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<center><b><big><big>Evaluate a Function Defined in Terms of an ODE</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>#&#xA0;include&#xA0;&lt;cppad/speed/ode_evaluate.hpp&gt;<br/>
</span></font></code>

<code><font color="blue"><span style='white-space: nowrap'>ode_evaluate(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>p</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>fp</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>


<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
This routine evaluates a function 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">:</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
<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>


defined by 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow></math>

where 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 solves the ordinary differential equation

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>x</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<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" >
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">]</mo>
</mtd></mtr></mtable>
</mrow></math>

where 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">:</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">&#x000D7;</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>


is an unspecified function.

<br/>
<br/>
<b><big><a name="Inclusion" id="Inclusion">Inclusion</a></big></b>
<br/>
The template function <code><font color="blue">ode_evaluate</font></code> 
is defined in the <code><font color="blue">CppAD</font></code> namespace by including 
the file <code><font color="blue">cppad/speed/ode_evaluate.hpp</font></code> 
(relative to the CppAD distribution directory).
It is only intended for example and testing purposes, 
so it is not automatically included by
<a href="cppad.xml" target="_top"><span style='white-space: nowrap'>cppad.hpp</span></a>
.

<br/>
<br/>
<b><big><a name="Float" id="Float">Float</a></big></b>


<br/>
<br/>
<b><a name="Float.Operation Sequence" id="Float.Operation Sequence">Operation Sequence</a></b>
<br/>
The type 
<code><i><font color="black"><span style='white-space: nowrap'>Float</span></font></i></code>
 must be a <a href="numerictype.xml" target="_top"><span style='white-space: nowrap'>NumericType</span></a>
.
The 
<code><i><font color="black"><span style='white-space: nowrap'>Float</span></font></i></code>
 
<a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation&#xA0;sequence</span></a>

for this routine does not depend on the value of the argument 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
,
hence it does not need to be retaped for each value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>

. 

<br/>
<br/>
<b><a name="Float.fabs" id="Float.fabs">fabs</a></b>
<br/>
If 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>z</span></font></i></code>
 are 
<code><i><font color="black"><span style='white-space: nowrap'>Float</span></font></i></code>
 objects, the syntax

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;fabs(</span></font><i><font color="black"><span style='white-space: nowrap'>z</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
must be supported. Note that it does not matter if the operation
sequence for <code><font color="blue">fabs</font></code> depends on 
<code><i><font color="black"><span style='white-space: nowrap'>z</span></font></i></code>
 because the 
corresponding results are not actually used by <code><font color="blue">ode_evaluate</font></code>;
see <code><font color="blue">fabs</font></code> in <a href="runge45.xml#Scalar.fabs" target="_top"><span style='white-space: nowrap'>Runge45</span></a>
.

<br/>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Float</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It contains he argument value for which the function,
or its derivative, is being evaluated.
The value 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>

 is determined by the size of the vector 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.

<br/>
<br/>
<b><big><a name="p" id="p">p</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>p</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
<br/>
<b><a name="p.p == 0" id="p.p == 0">p == 0</a></b>
<br/>
In this case a numerical method is used to solve the ode 
and obtain an accurate approximation for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow></math>

.
This numerical method has a fixed
that does not depend on 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.

<br/>
<br/>
<b><a name="p.p = 1" id="p.p = 1">p = 1</a></b>
<br/>
In this case an analytic solution for the partial derivative

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow></math>

 is returned.

<br/>
<br/>
<b><big><a name="fp" id="fp">fp</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>fp</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Float</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>fp</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The input value of the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>fp</span></font></i></code>
 does not matter.

<br/>
<br/>
<b><a name="fp.Function" id="fp.Function">Function</a></b>
<br/>
If 
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
 is zero, 
<code><i><font color="black"><span style='white-space: nowrap'>fp</span></font></i></code>
 has size equal to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>


and contains the value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><a name="fp.Gradient" id="fp.Gradient">Gradient</a></b>
<br/>
If 
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
 is one, 
<code><i><font color="black"><span style='white-space: nowrap'>fp</span></font></i></code>
 has size equal to 
<code><i><font color="black"><span style='white-space: nowrap'>n^2</span></font></i></code>
 
and for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mi mathvariant='italic'>and</mi>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

, 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>



<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>fp</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x000B7;</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow></math>

<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="ode_evaluate.cpp.xml" target="_top"><span style='white-space: nowrap'>ode_evaluate.cpp</span></a>

contains an example and test  of <code><font color="blue">ode_evaluate.hpp</font></code>.
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 file 
<a href="ode_evaluate.hpp.xml" target="_top"><span style='white-space: nowrap'>ode_evaluate.hpp</span></a>

contains the source code for this template function.


<hr/>Input File: cppad/speed/ode_evaluate.hpp

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