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<title>Computing a Jacobian With Constants that Change</title>
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<center><b><big><big>Computing a Jacobian With Constants that Change</big></big></b></center>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
In this example we use two levels of taping so that a derivative
can have constant parameters that can be changed. To be specific,
we consider 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>
<mn>2</mn>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mn>2</mn>
</msup>
</mrow></math>



<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'>p</mi>
<mrow><mo stretchy="true">(</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<mi>sin</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="center" >
<mi>sin</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>

were 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">&#x02208;</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
</mrow></math>

 is a parameter.
The Jacobian of this function is

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>p</mi>
<mrow><mo stretchy="true">(</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<mi>cos</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="center" >
<mn>0</mn>
</mtd><mtd columnalign="center" >
<mi>cos</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>

In this example we use two levels of AD to avoid computing
the partial of 
<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>

 with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>

,
but still allow for the evaluation of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow></math>


at different values of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>

.


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

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