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<title>Any Order Reverse Mode</title>
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<td>
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</td>
<td><a href="hes_times_dir.cpp.xml" target="_top">Prev</a>
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<option>Index-&gt;</option>
<option>contents</option>
<option>reference</option>
<option>index</option>
<option>search</option>
<option>external</option>
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<option>Up-&gt;</option>
<option>CppAD</option>
<option>ADFun</option>
<option>FunEval</option>
<option>Reverse</option>
<option>reverse_any</option>
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<td>
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<option>ADFun-&gt;</option>
<option>Independent</option>
<option>FunConstruct</option>
<option>Dependent</option>
<option>abort_recording</option>
<option>seq_property</option>
<option>FunEval</option>
<option>Drivers</option>
<option>FunCheck</option>
<option>optimize</option>
<option>check_for_nan</option>
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<td>
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<option>FunEval-&gt;</option>
<option>Forward</option>
<option>Reverse</option>
<option>Sparse</option>
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<td>
<select onchange='choose_down1(this)'>
<option>Reverse-&gt;</option>
<option>reverse_one</option>
<option>reverse_two</option>
<option>reverse_any</option>
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<td>
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<option>reverse_any-&gt;</option>
<option>reverse_three.cpp</option>
<option>reverse_any.cpp</option>
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<option>Headings-&gt;</option>
<option>Syntax</option>
<option>Purpose</option>
<option>Notation</option>
<option>---..x^(k)</option>
<option>---..X(t, u)</option>
<option>---..Y(t, u)</option>
<option>---..w^(k)</option>
<option>---..W(u)</option>
<option>f</option>
<option>q</option>
<option>w</option>
<option>dw</option>
<option>First Order</option>
<option>Second Order</option>
<option>Vector</option>
<option>Example</option>
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<center><b><big><big>Any Order Reverse Mode</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'>dw</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'>.Reverse(</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>w</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>
.
Reverse mode computes the derivative of the <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
 mode
<a href="glossary.xml#Taylor Coefficient" target="_top"><span style='white-space: nowrap'>Taylor&#xA0;coefficients</span></a>

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

.
To be specific, it computes the derivative 

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

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


which is specified by the following notation:

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


<br/>
<br/>
<b><a name="Notation.x^(k)" id="Notation.x^(k)">x^(k)</a></b>
<br/>
For 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mrow></math>

,
the vector 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>

 is defined as the value of 

<code><i><font color="black"><span style='white-space: nowrap'>x_k</span></font></i></code>
 in the previous calls of the form

<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'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>x_k</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
If there is no previous call with 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

 is the value of the independent variables when the 
corresponding 
AD of 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>

<a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation&#xA0;sequence</span></a>
 was recorded.


<br/>
<br/>
<b><a name="Notation.X(t, u)" id="Notation.X(t, u)">X(t, u)</a></b>
<br/>
The function

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">&#x000D7;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>

 is defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>q</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mrow>
</msup>
</mrow></math>

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

,

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

 is related to the <i>k</i>-th partial of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</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'>t</mi>
</mrow></math>

 by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mrow><mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mrow></math>

<br/>
<b><a name="Notation.Y(t, u)" id="Notation.Y(t, u)">Y(t, u)</a></b>
<br/>
The function

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">&#x000D7;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>

 is defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>

<br/>
<b><a name="Notation.w^(k)" id="Notation.w^(k)">w^(k)</a></b>
<br/>
If the argument 
<code><i><font color="black"><span style='white-space: nowrap'>w</span></font></i></code>
 has size 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i></code>
,
for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mrow></math>

 and 
<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>
<mn>-1</mn>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msubsup><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>w</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
</mrow></math>

If the argument 
<code><i><font color="black"><span style='white-space: nowrap'>w</span></font></i></code>
 has size 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
,
for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mrow></math>

 and 
<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>
<mn>-1</mn>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msubsup><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mi mathvariant='italic'>w</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mtd></mtr><mtr><mtd columnalign="left" >
<mn>0</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>otherwise</mi>
</mstyle></mrow>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mrow></math>

<br/>
<b><a name="Notation.W(u)" id="Notation.W(u)">W(u)</a></b>
<br/>
The function 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
<mo stretchy="false">&#x02192;</mo>
<mi mathvariant='italic'>B</mi>
</mrow></math>

 is defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mrow><mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>w</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mrow><mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</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;const&#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>
Before this call to <code><font color="blue">Reverse</font></code>, the value returned by

<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'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_order()<br/>
</span></font></code>
must be greater than or equal 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i></code>

(see <a href="size_order.xml" target="_top"><span style='white-space: nowrap'>size_order</span></a>
).

<br/>
<br/>
<b><big><a name="q" id="q">q</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>q</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'>q</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and specifies the number of Taylor coefficient orders to be differentiated
(for each variable).


<br/>
<br/>
<b><big><a name="w" id="w">w</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>w</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'>w</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="reverse_any.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'>m</span></font></i></code>
 or 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i></code>
, 
It specifies the weighting vector 
<code><i><font color="black"><span style='white-space: nowrap'>w</span></font></i></code>

in the definition of 
<a href="reverse_any.xml#Notation.W(u)" target="_top"><span style='white-space: nowrap'>W(u)</span></a>
.


<br/>
<br/>
<b><big><a name="dw" id="dw">dw</a></big></b>
<br/>
The return value 
<code><i><font color="black"><span style='white-space: nowrap'>dw</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'>dw</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="reverse_any.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
 below).
It is a vector with size 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow></math>

.
For 
<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>

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


If the argument 
<code><i><font color="black"><span style='white-space: nowrap'>w</span></font></i></code>
 has size 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i></code>
,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>dw</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>W</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msub><mo stretchy="false">)</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow>
</msub>
</mrow></math>

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

 is value of the Taylor coefficients where the 
derivative is evaluated.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>If the argument 
<code><i><font color="black"><span style='white-space: nowrap'>w</span></font></i></code>
 has size 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>dw</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">-</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>W</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msub><mo stretchy="false">)</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow>
</msub>
</mrow></math>

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

 is value of the Taylor coefficients where the 
derivative is evaluated.
Note the reverse order in which the order indices are stored.
This is an unfortunate consequence of keeping <code><font color="blue">Reverse</font></code> backward
compatible.

<br/>
<br/>
<b><big><a name="First Order" id="First Order">First Order</a></big></b>
<br/>
We consider the case where 

<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;1</span></font></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>w</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;==&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
In this case

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>Y</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mi mathvariant='italic'>Y</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>F</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>W</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>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msubsup><mi mathvariant='italic'>F</mi>
<mn>0</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

This is the same as the result documented in <a href="reverse_one.xml" target="_top"><span style='white-space: nowrap'>reverse_one</span></a>
.


<br/>
<br/>
<b><big><a name="Second Order" id="Second Order">Second Order</a></big></b>
<br/>
We consider the case where 

<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;2</span></font></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>w</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;==&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
In this case

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<msub><mi mathvariant='italic'>Y</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<msub><mi mathvariant='italic'>Y</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<mo stretchy="false">{</mo>
<msub><mi mathvariant='italic'>F</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<msub><mo stretchy="false">}</mo>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<mo stretchy="false">{</mo>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<msub><mo stretchy="false">}</mo>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msubsup><mi mathvariant='italic'>F</mi>
<mn>0</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mo stretchy="false">&#x02202;</mo>
<mrow><mi mathvariant='italic'>u</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<msubsup><mi mathvariant='italic'>F</mi>
<mn>0</mn>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mo stretchy="false">&#x02202;</mo>
<mrow><mi mathvariant='italic'>u</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msubsup><mi mathvariant='italic'>F</mi>
<mn>0</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

where 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">&#x02202;</mo>
<mrow><mi mathvariant='italic'>u</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</mrow></math>

 
denotes partial with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>u</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

.
These are the same as the result documented in <a href="reverse_two.xml" target="_top"><span style='white-space: nowrap'>reverse_two</span></a>
.


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



<ol type="1"><li>
The file <a href="reverse_three.cpp.xml" target="_top"><span style='white-space: nowrap'>reverse_three.cpp</span></a>

contains an example and test of using reverse mode
to compute third order derivatives.
</li><li>

The file <a href="reverse_any.cpp.xml" target="_top"><span style='white-space: nowrap'>reverse_any.cpp</span></a>

contains an example and test of the general reverse mode case.
</li></ol>



<hr/>Input File: omh/reverse/reverse_any.omh

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