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<option>col</option>
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<center><b><big><big>Speed Testing Sparse Hessian</big></big></b></center>
<br/>
<b><big><a name="Prototype" id="Prototype">Prototype</a></big></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>extern&#xA0;bool&#xA0;link_sparse_hessian(<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>size</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>repeat</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;double&gt;&amp;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&lt;size_t&gt;&amp;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>row</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&lt;size_t&gt;&amp;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>col</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;,&#xA0;<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;double&gt;&amp;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>hessian</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
);<br/>
</span></font></code>
<br/>
<b><big><a name="Method" id="Method">Method</a></big></b>
<br/>
Given a row index vector 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>row</mi>
</mrow></math>


and a second column vector 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>col</mi>
</mrow></math>

,
the corresponding function 

<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">&#x02192;</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
</mrow></math>

 
is defined by <a href="sparse_hes_fun.xml" target="_top"><span style='white-space: nowrap'>sparse_hes_fun</span></a>
.
The non-zero entries in the Hessian of this function have 
one of the following forms:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mn>2</mn>
</msup>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
</mrow>
</mfrac>
<mspace width='.3em'/>
<mo stretchy="false">,</mo>
<mspace width='.3em'/>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mn>2</mn>
</msup>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
</mfrac>
<mspace width='.3em'/>
<mo stretchy="false">,</mo>
<mspace width='.3em'/>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mn>2</mn>
</msup>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
</mrow>
</mfrac>
<mspace width='.3em'/>
<mo stretchy="false">,</mo>
<mspace width='.3em'/>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mn>2</mn>
</msup>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<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>
</mfrac>
</mrow></math>

for some 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
</mrow></math>

 between zero and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>K</mi>
<mn>-1</mn>
</mrow></math>

.
All the other terms of the Hessian are zero.

<br/>
<br/>
<b><big><a name="size" id="size">size</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>size</span></font></i></code>
, referred to as 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>

 below,
is the dimension of the domain space for 
<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>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="repeat" id="repeat">repeat</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>repeat</span></font></i></code>
 is the number of times
to repeat the test
(with a different value for 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 corresponding to
each repetition).

<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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;double&gt;&amp;&#xA0;</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>
and its size is 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>

; i.e., 
<code><i><font color="black"><span style='white-space: nowrap'>x</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'>size</span></font></i></code>
.
The input value of the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 does not matter.
On output, it has been set to the
argument value for which the function,
or its derivative, is being evaluated.
The value of this vector need not change with each repetition.

<br/>
<br/>
<b><big><a name="row" id="row">row</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>row</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;size_t&gt;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>row</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
Its size defines the value 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>K</mi>
</mrow></math>

.
It contains the row indices for the corresponding function 
<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>
</mrow></math>

.
All the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
 are between zero and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="col" id="col">col</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>col</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;size_t&gt;&#xA0;</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>
Its size must be the same as 
<code><i><font color="black"><span style='white-space: nowrap'>row</span></font></i></code>
; i.e., 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>K</mi>
</mrow></math>

.
It contains the column indices for the corresponding function 

<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>
</mrow></math>

.
All the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>col</span></font></i></code>
 are between zero and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>There are no duplicate row and column entires; i.e., if 
<code><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;!=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
,

<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'>row</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'>]&#xA0;!=&#xA0;</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'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;||&#xA0;</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'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;!=&#xA0;</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'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>]<br/>
</span></font></code>
Only the lower triangle of the Hessian is included in the indices; i.e.,

<code><font color="blue"><span style='white-space: nowrap'>row</span></font><i><font color="black"><span style='white-space: nowrap'>[</span></font></i><font color="blue"><span style='white-space: nowrap'>k</span></font><i><font color="black"><span style='white-space: nowrap'>]&#xA0;&gt;=&#xA0;</span></font></i><font color="blue"><span style='white-space: nowrap'>col</span></font><i><font color="black"><span style='white-space: nowrap'>[</span></font></i><font color="blue"><span style='white-space: nowrap'>k</span></font><i><font color="black"><span style='white-space: nowrap'>]</span></font></i></code>
.
Furthermore, for all the non-zero entries in the lower triangle
are included; i.e., if 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02265;</mo>
<mi mathvariant='italic'>j</mi>
</mrow></math>

 and

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mn>2</mn>
</msup>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow>
</mfrac>
<mo stretchy="false">&#x02260;</mo>
<mn>0</mn>
</mrow></math>

,
there is an index 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
 such that

<code><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</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'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 and

<code><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</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'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
.

<br/>
<br/>
<b><big><a name="hessian" id="hessian">hessian</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>hessian</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;double&gt;&amp;&#xA0;&#xA0;hessian<br/>
</span></font></code>
and its size is 
<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'>n</mi>
</mrow></math>

.
The input value of its elements does not matter. 
The output value of its elements is the Hessian of the function 
<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>
</mrow></math>

.
To be more specific, 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">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,

<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>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><msup><mo stretchy="false">&#x02202;</mo>
<mn>2</mn>
</msup>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>hessian</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow></math>

<br/>
<b><a name="hessian.double" id="hessian.double">double</a></b>
<br/>
In the case where 
<code><i><font color="black"><span style='white-space: nowrap'>package</span></font></i></code>
 is <code><font color="blue">double</font></code>,
only the first element of 
<code><i><font color="black"><span style='white-space: nowrap'>hessian</span></font></i></code>
 is used and it is actually 
the value 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>
</mrow></math>

 (derivatives are not computed).


<hr/>Input File: speed/src/link_sparse_hessian.cpp

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