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<center><b><big><big>Jacobian: Driver Routine</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<br/>

<code><i><font color="black"><span style='white-space: nowrap'>jac</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Jacobian(</span></font><i><font color="black"><span style='white-space: nowrap'>x</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/>
We use 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>

 to denote the
<a href="glossary.xml#AD Function" target="_top"><span style='white-space: nowrap'>AD&#xA0;function</span></a>
 corresponding to 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
The syntax above sets 
<code><i><font color="black"><span style='white-space: nowrap'>jac</span></font></i></code>
 to the
Jacobian of 
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i></code>
 evaluated at 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
; i.e.,

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

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;ADFun&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
Note that the <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
 object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 is not <code><font color="blue">const</font></code>
(see <a href="jacobian.xml#Forward or Reverse" target="_top"><span style='white-space: nowrap'>Forward&#xA0;or&#xA0;Reverse</span></a>
 below).

<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;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#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'><br/>
</span></font></code>
(see <a href="jacobian.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
 below)
and its size 
must be equal to 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
 space for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
It specifies
that point at which to evaluate the Jacobian.

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

<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'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>jac</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="jacobian.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
 below)
and its size is 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>

; i.e., the product of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>

and
<a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>

dimensions for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
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'>m</mi>
<mo stretchy="false">-</mo>
<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>
<mo stretchy="false">-</mo>
<mn>1</mn>
</mrow></math>



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

<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
The type 
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
 must be a <a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
 class with
<a href="simplevector.xml#Elements of Specified Type" target="_top"><span style='white-space: nowrap'>elements&#xA0;of&#xA0;type</span></a>


<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
.
The routine <a href="checksimplevector.xml" target="_top"><span style='white-space: nowrap'>CheckSimpleVector</span></a>
 will generate an error message
if this is not the case.

<br/>
<br/>
<b><big><a name="Forward or Reverse" id="Forward or Reverse">Forward or Reverse</a></big></b>
<br/>
This will use order zero Forward mode and either
order one Forward or order one Reverse to compute the Jacobian
(depending on which it estimates will require less work).
After each call to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
,
the object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 contains the corresponding 
<a href="glossary.xml#Taylor Coefficient" target="_top"><span style='white-space: nowrap'>Taylor&#xA0;coefficients</span></a>
.
After a call to <code><font color="blue">Jacobian</font></code>,
the zero order Taylor coefficients correspond to

<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(0,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>

and the other coefficients are unspecified.

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

<br/>
The routine 
<a href="jacobian.cpp.xml" target="_top"><span style='white-space: nowrap'>Jacobian</span></a>
 is both an example and test.
It returns <code><font color="blue">true</font></code>, if it succeeds and <code><font color="blue">false</font></code> otherwise.


<hr/>Input File: cppad/local/jacobian.hpp

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