<?xml version='1.0'?>
<?xml-stylesheet type='text/xsl' href='pmathml.xsl'?>
<html xmlns='http://www.w3.org/1999/xhtml'>
<head>
<title>Hessian: Easy Driver</title>
<meta name="description" id="description" content="Hessian: Easy Driver"/>
<meta name="keywords" id="keywords" content=" Hessian driver second derivative "/>
<style type='text/css'>
body { color : black }
body { background-color : white }
A:link { color : blue }
A:visited { color : purple }
A:active { color : purple }
</style>
<script type='text/javascript' language='JavaScript' src='_hessian_xml.js'>
</script>
</head>
<body>
<table><tr>
<td>
<a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a>
</td>
<td><a href="revone.cpp.xml" target="_top">Prev</a>
</td><td><a href="hessian.cpp.xml" target="_top">Next</a>
</td><td>
<select onchange='choose_across0(this)'>
<option>Index-></option>
<option>contents</option>
<option>reference</option>
<option>index</option>
<option>search</option>
<option>external</option>
</select>
</td>
<td>
<select onchange='choose_up0(this)'>
<option>Up-></option>
<option>CppAD</option>
<option>ADFun</option>
<option>Drivers</option>
<option>Hessian</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>CppAD-></option>
<option>Install</option>
<option>Introduction</option>
<option>AD</option>
<option>ADFun</option>
<option>library</option>
<option>Example</option>
<option>configure</option>
<option>Appendix</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>ADFun-></option>
<option>Independent</option>
<option>FunConstruct</option>
<option>Dependent</option>
<option>abort_recording</option>
<option>seq_property</option>
<option>FunEval</option>
<option>Drivers</option>
<option>FunCheck</option>
<option>omp_max_thread</option>
<option>optimize</option>
<option>FunDeprecated</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>Drivers-></option>
<option>Jacobian</option>
<option>ForOne</option>
<option>RevOne</option>
<option>Hessian</option>
<option>ForTwo</option>
<option>RevTwo</option>
<option>sparse_jacobian</option>
<option>sparse_hessian</option>
</select>
</td>
<td>
<select onchange='choose_down0(this)'>
<option>Hessian-></option>
<option>Hessian.cpp</option>
<option>HesLagrangian.cpp</option>
</select>
</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-></option>
<option>Syntax</option>
<option>Purpose</option>
<option>f</option>
<option>x</option>
<option>l</option>
<option>w</option>
<option>hes</option>
<option>Vector</option>
<option>Hessian Uses Forward</option>
<option>Example</option>
</select>
</td>
</tr></table><br/>
<center><b><big><big>Hessian: Easy Driver</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<br/>
<code><font color="blue"></font></code><i><span style='white-space: nowrap'>hes</span></i><code><font color="blue"><span style='white-space: nowrap'> = </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><font color="blue"><span style='white-space: nowrap'>.Hessian(</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>w</span></i><code><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code><code><font color="blue"></font></code><i><span style='white-space: nowrap'>hes</span></i><code><font color="blue"><span style='white-space: nowrap'> = </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><font color="blue"><span style='white-space: nowrap'>.Hessian(</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>l</span></i><code><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'>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 <i>f</i>.
The syntax above sets <i>hes</i> to the Hessian
The syntax above sets <i>h</i> to the Hessian
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>hes</mi>
<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>
The routine <a href="sparse_hessian.xml" target="_top"><span style='white-space: nowrap'>sparse_hessian</span></a>
may be faster in the case
where the Hessian is sparse.
<br/>
<br/>
<b><big><a name="f" id="f">f</a></big></b>
<br/>
The object <i>f</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     ADFun<</span></font></code><i><span style='white-space: nowrap'>Base</span></i><code><font color="blue"><span style='white-space: nowrap'>> </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><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 <i>f</i> is not <code><font color="blue">const</font></code>
(see <a href="hessian.xml#Hessian Uses Forward" target="_top"><span style='white-space: nowrap'>Hessian Uses Forward</span></a>
below).
<br/>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The argument <i>x</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>Vector</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>(see <a href="hessian.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below)
and its size
must be equal to <i>n</i>, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
space for <i>f</i>.
It specifies
that point at which to evaluate the Hessian.
<br/>
<br/>
<b><big><a name="l" id="l">l</a></big></b>
<br/>
If the argument <i>l</i> is present, it has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     size_t </span></font></code><i><span style='white-space: nowrap'>l</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>and is less than <i>m</i>, the dimension of the
<a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>
space for <i>f</i>.
It specifies the component of <i>F</i>
for which we are evaluating the Hessian.
To be specific, in the case where the argument <i>l</i> is present,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>w</mi>
<mi mathvariant='italic'>i</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" >
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>l</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>
<br/>
<b><big><a name="w" id="w">w</a></big></b>
<br/>
If the argument <i>w</i> is present, it has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>Vector</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>w</span></i><code><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 <i>h</i>.
<br/>
<br/>
<b><big><a name="hes" id="hes">hes</a></big></b>
<br/>
The result <i>hes</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>Vector</span></i><code><font color="blue"><span style='white-space: nowrap'> </span></font></code><i><span style='white-space: nowrap'>hes</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>(see <a href="hessian.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below)
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>
.
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">…</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>
<mo stretchy="false">ℓ</mo>
<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>
<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
The type <i>Vector</i> 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>
<i>Base</i>.
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="Hessian Uses Forward" id="Hessian Uses Forward">Hessian 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 <i>f</i> contains the corresponding
<a href="glossary.xml#Taylor Coefficient" target="_top"><span style='white-space: nowrap'>Taylor coefficients</span></a>
.
After <code><font color="blue">Hessian</font></code>,
the previous calls to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
are undefined.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The routines
<a href="hessian.cpp.xml" target="_top"><span style='white-space: nowrap'>Hessian.cpp</span></a>
and
<a href="heslagrangian.cpp.xml" target="_top"><span style='white-space: nowrap'>HesLagrangian.cpp</span></a>
are examples and tests of <code><font color="blue">Hessian</font></code>.
They return <code><font color="blue">true</font></code>, if they succeed and <code><font color="blue">false</font></code> otherwise.
<hr/>Input File: cppad/local/hessian.hpp
</body>
</html>
