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<center><b><big><big>One DimensionalRomberg Integration</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<code><font color="blue"><br/>
# include <cppad/romberg_one.hpp></font></code>
<code><span style='white-space: nowrap'><br/>
</span></code><code><font color="blue"></font></code><i><span style='white-space: nowrap'>r</span></i><code><font color="blue"><span style='white-space: nowrap'> = RombergOne(</span></font></code><i><span style='white-space: nowrap'>F</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>a</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>b</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>n</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>e</span></i><code><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
<br/>
<br/>
<b><big><a name="Description" id="Description">Description</a></big></b>
<br/>
Returns the Romberg integration estimate
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>r</mi>
</mrow></math>
for a one dimensional integral
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>r</mi>
<mo stretchy="false">=</mo>
<msubsup><mo stretchy="false">∫</mo>
<mi mathvariant='italic'>a</mi>
<mi mathvariant='italic'>b</mi>
</msubsup>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>d</mi>
</mstyle></mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>O</mi>
<msup><mrow><mo stretchy="true">[</mo><mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>b</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>a</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<msup><mn>2</mn>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow>
</msup>
</mrow><mo stretchy="true">]</mo></mrow>
<mrow><mn>2</mn>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
<br/>
<b><big><a name="Include" id="Include">Include</a></big></b>
<br/>
The file <code><font color="blue">cppad/romberg_one.hpp</font></code> is included by <code><font color="blue">cppad/cppad.hpp</font></code>
but it can also be included separately with out the rest of
the <code><font color="blue">CppAD</font></code> routines.
<br/>
<br/>
<b><big><a name="r" id="r">r</a></big></b>
<br/>
The return value <i>r</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'> </span></font></code><i><span style='white-space: nowrap'>r</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>It is the estimate computed by <code><font color="blue">RombergOne</font></code> for the integral above.
<br/>
<br/>
<b><big><a name="F" id="F">F</a></big></b>
<br/>
The object <i>F</i> can be of any type, but it must support
the operation
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>F</span></i><code><font color="blue"><span style='white-space: nowrap'>(</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>The argument <i>x</i> to <i>F</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>The return value of <i>F</i> is a <i>Float</i> object
(see description of <a href="rombergone.xml#Float" target="_top"><span style='white-space: nowrap'>Float</span></a>
below).
<br/>
<br/>
<b><big><a name="a" id="a">a</a></big></b>
<br/>
The argument <i>a</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>a</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>It specifies the lower limit for the integration.
<br/>
<br/>
<b><big><a name="b" id="b">b</a></big></b>
<br/>
The argument <i>b</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>b</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>It specifies the upper limit for the integration.
<br/>
<br/>
<b><big><a name="n" id="n">n</a></big></b>
<br/>
The argument <i>n</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     size_t </span></font></code><i><span style='white-space: nowrap'>n</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>A total number of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mn>2</mn>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow></math>
evaluations of <code><font color="blue"></font></code><i><span style='white-space: nowrap'>F</span></i><code><font color="blue"><span style='white-space: nowrap'>(</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
are used to estimate the integral.
<br/>
<br/>
<b><big><a name="p" id="p">p</a></big></b>
<br/>
The argument <i>p</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     size_t </span></font></code><i><span style='white-space: nowrap'>p</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>It must be less than or equal
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>
and determines the accuracy order in the approximation for the integral
that is returned by <code><font color="blue">RombergOne</font></code>.
To be specific
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>r</mi>
<mo stretchy="false">=</mo>
<msubsup><mo stretchy="false">∫</mo>
<mi mathvariant='italic'>a</mi>
<mi mathvariant='italic'>b</mi>
</msubsup>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>d</mi>
</mstyle></mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>O</mi>
<msup><mrow><mo stretchy="true">[</mo><mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>b</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>a</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<msup><mn>2</mn>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow>
</msup>
</mrow><mo stretchy="true">]</mo></mrow>
<mrow><mn>2</mn>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
<br/>
<b><big><a name="e" id="e">e</a></big></b>
<br/>
The argument <i>e</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>e</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>The input value of <i>e</i> does not matter
and its output value is an approximation for the error in
the integral estimates; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>e</mi>
<mo stretchy="false">≈</mo>
<mrow><mo stretchy="true">|</mo><mrow><mi mathvariant='italic'>r</mi>
<mo stretchy="false">-</mo>
<msubsup><mo stretchy="false">∫</mo>
<mi mathvariant='italic'>a</mi>
<mi mathvariant='italic'>b</mi>
</msubsup>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>d</mi>
</mstyle></mrow>
<mi mathvariant='italic'>x</mi>
</mrow><mo stretchy="true">|</mo></mrow>
</mrow></math>
<br/>
<b><big><a name="Float" id="Float">Float</a></big></b>
<br/>
The type <i>Float</i> must satisfy the conditions
for a <a href="numerictype.xml" target="_top"><span style='white-space: nowrap'>NumericType</span></a>
type.
The routine <a href="checknumerictype.xml" target="_top"><span style='white-space: nowrap'>CheckNumericType</span></a>
will generate an error message
if this is not the case.
In addition, if <i>x</i> and <i>y</i> are <i>Float</i> objects,
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'> < </span></font></code><i><span style='white-space: nowrap'>y</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>returns the <code><font color="blue">bool</font></code> value true if <i>x</i> is less than
<i>y</i> and false otherwise.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="rombergone.cpp.xml" target="_top"><span style='white-space: nowrap'>RombergOne.cpp</span></a>
contains an example and test a test of using this routine.
It returns true if it succeeds and false otherwise.
<br/>
<br/>
<b><big><a name="Source Code" id="Source Code">Source Code</a></big></b>
<br/>
The source code for this routine is in the file
<code><font color="blue">cppad/romberg_one.hpp</font></code>.
<hr/>Input File: cppad/romberg_one.hpp
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