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<center><b><big><big>Rosen34: Example and Test</big></big></b></center>
Define 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">:</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mo stretchy="false">&#x02192;</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>

 by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>X</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow></math>

for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

.
It follows that

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>X</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>0</mn>
</mtd><mtd columnalign="right" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>for</mi>
<mspace width='.3em'/>
<mi mathvariant='normal'>all</mi>
<mspace width='.3em'/>
</mstyle></mrow>
<mi mathvariant='italic'>i</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>X</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">'</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>1</mn>
</mtd><mtd columnalign="right" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
<mspace width='.3em'/>
</mstyle></mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>X</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">'</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msup>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msub><mi mathvariant='italic'>X</mi>
<mrow><mi mathvariant='italic'>i</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="right" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
<mspace width='.3em'/>
</mstyle></mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&gt;</mo>
<mn>0</mn>
</mtd></mtr></mtable>
</mrow></math>

The example tests Rosen34 using the relations above:

<code><font color="blue">
<pre style='display:inline'> 

# include &lt;cppad/cppad.hpp&gt;        // For automatic differentiation

namespace {
	class Fun {
	public:
		// constructor
		Fun(bool use_x_) : use_x(use_x_) 
		{ }

		// compute f(t, x) both for double and <a href="ad.xml" target="_top">AD</a>&lt;double&gt;
		template &lt;typename Scalar&gt;
		void Ode(
			const Scalar                    &amp;t, 
			const <a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(Scalar) &amp;x, 
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(Scalar)       &amp;f)
		{	size_t n  = x.size();	
			Scalar ti(1);
			f[0]   = Scalar(1);
			size_t i;
			for(i = 1; i &lt; n; i++)
			{	ti *= t;
				// convert int(size_t) to avoid warning
				// on _MSC_VER systems
				if( use_x )
					f[i] = int(i+1) * x[i-1];
				else	f[i] = int(i+1) * ti;
			}
		}

		// compute partial of f(t, x) w.r.t. t using AD
		void Ode_ind(
			const double                    &amp;t, 
			const <a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double) &amp;x, 
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double)       &amp;f_t)
		{	using namespace CppAD;

			size_t n  = x.size();	
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(AD&lt;double&gt;) T(1);
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(AD&lt;double&gt;) X(n);
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(AD&lt;double&gt;) F(n);

			// set argument values
			T[0] = t;
			size_t i;
			for(i = 0; i &lt; n; i++)
				X[i] = x[i];

			// declare independent variables
			<a href="independent.xml" target="_top">Independent</a>(T);

			// compute f(t, x)
			this-&gt;Ode(T[0], X, F);

			// define AD function object
			<a href="funconstruct.xml" target="_top">ADFun</a>&lt;double&gt; fun(T, F);

			// compute partial of f w.r.t t
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double) dt(1);
			dt[0] = 1.;
			f_t = fun.<a href="forward.xml" target="_top">Forward</a>(1, dt);
		}

		// compute partial of f(t, x) w.r.t. x using AD
		void Ode_dep(
			const double                    &amp;t, 
			const <a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double) &amp;x, 
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double)       &amp;f_x)
		{	using namespace CppAD;

			size_t n  = x.size();	
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(AD&lt;double&gt;) T(1);
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(AD&lt;double&gt;) X(n);
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(AD&lt;double&gt;) F(n);

			// set argument values
			T[0] = t;
			size_t i, j;
			for(i = 0; i &lt; n; i++)
				X[i] = x[i];

			// declare independent variables
			<a href="independent.xml" target="_top">Independent</a>(X);

			// compute f(t, x)
			this-&gt;Ode(T[0], X, F);

			// define AD function object
			<a href="funconstruct.xml" target="_top">ADFun</a>&lt;double&gt; fun(X, F);

			// compute partial of f w.r.t x
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double) dx(n);
			<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double) df(n);
			for(j = 0; j &lt; n; j++)
				dx[j] = 0.;
			for(j = 0; j &lt; n; j++)
			{	dx[j] = 1.;
				df = fun.<a href="forward.xml" target="_top">Forward</a>(1, dx);
				for(i = 0; i &lt; n; i++)
					f_x [i * n + j] = df[i];
				dx[j] = 0.;
			}
		}

	private:
		const bool use_x;

	};
}

bool Rosen34(void)
{	bool ok = true;     // initial return value
	size_t i;           // temporary indices

	size_t  n = 4;      // number components in X(t) and order of method
	size_t  M = 2;      // number of Rosen34 steps in [ti, tf]
	double ti = 0.;     // initial time
	double tf = 2.;     // final time 

	// xi = X(0)
	<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double) xi(n); 
	for(i = 0; i &lt;n; i++)
		xi[i] = 0.;

	size_t use_x;
	for( use_x = 0; use_x &lt; 2; use_x++)
	{	// function object depends on value of use_x
		Fun F(use_x &gt; 0); 

		// compute Rosen34 approximation for X(tf)
		<a href="testvector.xml" target="_top">CPPAD_TESTVECTOR</a>(double) xf(n), e(n); 
		xf = CppAD::Rosen34(F, M, ti, tf, xi, e);

		double check = tf;
		for(i = 0; i &lt; n; i++)
		{	// check that error is always positive
			ok    &amp;= (e[i] &gt;= 0.);
			// 4th order method is exact for i &lt; 4
			if( i &lt; 4 ) ok &amp;=
				CppAD::<a href="nearequal.xml" target="_top">NearEqual</a>(xf[i], check, 1e-10, 1e-10);
			// 3rd order method is exact for i &lt; 3
			if( i &lt; 3 )
				ok &amp;= (e[i] &lt;= 1e-10);

			// check value for next i
			check *= tf;
		}
	}
	return ok;
}

</pre>

</font></code>


<hr/>Input File: example/rosen_34.cpp

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