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<center><b><big><big>Jacobian Sparsity Pattern: Forward 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'>s</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'>.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>
<code><i><font color="black"><span style='white-space: nowrap'>s</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'>.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'>, </span></font><i><font color="black"><span style='white-space: nowrap'>transpose</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">→</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 function</span></a>
corresponding to
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
For a fixed
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>q</mi>
</mrow></math>
matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
</mrow></math>
,
the Jacobian 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>
<mi mathvariant='italic'>R</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'>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>
is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><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'>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>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>R</mi>
</mrow></math>
Given a
<a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
</mrow></math>
,
<code><font color="blue">ForSparseJac</font></code> returns a sparsity pattern for the
<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>
</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/>
     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>
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>.
After a call to <code><font color="blue">ForSparseJac</font></code>, the sparsity pattern
for each of the variables in the operation sequence
is held in
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
(for possible later use by <a href="revsparsehes.xml" target="_top"><span style='white-space: nowrap'>RevSparseHes</span></a>
).
These sparsity patterns are stored with elements of type <code><font color="blue">bool</font></code>
or elements of type <code><font color="blue">std::set<size_t></font></code>
(see <a href="forsparsejac.xml#VectorSet" target="_top"><span style='white-space: nowrap'>VectorSet</span></a>
below).
<br/>
<br/>
<b><a name="f.size_forward_bool" id="f.size_forward_bool">size_forward_bool</a></b>
<br/>
After <code><font color="blue">ForSparseJac</font></code>, if
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
is a <code><font color="blue">size_t</font></code> object,
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>k</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'>.size_forward_bool()<br/>
</span></font></code>
sets
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
to the amount of memory (in unsigned character units)
used to store the sparsity pattern with elements of type <code><font color="blue">bool</font></code>
in the function object
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
If the sparsity patterns for the previous <code><font color="blue">ForSparseJac</font></code> used
elements of type <code><font color="blue">bool</font></code>,
the return value for <code><font color="blue">size_forward_bool</font></code> will be non-zero.
Otherwise, its return value will be zero.
This sparsity pattern is stored for use by <a href="revsparsehes.xml" target="_top"><span style='white-space: nowrap'>RevSparseHes</span></a>
and
when it is not longer needed, it can be deleted
(and the corresponding memory freed) using
<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'>.size_forward_bool(0)<br/>
</span></font></code>
After this call,
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_forward_bool()</span></font></code>
will return zero.
<br/>
<br/>
<b><a name="f.size_forward_set" id="f.size_forward_set">size_forward_set</a></b>
<br/>
After <code><font color="blue">ForSparseJac</font></code>, if
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
is a <code><font color="blue">size_t</font></code> object,
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>k</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'>.size_forward_set()<br/>
</span></font></code>
sets
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
to the total number of elements in all the sets corresponding
to the sparsity pattern stored in the function object
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
If the sparsity patterns for this operation use elements of type <code><font color="blue">bool</font></code>,
the return value for <code><font color="blue">size_forward_set</font></code> will be zero.
Otherwise, its return value will be non-zero
(unless the entire sparsity pattern is false).
This sparsity pattern is stored for use by <a href="revsparsehes.xml" target="_top"><span style='white-space: nowrap'>RevSparseHes</span></a>
and
when it is not longer needed, it can be deleted
(and the corresponding memory freed) using
<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'>.size_forward_set(0)<br/>
</span></font></code>
After this call,
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_forward_set()</span></font></code>
will return zero.
<br/>
<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><mi mathvariant='italic'>B</mi>
<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>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>
and the Jacobian
<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'>B</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="transpose" id="transpose">transpose</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>transpose</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     bool </span></font><i><font color="black"><span style='white-space: nowrap'>transpose</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The default value <code><font color="blue">false</font></code> is used when
<code><i><font color="black"><span style='white-space: nowrap'>transpose</span></font></i></code>
is not present.
<br/>
<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>
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'>r</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
see <a href="forsparsejac.xml#VectorSet" target="_top"><span style='white-space: nowrap'>VectorSet</span></a>
below.
<br/>
<br/>
<b><a name="r.transpose false" id="r.transpose false">transpose false</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i></code>
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'>n</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>q</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'>n</mi>
</mrow></math>
and all the set elements must be between
zero and
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
inclusive.
It specifies a
<a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for the matrix
<code><i><font color="black"><span style='white-space: nowrap'>R in B^{n times q}</span></font></i></code>
.
<br/>
<br/>
<b><a name="r.transpose true" id="r.transpose true">transpose true</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i></code>
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>
and all the set elements must be between
zero and
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
inclusive.
It specifies a
<a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity pattern</span></a>
for the matrix
<code><i><font color="black"><span style='white-space: nowrap'>R^R{T} in B^{q times n}</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="s" id="s">s</a></big></b>
<br/>
The return value
<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/>
     </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="forsparsejac.xml#VectorSet" target="_top"><span style='white-space: nowrap'>VectorSet</span></a>
below.
<br/>
<br/>
<b><a name="s.transpose false" id="s.transpose false">transpose false</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
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>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>q</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'>m</mi>
</mrow></math>
and all its set elements are between
zero and
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
inclusive.
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'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>
.
<br/>
<br/>
<b><a name="s.transpose true" id="s.transpose true">transpose true</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
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'>m</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>
and all its set elements are between
zero and
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
inclusive.
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'>S</mi>
<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>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>q</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>m</mi>
</mrow>
</msup>
</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.
<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>
</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.
In this case,
the corresponding value for
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
is a
sparsity pattern for the Jacobian
<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'>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/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="for_sparse_jac.cpp.xml" target="_top"><span style='white-space: nowrap'>for_sparse_jac.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/for_sparse_jac.hpp
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