------------------------------
	// forward mode routine called by CppAD
	bool solve_ode_forward(
		size_t                   id ,
		size_t                    k ,
		size_t                    n ,
		size_t                    m ,
		const vector&lt;bool&gt;&amp;      vx ,
		vector&lt;bool&gt;&amp;            vy ,
		const vector&lt;double&gt;&amp;    tx ,
		vector&lt;double&gt;&amp;          ty
	)
	{	assert( id == 0 );
		assert( n == 3 );
		assert( m == 2 );
		assert( k == 0 || vx.size() == 0 );
		bool ok = true;	
		vector&lt;double&gt; xp(n), yp(m);
		size_t i, j;

		// check for special case
		if( vx.size() &gt; 0 )
		{	//Compute r, a Jacobian sparsity pattern.
			// Use reverse mode because m &lt; n.
			vector&lt; std::set&lt;size_t&gt; &gt; s(m), r(m);
			for(i = 0; i &lt; m; i++)
				s[i].insert(i);
			r = r_ptr_-&gt;RevSparseJac(m, s);
			std::set&lt;size_t&gt;::const_iterator itr;
			for(i = 0; i &lt; m; i++)
			{	vy[i] = false;
				for(itr = s[i].begin(); itr != s[i].end(); itr++)
				{	j = *itr;
					assert( j &lt; n );
					// y[i] depends on the value of x[j]
					vy[i] |= vx[j];
				}
			}
		}
		// make sure r_ has proper lower order Taylor coefficients stored
		// then compute ty[k]
		for(size_t q = 0; q &lt;= k; q++)
		{	for(j = 0; j &lt; n; j++)
				xp[j] = tx[j * (k+1) + q];
			yp    = r_ptr_-&gt;Forward(q, xp);
			if( q == k )
			{	for(i = 0; i &lt; m; i++)
					ty[i * (k+1) + q] = yp[i];
			}
# ifndef NDEBUG
			else
			{	for(i = 0; i &lt; m; i++)
					assert( ty[i * (k+1) + q] == yp[i] );
			}
# endif
		}
		// no longer need the Taylor coefficients in r_ptr_
		// (have to reconstruct them every time)
		r_ptr_-&gt;capacity_order(0);
		return ok;
	}
	// ----------------------------------------------------------------------
	// reverse mode routine called by CppAD
	bool solve_ode_reverse(
		size_t                   id ,
		size_t                    k ,
		size_t                    n ,
		size_t                    m ,
		const vector&lt;double&gt;&amp;    tx ,
		const vector&lt;double&gt;&amp;    ty ,
		vector&lt;double&gt;&amp;          px ,
		const vector&lt;double&gt;&amp;    py
	)
	{	assert( id == 0 );
		assert( n == 3 );
		assert( m == 2 );
		bool ok = true;	
		vector&lt;double&gt; xp(n), w( (k+1) * m ), dw( (k+1) * n );

		// make sure r_ has proper forward mode coefficients 
		size_t i, j, q;
		for(q = 0; q &lt;= k; q++)
		{	for(j = 0; j &lt; n; j++)
				xp[j] = tx[j * (k+1) + q];
# ifdef NDEBUG
			r_ptr_-&gt;Forward(q, xp);
# else
			vector&lt;double&gt; yp(m);
			yp = r_ptr_-&gt;Forward(q, xp);
			for(i = 0; i &lt; m; i++)
				assert( ty[i * (k+1) + q] == yp[i] );
# endif
		}
		for(i = 0; i &lt; m; i++)
		{	for(q = 0; q &lt;=k; q++)
				w[ i * (k+1) + q] = py[ i * (k+1) + q];
		}
		dw = r_ptr_-&gt;Reverse(k+1, w);
		for(j = 0; j &lt; n; j++)
		{	for(q = 0; q &lt;=k; q++)
				px[ j * (k+1) + q] = dw[ j * (k+1) + q];
		}
		// no longer need the Taylor coefficients in r_ptr_
		// (have to reconstruct them every time)
		r_ptr_-&gt;capacity_order(0);

		return ok;
	}
	// ----------------------------------------------------------------------
	// forward Jacobian sparsity routine called by CppAD
	bool solve_ode_for_jac_sparse(
		size_t                               id ,             
		size_t                                n ,
		size_t                                m ,
		size_t                                p ,
		const vector&lt; std::set&lt;size_t&gt; &gt;&amp;     r ,
		vector&lt; std::set&lt;size_t&gt; &gt;&amp;           s )
	{	assert( id == 0 );
		assert( n == 3 );
		assert( m == 2 );
		bool ok = true;

		vector&lt; std::set&lt;size_t&gt; &gt; R(n), S(m);
		for(size_t j = 0; j &lt; n; j++)
			R[j] = r[j];
		S = r_ptr_-&gt;ForSparseJac(p, R);
		for(size_t i = 0; i &lt; m; i++)
			s[i] = S[i];

		// no longer need the forward mode sparsity pattern
		// (have to reconstruct them every time)
		r_ptr_-&gt;size_forward_set(0);
		
		return ok; 
	}
	// ----------------------------------------------------------------------
	// reverse Jacobian sparsity routine called by CppAD
	bool solve_ode_rev_jac_sparse(
		size_t                               id ,             
		size_t                                n ,
		size_t                                m ,
		size_t                                p ,
		vector&lt; std::set&lt;size_t&gt; &gt;&amp;           r ,
		const vector&lt; std::set&lt;size_t&gt; &gt;&amp;     s )
	{
		assert( id == 0 );
		assert( n == 3 );
		assert( m == 2 );
		bool ok = true;

		vector&lt; std::set&lt;size_t&gt; &gt; R(p), S(p);
		std::set&lt;size_t&gt;::const_iterator itr;
		size_t i;
		// untranspose s
		for(i = 0; i &lt; m; i++)
		{	for(itr = s[i].begin(); itr != s[i].end(); itr++)
				S[*itr].insert(i);
		}
		R = r_ptr_-&gt;RevSparseJac(p, S);
		// transpose r
		for(i = 0; i &lt; m; i++)
			r[i].clear();
		for(i = 0; i &lt; p; i++)
		{	for(itr = R[i].begin(); itr != R[i].end(); itr++)
				r[*itr].insert(i);
		}
		return ok; 
	}
	// ----------------------------------------------------------------------
	// reverse Hessian sparsity routine called by CppAD
	bool solve_ode_rev_hes_sparse(
		size_t                               id ,             
		size_t                                n ,
		size_t                                m ,
		size_t                                p ,
		const vector&lt; std::set&lt;size_t&gt; &gt;&amp;     r ,
		const vector&lt;bool&gt;&amp;                   s ,
		vector&lt;bool&gt;&amp;                         t ,
		const vector&lt; std::set&lt;size_t&gt; &gt;&amp;     u ,
		vector&lt; std::set&lt;size_t&gt; &gt;&amp;           v )
	{	// Can just return false if not use RevSparseHes.
		assert( id == 0 );
		assert( n == 3 );
		assert( m == 2 );
		bool ok = true;
		std::set&lt;size_t&gt;::const_iterator itr;

		// compute sparsity pattern for T(x) = S(x) * f'(x)
		vector&lt; std::set&lt;size_t&gt; &gt; S(1);
		size_t i, j;
		S[0].clear();
		for(i = 0; i &lt; m; i++)
			if( s[i] )
				S[0].insert(i);
		t = r_ptr_-&gt;RevSparseJac(1, s);

		// compute sparsity pattern for A(x)^T = U(x)^T * f'(x)
		vector&lt; std::set&lt;size_t&gt; &gt; Ut(p), At(p);
		for(i = 0; i &lt; m; i++)
		{	for(itr = u[i].begin(); itr != u[i].end(); itr++)
				Ut[*itr].insert(i);
		}
		At = r_ptr_-&gt;RevSparseJac(p, Ut);

		// compute sparsity pattern for H(x)^T = R^T * (S * F)''(x)
		vector&lt; std::set&lt;size_t&gt; &gt; R(n), Ht(p);
		for(j = 0; j &lt; n; j++)
			R[j] = r[j];
		r_ptr_-&gt;ForSparseJac(p, R);
		Ht = r_ptr_-&gt;RevSparseHes(p, S);

		// compute sparsity pattern for V(x) = A(x) + H(x)^T
		for(j = 0; j &lt; n; j++)
			v[j].clear();
		for(i = 0; i &lt; p; i++)
		{	for(itr = At[i].begin(); itr != At[i].end(); itr++)
				v[*itr].insert(i);
			for(itr = Ht[i].begin(); itr != Ht[i].end(); itr++)
				v[*itr].insert(i);
		}

		// no longer need the forward mode sparsity pattern
		// (have to reconstruct them every time)
		r_ptr_-&gt;size_forward_set(0);

		return ok;
	}
	// ---------------------------------------------------------------------
	// Declare the <a href="ad.xml" target="_top">AD</a>&lt;double&gt; routine solve_ode(id, ax, ay)
	CPPAD_USER_ATOMIC(
		solve_ode                 , 
		CppAD::vector             ,
		double                    , 
		solve_ode_forward         , 
		solve_ode_reverse         ,
		solve_ode_for_jac_sparse  ,
		solve_ode_rev_jac_sparse  ,
		solve_ode_rev_hes_sparse  
	)
} // End empty namespace

bool old_usead_2(void)
{	bool ok = true;
	using CppAD::NearEqual;
	double eps = 10. * CppAD::numeric_limits&lt;double&gt;::epsilon();

	// --------------------------------------------------------------------
	// Create the <a href="funconstruct.xml" target="_top">ADFun</a>&lt;doulbe&gt; r_
	create_r();

	// --------------------------------------------------------------------
	// domain and range space vectors
	size_t n = 3, m = 2;
	vector&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt; au(n), ax(n), ay(m);
	au[0]         = 0.0;        // value of z_0 (t) = t, at t = 0
	ax[1]         = 0.0;        // value of z_1 (t) = t^2/2, at t = 0
	au[2]         = 1.0;        // final t
	CppAD::<a href="independent.xml" target="_top">Independent</a>(au);
	size_t M      = 2;          // number of r steps to take
	ax[0]         = au[0];      // value of z_0 (t) = t, at t = 0
	ax[1]         = au[1];      // value of z_1 (t) = t^2/2, at t = 0
	<a href="ad.xml" target="_top">AD</a>&lt;double&gt; dt = au[2] / M;  // size of each r step
	ax[2]         = dt;
	for(size_t i_step = 0; i_step &lt; M; i_step++)
	{	size_t id = 0;               // not used
		solve_ode(id, ax, ay); 
		ax[0] = ay[0];
		ax[1] = ay[1];
	}

	// create f: u -&gt; y and stop tape recording
	// y_0(t) = u_0 + t                   = u_0 + u_2
	// y_1(t) = u_1 + u_0 * t + t^2 / 2   = u_1 + u_0 * u_2 + u_2^2 / 2
	// where t = u_2
	<a href="funconstruct.xml" target="_top">ADFun</a>&lt;double&gt; f;
	f.Dependent(au, ay); 

	// --------------------------------------------------------------------
	// Check forward mode results
	//
	// zero order forward
	vector&lt;double&gt; up(n), yp(m);
	size_t q  = 0;
	double u0 = 0.5;
	double u1 = 0.25;
	double u2 = 0.75;
	double check;
	up[0]     = u0;
	up[1]     = u1;
	up[2]     = u2;
	yp        = f.<a href="forward.xml" target="_top">Forward</a>(q, up);
	check     = u0 + u2;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( yp[0], check,  eps, eps);
	check     = u1 + u0 * u2 + u2 * u2 / 2.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( yp[1], check,  eps, eps);
	//
	// forward mode first derivative w.r.t t
	q         = 1;
	up[0]     = 0.0;
	up[1]     = 0.0;
	up[2]     = 1.0;
	yp        = f.<a href="forward.xml" target="_top">Forward</a>(q, up);
	check     = 1.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( yp[0], check,  eps, eps);
	check     = u0 + u2;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( yp[1], check,  eps, eps);
	//
	// forward mode second order Taylor coefficient w.r.t t
	q         = 2;
	up[0]     = 0.0;
	up[1]     = 0.0;
	up[2]     = 0.0;
	yp        = f.<a href="forward.xml" target="_top">Forward</a>(q, up);
	check     = 0.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( yp[0], check,  eps, eps);
	check     = 1.0 / 2.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( yp[1], check,  eps, eps);
	// --------------------------------------------------------------------
	// reverse mode derivatives of \partial_t y_1 (t)
	vector&lt;double&gt; w(m * q), dw(n * q);
	w[0 * q + 0]  = 0.0;
	w[1 * q + 0]  = 0.0;
	w[0 * q + 1]  = 0.0;
	w[1 * q + 1]  = 1.0;
	dw        = f.<a href="reverse.xml" target="_top">Reverse</a>(q, w);
	// derivative of y_1(u) = u_1 + u_0 * u_2 + u_2^2 / 2,  w.r.t. u
	// is equal deritative of \partial_u2 y_1(u) w.r.t \partial_u2 u
	check     = u2;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( dw[0 * q + 1], check,  eps, eps);
	check     = 1.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( dw[1 * q + 1], check,  eps, eps);
	check     = u0 + u2;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( dw[2 * q + 1], check,  eps, eps);
	// derivative of \partial_t y_1 w.r.t u = u_0 + t,  w.r.t u
	check     = 1.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( dw[0 * q + 0], check,  eps, eps);
	check     = 0.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( dw[1 * q + 0], check,  eps, eps);
	check     = 1.0;
	ok       &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>( dw[2 * q + 0], check,  eps, eps);
	// --------------------------------------------------------------------
	// forward mode sparsity pattern for the Jacobian
	// f_u = [   1, 0,   1 ]
	//       [ u_2, 1, u_2 ] 
	size_t i, j, p = n;
	CppAD::vectorBool r(n * p), s(m * p);
	// r = identity sparsity pattern
	for(i = 0; i &lt; n; i++)
		for(j = 0; j &lt; p; j++)
			r[i*n +j] = (i == j); 
	s   = f.ForSparseJac(p, r);
	ok &amp;= s[ 0 * p + 0] == true;
	ok &amp;= s[ 0 * p + 1] == false;
	ok &amp;= s[ 0 * p + 2] == true;
	ok &amp;= s[ 1 * p + 0] == true;
	ok &amp;= s[ 1 * p + 1] == true;
	ok &amp;= s[ 1 * p + 2] == true;
	// --------------------------------------------------------------------
	// reverse mode sparsity pattern for the Jacobian
	q = m;
	s.resize(q * m);
	r.resize(q * n);
	// s = identity sparsity pattern
	for(i = 0; i &lt; q; i++)
		for(j = 0; j &lt; m; j++)
			s[i*m +j] = (i == j); 
	r   = f.RevSparseJac(q, s);
	ok &amp;= r[ 0 * n + 0] == true;
	ok &amp;= r[ 0 * n + 1] == false;
	ok &amp;= r[ 0 * n + 2] == true;
	ok &amp;= r[ 1 * n + 0] == true;
	ok &amp;= r[ 1 * n + 1] == true;
	ok &amp;= r[ 1 * n + 2] == true;

	// --------------------------------------------------------------------
	// Hessian sparsity for y_1 (u) = u_1 + u_0 * u_2 + u_2^2 / 2
	s.resize(m);
	s[0] = false;
	s[1] = true;
	r.resize(n * n);
	for(i = 0; i &lt; n; i++)
		for(j = 0; j &lt; n; j++)
			r[ i * n + j ] = (i == j);
	CppAD::vectorBool h(n * n);
	h   = f.RevSparseHes(n, s);
	ok &amp;= h[0 * n + 0] == false;
	ok &amp;= h[0 * n + 1] == false;
	ok &amp;= h[0 * n + 2] == true;
	ok &amp;= h[1 * n + 0] == false;
	ok &amp;= h[1 * n + 1] == false;
	ok &amp;= h[1 * n + 2] == false;
	ok &amp;= h[2 * n + 0] == true;
	ok &amp;= h[2 * n + 1] == false;
	ok &amp;= h[2 * n + 2] == true;
	
	// --------------------------------------------------------------------
	destroy_r();

	// Free all temporary work space associated with old_atomic objects. 
	// (If there are future calls to user atomic functions, they will 
	// create new temporary work space.)
	CppAD::user_atomic&lt;double&gt;::clear();

	return ok;
}
</pre>

</font></code>


<hr/>Input File: example/atomic/old_usead_2.cpp

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