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<center><b><big><big>Atomic Reverse Hessian Sparsity Patterns</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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>.rev_sparse_hes(</span></font><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</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'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</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 function is used by <a href="revsparsehes.xml" target="_top"><span style='white-space: nowrap'>RevSparseHes</span></a>
 to compute
Hessian sparsity patterns.
There is an unspecified scalar valued function 

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

.
Given a <a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity&#xA0;pattern</span></a>
 for 

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

,
and information about the function 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>

,
this routine computes the sparsity pattern for

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>V</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mrow></math>

<br/>
<b><big><a name="Implementation" id="Implementation">Implementation</a></big></b>
<br/>
If you are using and <a href="revsparsehes.xml" target="_top"><span style='white-space: nowrap'>RevSparseHes</span></a>
,
this virtual function must be defined by the
<a href="atomic_ctor.xml#atomic_user" target="_top"><span style='white-space: nowrap'>atomic_user</span></a>
 class.

<br/>
<br/>
<b><a name="Implementation.vx" id="Implementation.vx">vx</a></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>vx</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;bool&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>

<code><i><font color="black"><span style='white-space: nowrap'>vx</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'>n</span></font></i></code>
, and
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>

,

<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 is true if and only if

<code><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 is a <a href="glossary.xml#Variable" target="_top"><span style='white-space: nowrap'>variable</span></a>
 
in the corresponding call to 

<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'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><a name="Implementation.s" id="Implementation.s">s</a></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>s</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;bool&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size is 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
. 
It is a sparsity pattern for 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mn>1</mn>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>m</mi>
</mrow>
</msup>
</mrow></math>

.

<br/>
<br/>
<b><a name="Implementation.t" id="Implementation.t">t</a></b>
<br/>
This argument has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD:vector&lt;bool&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size is 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
The input values of its elements 
are not specified (must not matter).
Upon return, 
<code><i><font color="black"><span style='white-space: nowrap'>t</span></font></i></code>
 is a 
sparsity pattern for 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mn>1</mn>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>

 where

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<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>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<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><a name="Implementation.q" id="Implementation.q">q</a></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>
It specifies the number of columns in 

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

,

<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>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>

, and

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

. 

<br/>
<br/>
<b><a name="Implementation.r" id="Implementation.r">r</a></b>
<br/>
This argument 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'>atomic_sparsity</span></font></i><font color="blue"><span style='white-space: nowrap'>&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and is a <a href="atomic_option.xml#atomic_sparsity" target="_top"><span style='white-space: nowrap'>atomic_sparsity</span></a>
 pattern for 

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

.

<br/>
<br/>
<b><big><a name="u" id="u">u</a></big></b>
<br/>
This argument 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'>atomic_sparsity</span></font></i><font color="blue"><span style='white-space: nowrap'>&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and is a <a href="atomic_option.xml#atomic_sparsity" target="_top"><span style='white-space: nowrap'>atomic_sparsity</span></a>
 pattern for 

<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>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>

 which is defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>U</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><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">{</mo>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>y</mi>
</msub>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>y</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>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">]</mo>
<msub><mo stretchy="false">}</mo>
<mrow><mi mathvariant='italic'>u</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">{</mo>
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>y</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>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">]</mo>
<msub><mo stretchy="false">}</mo>
<mrow><mi mathvariant='italic'>u</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</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>
<mi mathvariant='italic'>R</mi>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="u.v" id="u.v">v</a></b>
<br/>
This argument 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'>atomic_sparsity</span></font></i><font color="blue"><span style='white-space: nowrap'>&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The input value of its elements 
are not specified (must not matter).
Upon return, 
<code><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>
 is a 
<a href="atomic_option.xml#atomic_sparsity" target="_top"><span style='white-space: nowrap'>atomic_sparsity</span></a>
 pattern for 

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

 which is defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>V</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><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">[</mo>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<msub><mo stretchy="false">]</mo>
<mrow><mi mathvariant='italic'>u</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">[</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<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>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<msub><mo stretchy="false">]</mo>
<mrow><mi mathvariant='italic'>u</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<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>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</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>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">+</mo>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant='italic'>m</mi>
</munderover>
<msubsup><mi mathvariant='italic'>g</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<mspace width='.3em'/>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<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>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<mi mathvariant='italic'>U</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant='italic'>m</mi>
</munderover>
<msub><mi mathvariant='italic'>S</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mspace width='.3em'/>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mtd></mtr></mtable>
</mrow></math>

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


<br/>
<br/>
<b><a name="Examples.Define rev_sparse_hes" id="Examples.Define rev_sparse_hes">Define rev_sparse_hes</a></b>
<br/>
The following files contain example atomic <code><font color="blue">rev_sparse_hes</font></code> functions:
<a href="atomic_norm_sq.cpp.xml#rev_sparse_hes" target="_top"><span style='white-space: nowrap'>norm_sq.cpp</span></a>
,
<a href="atomic_reciprocal.cpp.xml#rev_sparse_hes" target="_top"><span style='white-space: nowrap'>reciprocal.cpp</span></a>
,
<a href="atomic_tangent.cpp.xml#rev_sparse_hes" target="_top"><span style='white-space: nowrap'>tangent.cpp</span></a>
,
<a href="atomic_matrix_mul.hpp.xml#rev_sparse_hes" target="_top"><span style='white-space: nowrap'>matrix_mul.hpp</span></a>
.

<br/>
<br/>
<b><a name="Examples.Use rev_sparse_hes" id="Examples.Use rev_sparse_hes">Use rev_sparse_hes</a></b>
<br/>
The following are links to user atomic function constructor uses:
<a href="atomic_norm_sq.cpp.xml#Use Atomic Function.rev_sparse_hes" target="_top"><span style='white-space: nowrap'>norm_sq.cpp</span></a>
,
<a href="atomic_reciprocal.cpp.xml#Use Atomic Function.rev_sparse_hes" target="_top"><span style='white-space: nowrap'>reciprocal.cpp</span></a>
,
<a href="atomic_tangent.cpp.xml#Use Atomic Function.rev_sparse_hes" target="_top"><span style='white-space: nowrap'>tangent.cpp</span></a>
,
<a href="atomic_mat_mul.cpp.xml#Use Atomic Function.rev_sparse_hes" target="_top"><span style='white-space: nowrap'>mat_mul.cpp</span></a>
.


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

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