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<option>x</option>
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<center><b><big><big>Speed Testing the Jacobian of Ode Solution</big></big></b></center>
<br/>
<b><big><a name="Prototype" id="Prototype">Prototype</a></big></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>extern&#xA0;bool&#xA0;link_ode(<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>size</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>repeat</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;double&gt;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;double&gt;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>jacobian</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
);<br/>
</span></font></code>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
Each <a href="speed_main.xml#package" target="_top"><span style='white-space: nowrap'>package</span></a>

must define a version of this routine as specified below.
This is used by the <a href="speed_main.xml" target="_top"><span style='white-space: nowrap'>speed_main</span></a>
 program
to run the corresponding speed and correctness tests.

<br/>
<br/>
<b><big><a name="Method" id="Method">Method</a></big></b>
<br/>
The same template routine <a href="ode_evaluate.xml" target="_top"><span style='white-space: nowrap'>ode_evaluate</span></a>
 is used
by th different AD packages.

<br/>
<br/>
<b><big><a name="f" id="f">f</a></big></b>
<br/>
The 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>

 that is defined and computed by
evaluating <a href="ode_evaluate.xml" target="_top"><span style='white-space: nowrap'>ode_evaluate</span></a>
 with a call of the form

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;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'>)<br/>
</span></font></code>
with 
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
 equal to zero.
Calls with the value 
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
 equal to one are used to check
the derivative values.

<br/>
<br/>
<b><big><a name="Return Value" id="Return Value">Return Value</a></big></b>
<br/>
If this speed test is not yet
supported by a particular 
<code><i><font color="black"><span style='white-space: nowrap'>package</span></font></i></code>
,
the corresponding return value for <code><font color="blue">link_ode</font></code>
should be <code><font color="blue">false</font></code>.

<br/>
<br/>
<b><big><a name="size" id="size">size</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>size</span></font></i></code>
 
is the number of variables in the ordinary differential equations
which is also equal to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="repeat" id="repeat">repeat</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>repeat</span></font></i></code>
 is the number of times the
Jacobian is computed. 

<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>
 is a vector with 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>

 elements.
The input value of the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 does not matter.
On output, it has been set to the
argument value for which the function,
or its derivative, is being evaluated.
The value of this vector must change with each repetition.

<br/>
<br/>
<b><big><a name="jacobian" id="jacobian">jacobian</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>jacobian</span></font></i></code>
 is a vector with 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>n</mi>
<mn>2</mn>
</msup>
</mrow></math>

 elements.
The input value of its elements does not matter. 
The output value of its elements is the Jacobian of the function 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>


that corresponds to output value of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.
To be more specific, 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>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

 and 
<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'>f</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>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>jacobian</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><a name="jacobian.double" id="jacobian.double">double</a></b>
<br/>
In the case where 
<code><i><font color="black"><span style='white-space: nowrap'>package</span></font></i></code>
 is <code><font color="blue">double</font></code>,
only the first 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>

 element of 
<code><i><font color="black"><span style='white-space: nowrap'>jacobian</span></font></i></code>

are modified and they are to the function value

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

 corresponding to the output value of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.


<hr/>Input File: speed/src/link_ode.cpp

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