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<option>RevSparseHes-></option>
<option>RevSparseHes.cpp</option>
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<option>f</option>
<option>x</option>
<option>q</option>
<option>r</option>
<option>s</option>
<option>h</option>
<option>VectorSet</option>
<option>Entire Sparsity Pattern</option>
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<center><b><big><big>Hessian Sparsity Pattern: 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'>h</span></font></i><font color="blue"><span style='white-space: nowrap'> = </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.RevSparseHes(</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>s</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><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">→</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<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 function</span></a>
corresponding to
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
We define the matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>q</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>
as the partial with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>
,
of the partial with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>u</mi>
</mrow></math>
(at
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
),
of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<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'>R</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">]</mo>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>q</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mn>1</mn>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>m</mi>
</mrow>
</msup>
</mrow></math>
; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>R</mi>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">*</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>
</mrow></math>
Given a
<a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for the matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
</mrow></math>
and the vector
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
</mrow></math>
,
<code><font color="blue">RevSparseHes</font></code> returns a sparsity pattern for the
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
.
<br/>
<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/>
     const ADFun<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>> </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>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
the sparsity pattern is valid for all values of the independent
variables in
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
(even if it has <a href="condexp.xml" target="_top"><span style='white-space: nowrap'>CondExp</span></a>
or <a href="vecad.xml" target="_top"><span style='white-space: nowrap'>VecAD</span></a>
operations).
<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/>
     size_t </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">∈</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</mrow></math>
and the number of rows in
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>q</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>
.
It must be the same value as in the previous <a href="forsparsejac.xml" target="_top"><span style='white-space: nowrap'>ForSparseJac</span></a>
call
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.ForSparseJac(</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>, </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>
<br/>
<b><big><a name="r" id="r">r</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i></code>
in the previous call
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.ForSparseJac(</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>, </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>
is a <a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for the matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
</mrow></math>
above.
The type of the elements of
<a href="revsparsehes.xml#VectorSet" target="_top"><span style='white-space: nowrap'>VectorSet</span></a>
must be the
same as the type of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="s" id="s">s</a></big></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/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>VectorSet</span></font></i><font color="blue"><span style='white-space: nowrap'>& </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>
(see <a href="revsparsehes.xml#VectorSet" target="_top"><span style='white-space: nowrap'>VectorSet</span></a>
below)
If it has elements of type <code><font color="blue">bool</font></code>,
its size is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
</mrow></math>
.
If it has elements of type <code><font color="blue">std::set<size_t></font></code>,
its size is one and all the elements of
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>[0]</span></font></code>
are between zero and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">-</mo>
<mn>1</mn>
</mrow></math>
.
It specifies a
<a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for the vector
<code><i><font color="black"><span style='white-space: nowrap'>S</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="h" id="h">h</a></big></b>
<br/>
The result
<code><i><font color="black"><span style='white-space: nowrap'>h</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>VectorSet</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>h</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="revsparsehes.xml#VectorSet" target="_top"><span style='white-space: nowrap'>VectorSet</span></a>
below).
If it has elements of type <code><font color="blue">bool</font></code>,
its size is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
.
If it has elements of type <code><font color="blue">std::set<size_t></font></code>,
its size is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
</mrow></math>
.
It specifies a
<a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for the matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="VectorSet" id="VectorSet">VectorSet</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>VectorSet</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 of type</span></a>
<code><font color="blue">bool</font></code> or <code><font color="blue">std::set<size_t></font></code>;
see <a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for a discussion
of the difference.
The type of the elements of
<a href="revsparsehes.xml#VectorSet" target="_top"><span style='white-space: nowrap'>VectorSet</span></a>
must be the
same as the type of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="Entire Sparsity Pattern" id="Entire Sparsity Pattern">Entire Sparsity Pattern</a></big></b>
<br/>
Suppose that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>
is the
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
identity matrix.
Further suppose that the
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
</mrow></math>
is the <i>k</i>-th
<a href="glossary.xml#Elementary Vector" target="_top"><span style='white-space: nowrap'>elementary vector</span></a>
; i.e.
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>S</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mn>1</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>k</mi>
</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>
In this case,
the corresponding value
<code><i><font color="black"><span style='white-space: nowrap'>h</span></font></i></code>
is a
sparsity pattern for the Hessian matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>k</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>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="revsparsehes.cpp.xml" target="_top"><span style='white-space: nowrap'>RevSparseHes.cpp</span></a>
contains an example and test of this operation.
It returns true if it succeeds and false otherwise.
<hr/>Input File: cppad/local/rev_sparse_hes.hpp
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