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<center><b><big><big>Computing Jacobian and Hessian of Bender's Reduced Objective</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<br/>
<code><font color="blue"><span style='white-space: nowrap'><br/>
# include <cppad/cppad.hpp><br/>
BenderQuad(</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'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>g</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>gx</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>gxx</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
<br/>
<br/>
<b><big><a name="See Also" id="See Also">See Also</a></big></b>
<br/>
<a href="opt_val_hes.xml" target="_top"><span style='white-space: nowrap'>opt_val_hes</span></a>
<br/>
<br/>
<b><big><a name="Problem" id="Problem">Problem</a></big></b>
<br/>
The type <a href="benderquad.xml#ADvector" target="_top"><span style='white-space: nowrap'>ADvector</span></a>
cannot be determined
form the arguments above
(currently the type
<code><i><font color="black"><span style='white-space: nowrap'>ADvector</span></font></i></code>
must be
<code><font color="blue"><span style='white-space: nowrap'>CPPAD_TESTVECTOR(</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></code>
.)
This will be corrected in the future by requiring
<code><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i></code>
to define
<code><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i><font color="blue"><span style='white-space: nowrap'>::vector_type</span></font></code>
which will specify the
type
<code><i><font color="black"><span style='white-space: nowrap'>ADvector</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
We are given the optimization problem
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>minimize</mi>
</mstyle></mrow>
</mtd><mtd columnalign="center" >
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>w</mi>
<mo stretchy="false">.</mo>
<mi mathvariant='normal'>r</mi>
<mo stretchy="false">.</mo>
<mi mathvariant='normal'>t</mi>
<mo stretchy="false">.</mo>
</mstyle></mrow>
<mspace width='.3em'/>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<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>
</mtd></mtr></mtable>
</mrow></math>
that is convex with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
</mrow></math>
.
In addition, we are given a set of equations
<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>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>
such that
<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>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mspace width='.3em'/>
<mspace width='.3em'/>
<mo stretchy="false">⇒</mo>
<mspace width='.3em'/>
<mspace width='.3em'/>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>y</mi>
</msub>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
(In fact, it is often the case that
<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>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>y</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>
.)
Furthermore, it is easy to calculate a Newton step for these equations; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>dy</mi>
<mo stretchy="false">=</mo>
<mo stretchy="false">-</mo>
<mo stretchy="false">[</mo>
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>y</mi>
</msub>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<msup><mo stretchy="false">]</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>
The purpose of this routine is to compute the
value, Jacobian, and Hessian of the reduced objective function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<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'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
Note that if only the value and Jacobian are needed, they can be
computed more quickly using the relations
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>G</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>
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The <code><font color="blue">BenderQuad</font></code> 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'>BAvector</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="benderquad.xml#BAvector" target="_top"><span style='white-space: nowrap'>BAvector</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>
.
It specifies the point at which we evaluating
the reduced objective function and its derivatives.
<br/>
<br/>
<b><big><a name="y" id="y">y</a></big></b>
<br/>
The <code><font color="blue">BenderQuad</font></code> argument
<code><i><font color="black"><span style='white-space: nowrap'>y</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'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
It must be equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
; i.e.,
it must solve the problem in
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
</mrow></math>
for this given value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>minimize</mi>
</mstyle></mrow>
</mtd><mtd columnalign="center" >
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>w</mi>
<mo stretchy="false">.</mo>
<mi mathvariant='normal'>r</mi>
<mo stretchy="false">.</mo>
<mi mathvariant='normal'>t</mi>
<mo stretchy="false">.</mo>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>m</mi>
</msup>
</mtd></mtr></mtable>
</mrow></math>
<br/>
<b><big><a name="fun" id="fun">fun</a></big></b>
<br/>
The <code><font color="blue">BenderQuad</font></code> object
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i></code>
must support the member functions listed below.
The
<code><font color="blue"><span style='white-space: nowrap'>AD<</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></code>
arguments will be variables for
a tape created by a call to <a href="independent.xml" target="_top"><span style='white-space: nowrap'>Independent</span></a>
from <code><font color="blue">BenderQuad</font></code>
(hence they can not be combined with variables corresponding to a
different tape).
<br/>
<br/>
<b><a name="fun.fun.f" id="fun.fun.f">fun.f</a></b>
<br/>
The <code><font color="blue">BenderQuad</font></code> argument
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i></code>
supports the syntax
<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'> = </span></font><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.f(</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'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.f</span></font></code>
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'>ADvector</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="benderquad.xml#ADvector" target="_top"><span style='white-space: nowrap'>ADvector</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
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.f</span></font></code>
argument
<code><i><font color="black"><span style='white-space: nowrap'>y</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'>ADvector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.f</span></font></code>
result
<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/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>ADvector</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>
and its size must be equal to one.
The value of
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>f</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'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>
.
<br/>
<br/>
<b><a name="fun.fun.h" id="fun.fun.h">fun.h</a></b>
<br/>
The <code><font color="blue">BenderQuad</font></code> argument
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i></code>
supports the syntax
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><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'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.h(</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'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.h</span></font></code>
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'>ADvector</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>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.h</span></font></code>
argument
<code><i><font color="black"><span style='white-space: nowrap'>y</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'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.h</span></font></code>
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'>ADvector</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>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
The value of
<code><i><font color="black"><span style='white-space: nowrap'>h</span></font></i></code>
is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>
.
<br/>
<br/>
<b><a name="fun.fun.dy" id="fun.fun.dy">fun.dy</a></b>
<br/>
The <code><font color="blue">BenderQuad</font></code> argument
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i></code>
supports the syntax
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i><font color="blue"><span style='white-space: nowrap'> = </span></font><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.dy(</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'>y</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/>
<br/>
</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>
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.dy</span></font></code>
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'>BAvector</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>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
Its value will be exactly equal to the <code><font color="blue">BenderQuad</font></code> argument
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
and values depending on it can be stored as private objects
in
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
and need not be recalculated.
<code><font color="blue"><span style='white-space: nowrap'><br/>
<br/>
</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.dy</span></font></code>
argument
<code><i><font color="black"><span style='white-space: nowrap'>y</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'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
Its value will be exactly equal to the <code><font color="blue">BenderQuad</font></code> argument
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
and values depending on it can be stored as private objects
in
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
and need not be recalculated.
<code><font color="blue"><span style='white-space: nowrap'><br/>
<br/>
</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>
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.dy</span></font></code>
argument
<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/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>ADvector</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>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
<code><font color="blue"><span style='white-space: nowrap'><br/>
<br/>
</span></font><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>.dy</span></font></code>
result
<code><i><font color="black"><span style='white-space: nowrap'>dy</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'>ADvector</span></font></i><font color="blue"><span style='white-space: nowrap'> </span></font><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
The return value
<code><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i></code>
is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>dy</mi>
<mo stretchy="false">=</mo>
<mo stretchy="false">-</mo>
<mo stretchy="false">[</mo>
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>y</mi>
</msub>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<msup><mo stretchy="false">]</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
<mi mathvariant='italic'>h</mi>
</mrow></math>
Note that if
<code><i><font color="black"><span style='white-space: nowrap'>h</span></font></i></code>
is equal to
<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>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>dy</mi>
</mrow></math>
is the Newton step for finding a zero
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>
<mi mathvariant='italic'>y</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'>y</mi>
</mrow></math>
;
i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>dy</mi>
</mrow></math>
is an approximate solution for the equation
<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>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>dy</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="g" id="g">g</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>g</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'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>g</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and has size one.
The input value of its element does not matter.
On output,
it contains the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<b><big><a name="gx" id="gx">gx</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>gx</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'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>gx</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and has size
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>
.
The input values of its elements do not matter.
On output,
it contains the Jacobian of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
; i.e.,
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>
<mn>-1</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>gx</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>G</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msub><mo stretchy="false">)</mo>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
<br/>
<b><big><a name="gxx" id="gxx">gxx</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>gx</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'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>gxx</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and has size
<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>
.
The input values of its elements do not matter.
On output,
it contains the Hessian of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
; i.e.,
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>
<mn>-1</mn>
</mrow></math>
, and
<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>
<mn>-1</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>gxx</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>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>G</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msub><mo stretchy="false">)</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</msub>
</mrow></math>
<br/>
<b><big><a name="BAvector" id="BAvector">BAvector</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>BAvector</span></font></i></code>
must be a
<a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
class.
We use
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
to refer to the type of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>BAvector</span></font></i></code>
; i.e.,
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'>::value_type<br/>
</span></font></code>
<br/>
<b><big><a name="ADvector" id="ADvector">ADvector</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>ADvector</span></font></i></code>
must be a
<a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
class with elements of type
<code><font color="blue"><span style='white-space: nowrap'>AD<</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></code>
; i.e.,
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>ADvector</span></font></i><font color="blue"><span style='white-space: nowrap'>::value_type<br/>
</span></font></code>
must be the same type as
<code><font color="blue"><span style='white-space: nowrap'><br/>
     AD< </span></font><i><font color="black"><span style='white-space: nowrap'>BAvector</span></font></i><font color="blue"><span style='white-space: nowrap'>::value_type ><br/>
</span></font></code>
.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="bender_quad.cpp.xml" target="_top"><span style='white-space: nowrap'>bender_quad.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/bender_quad.hpp
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