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<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
      xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>One Dimensional Romberg Integration: Example and Test</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="One Dimensional Romberg Integration: Example and Test"/>
<meta name="keywords" id="keywords" content=" one dimensional romberg integration: example and test Romberg "/>
<style type='text/css'>
body { color : black }
body { background-color : white }
A:link { color : blue }
A:visited { color : purple }
A:active { color : purple }
</style>
<script type='text/javascript' language='JavaScript' src='_romberg_one.cpp_xml.js'>
</script>
</head>
<body>
<table><tr>
<td>
<a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a>
</td>
<td><a href="rombergone.xml" target="_top">Prev</a>
</td><td><a href="rombergmul.xml" target="_top">Next</a>
</td><td>
<select onchange='choose_across0(this)'>
<option>Index-&gt;</option>
<option>contents</option>
<option>reference</option>
<option>index</option>
<option>search</option>
<option>external</option>
</select>
</td>
<td>
<select onchange='choose_up0(this)'>
<option>Up-&gt;</option>
<option>CppAD</option>
<option>library</option>
<option>RombergOne</option>
<option>romberg_one.cpp</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>CppAD-&gt;</option>
<option>Install</option>
<option>Introduction</option>
<option>AD</option>
<option>ADFun</option>
<option>preprocessor</option>
<option>multi_thread</option>
<option>library</option>
<option>ipopt_solve</option>
<option>Example</option>
<option>speed</option>
<option>Appendix</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>library-&gt;</option>
<option>ErrorHandler</option>
<option>NearEqual</option>
<option>speed_test</option>
<option>SpeedTest</option>
<option>time_test</option>
<option>NumericType</option>
<option>CheckNumericType</option>
<option>SimpleVector</option>
<option>CheckSimpleVector</option>
<option>nan</option>
<option>pow_int</option>
<option>Poly</option>
<option>LuDetAndSolve</option>
<option>RombergOne</option>
<option>RombergMul</option>
<option>Runge45</option>
<option>Rosen34</option>
<option>OdeErrControl</option>
<option>OdeGear</option>
<option>OdeGearControl</option>
<option>CppAD_vector</option>
<option>thread_alloc</option>
<option>index_sort</option>
<option>BenderQuad</option>
<option>opt_val_hes</option>
<option>LuRatio</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>RombergOne-&gt;</option>
<option>romberg_one.cpp</option>
</select>
</td>
<td>romberg_one.cpp</td>
<td>Headings</td>
</tr></table><br/>



<center><b><big><big>One Dimensional Romberg Integration: Example and Test</big></big></b></center>
<code><font color="blue"><pre style='display:inline'> 

# include &lt;cppad/romberg_one.hpp&gt;
# include &lt;cppad/vector.hpp&gt;
# include &lt;cppad/near_equal.hpp&gt;

namespace {
	class Fun {
	private:
		const size_t degree;
	public:
		// constructor
		Fun(size_t degree_) : degree(degree_) 
		{ }

		// function F(x) = x^degree
		template &lt;class Type&gt;
		Type operator () (const Type &amp;x)
		{	size_t i;
			Type   f = 1;
			for(i = 0; i &lt; degree; i++)
				f *= x;
			return f;
		}
	};
}

bool RombergOne(void)
{	bool ok = true;
	size_t i;

	size_t degree = 4;
	Fun F(degree);

	// arguments to RombergOne
	double a = 0.;
	double b = 1.;
	size_t n = 4;
	size_t p;
	double r, e;

	// int_a^b F(x) dx = [ b^(degree+1) - a^(degree+1) ] / (degree+1) 
	double bpow = 1.;
	double apow = 1.;
	for(i = 0; i &lt;= degree; i++)
	{	bpow *= b;
		apow *= a;
	}  
	double check = (bpow - apow) / (degree+1);

	// step size corresponding to r
	double step = (b - a) / exp(log(2.)*(n-1));
	// step size corresponding to error estimate
	step *= 2.;
	// step size raised to a power
	double spow = 1;

	for(p = 0; p &lt; n; p++)
	{	spow = spow * step * step;

		r = CppAD::RombergOne(F, a, b, n, p, e);

		ok  &amp;= e &lt; (degree+1) * spow;
		ok  &amp;= CppAD::<a href="nearequal.xml" target="_top">NearEqual</a>(check, r, 0., e);	
	}

	return ok;
}

</pre>

</font></code>


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

</body>
</html>