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<option>seq_property</option>
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<option>sparse_hessian</option>
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<option>sparse_hessian-></option>
<option>sparse_hessian.cpp</option>
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<option>f</option>
<option>x</option>
<option>w</option>
<option>p</option>
<option>row, col</option>
<option>hes</option>
<option>work</option>
<option>n_sweep</option>
<option>VectorBase</option>
<option>VectorSet</option>
<option>---..Restrictions</option>
<option>VectorSize</option>
<option>Uses Forward</option>
<option>Example</option>
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<center><b><big><big>Sparse Hessian: Easy Driver</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'>hes</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'>.SparseHessian(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>w</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font><i><font color="black"><span style='white-space: nowrap'>hes</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'>.SparseHessian(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>w</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>p</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font><i><font color="black"><span style='white-space: nowrap'>n_sweep</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'>.SparseHessian(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>w</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>p</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>row</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>col</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>work</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<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'>n</mi>
</mrow></math>
for the <a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
size,
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
</mrow></math>
for the <a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>
size of
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
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>
do 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>
.
The syntax above sets
<code><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i></code>
to the Hessian
<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>
<mfrac><mrow><msup><mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>d</mi>
</mstyle></mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>d</mi>
</mstyle></mrow>
<mspace width='.18em'/>
<msup><mrow><mi mathvariant='italic'>x</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<munderover><mo displaystyle='true' largeop='true'>∑</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'>w</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
This routine takes advantage of the sparsity of the Hessian
in order to reduce the amount of computation necessary.
If
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>col</span></font></i></code>
are present, it also takes
advantage of the reduced set of elements of the Hessian that
need to be computed.
One can use speed tests (e.g. <a href="speed_test.xml" target="_top"><span style='white-space: nowrap'>speed_test</span></a>
)
to verify that results are computed faster
than when using the routine <a href="hessian.xml" target="_top"><span style='white-space: nowrap'>Hessian</span></a>
.
<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>
(see <a href="sparse_hessian.xml#Uses Forward" target="_top"><span style='white-space: nowrap'>Uses Forward</span></a>
below).
<br/>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>VectorBase</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="sparse_hessian.xml#VectorBase" target="_top"><span style='white-space: nowrap'>VectorBase</span></a>
below)
and its size
must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
space for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
It specifies
that point at which to evaluate the Hessian.
<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/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>VectorBase</span></font></i><font color="blue"><span style='white-space: nowrap'>& </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>
and size
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
</mrow></math>
.
It specifies the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow></math>
in the expression
for
<code><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i></code>
.
The more components of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>w</mi>
</mrow></math>
that are identically zero,
the more sparse the resulting Hessian may be (and hence the more efficient
the calculation of
<code><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i></code>
may be).
<br/>
<br/>
<b><big><a name="p" id="p">p</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
is optional and 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'>p</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="sparse_hessian.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'>n</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'>n</mi>
</mrow></math>
and all its set elements are between
zero and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</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 Hessian
<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>
.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>If this sparsity pattern does not change between calls to
<code><font color="blue"><span style='white-space: nowrap'>SparseHessian</span></font></code>
, it should be faster to calculate
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
once and
pass this argument to
<code><font color="blue"><span style='white-space: nowrap'>SparseHessian</span></font></code>
.
In addition,
if you specify
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
, CppAD will use the same
type of sparsity representation
(vectors of <code><font color="blue">bool</font></code> or vectors of <code><font color="blue">std::set<size_t></font></code>)
for its internal calculations.
Otherwise, the representation
for the internal calculations is unspecified.
<br/>
<br/>
<b><big><a name="row, col" id="row, col">row, col</a></big></b>
<br/>
The arguments
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>col</span></font></i></code>
are optional and have prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>VectorSize</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>row</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>VectorSize</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>col</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="sparse_hessian.xml#VectorSize" target="_top"><span style='white-space: nowrap'>VectorSize</span></a>
below).
They specify which rows and columns of
<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>
are
returned and in what order.
We use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>K</mi>
</mrow></math>
to denote the value
<code><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()</span></font></code>
which must also equal the size of
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>col</span></font></i></code>
.
Furthermore,
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">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>K</mi>
<mn>-1</mn>
</mrow></math>
, it must hold that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>row</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<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'>col</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false"><</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
.
In addition,
all of the
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>row</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>col</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
</mrow></math>
pairs must correspond to a true value
in the sparsity pattern
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="hes" id="hes">hes</a></big></b>
<br/>
The result
<code><i><font color="black"><span style='white-space: nowrap'>hes</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'>VectorBase</span></font></i><font color="blue"><span style='white-space: nowrap'> </span></font><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
In the case where
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>col</span></font></i></code>
are not present,
the size of
<code><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i></code>
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'>n</mi>
</mrow></math>
and
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'>n</mi>
</mrow></math>
.
In this case, for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">-</mo>
<mn>1</mn>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>ell</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">-</mo>
<mn>1</mn>
</mrow></math>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>hes</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">ℓ</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msup><mi mathvariant='italic'>w</mi>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<mi mathvariant='italic'>F</mi>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>In the case where the arguments
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>col</span></font></i></code>
are present,
we use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>K</mi>
</mrow></math>
to denote the size of
<code><i><font color="black"><span style='white-space: nowrap'>hes</span></font></i></code>
.
The input value of its elements does not matter.
Upon return, 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">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>K</mi>
<mo stretchy="false">-</mo>
<mn>1</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>hes</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msup><mi mathvariant='italic'>w</mi>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<mi mathvariant='italic'>F</mi>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mspace width='.3em'/>
<mo stretchy="false">,</mo>
<mspace width='.3em'/>
<mspace width='.3em'/>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>where</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>row</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<mspace width='.3em'/>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>and</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mo stretchy="false">ℓ</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>col</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
</mrow></math>
<br/>
<b><big><a name="work" id="work">work</a></big></b>
<br/>
If this argument is present, it has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     sparse_hessian_work& </span></font><i><font color="black"><span style='white-space: nowrap'>work</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
This object can only be used with the routines <code><font color="blue">SparseHessian</font></code>.
During its the first use, information is stored in
<code><i><font color="black"><span style='white-space: nowrap'>work</span></font></i></code>
.
This is used to reduce the work done by future calls to <code><font color="blue">SparseHessian</font></code>
with the same
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
,
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i></code>
,
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
, and
<code><i><font color="black"><span style='white-space: nowrap'>col</span></font></i></code>
.
If a future call is make where any of these values have changed,
you must first call
<code><i><font color="black"><span style='white-space: nowrap'>work</span></font></i><font color="blue"><span style='white-space: nowrap'>.clear()</span></font></code>
to inform CppAD that this information needs to be recomputed.
<br/>
<br/>
<b><big><a name="n_sweep" id="n_sweep">n_sweep</a></big></b>
<br/>
The return value
<code><i><font color="black"><span style='white-space: nowrap'>n_sweep</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'>n_sweep</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It is the number of first order forward sweeps
used to compute the requested Hessian values.
Each first forward sweep is followed by a second order reverse sweep
so it is also the number of reverse sweeps.
This is proportional to the total work that <code><font color="blue">SparseHessian</font></code> does,
not counting the zero order forward sweep,
or the work to combine multiple columns into a single
forward-reverse sweep pair.
<br/>
<br/>
<b><big><a name="VectorBase" id="VectorBase">VectorBase</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>VectorBase</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><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="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 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><a name="VectorSet.Restrictions" id="VectorSet.Restrictions">Restrictions</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>VectorSet</span></font></i></code>
has elements of <code><font color="blue">std::set<size_t></font></code>,
then
<code><i><font color="black"><span style='white-space: nowrap'>p</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
must return a reference (not a copy) to the
corresponding set.
According to section 26.3.2.3 of the 1998 C++ standard,
<code><font color="blue">std::valarray< std::set<size_t> ></font></code> does not satisfy
this condition.
<br/>
<br/>
<b><big><a name="VectorSize" id="VectorSize">VectorSize</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>VectorSize</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">size_t</font></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="Uses Forward" id="Uses Forward">Uses Forward</a></big></b>
<br/>
After each call to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
,
the object
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
contains the corresponding
<a href="glossary.xml#Taylor Coefficient" target="_top"><span style='white-space: nowrap'>Taylor coefficients</span></a>
.
After a call to any of the sparse Hessian routines,
the zero order Taylor coefficients correspond to
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(0, </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
and the other coefficients are unspecified.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The routine
<a href="sparse_hessian.cpp.xml" target="_top"><span style='white-space: nowrap'>sparse_hessian.cpp</span></a>
is examples and tests of <code><font color="blue">sparse_hessian</font></code>.
It return <code><font color="blue">true</font></code>, if it succeeds and <code><font color="blue">false</font></code> otherwise.
<hr/>Input File: cppad/local/sparse_hessian.hpp
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