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<center><b><big><big>AD Absolute Value Function</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'>y</span></i><code><font color="blue"><span style='white-space: nowrap'> = abs(</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
Evaluates the absolute value function.
<br/>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The argument <i>x</i> has one of the following prototypes
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const AD<</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'>x</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
     const VecAD<</span></font></code><i><span style='white-space: nowrap'>Base</span></i><code><font color="blue"><span style='white-space: nowrap'>>::reference &</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><br/>
<b><big><a name="y" id="y">y</a></big></b>
<br/>
The result <i>y</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     AD<</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'>y</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code><br/>
<b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b>
<br/>
This is an AD of <i>Base</i>
<a href="glossary.xml#Operation.Atomic" target="_top"><span style='white-space: nowrap'>atomic operation</span></a>
and hence is part of the current
AD of <i>Base</i>
<a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation sequence</span></a>
.
<br/>
<br/>
<b><big><a name="Complex Types" id="Complex Types">Complex Types</a></big></b>
<br/>
The function <code><font color="blue">abs</font></code> is not defined for the AD type sequences
above <code><font color="blue">std::complex<float></font></code> or <code><font color="blue">std::complex<double></font></code>
because the complex <code><font color="blue">abs</font></code> function is not complex differentiable
(see <a href="faq.xml#Complex Types" target="_top"><span style='white-space: nowrap'>complex types faq</span></a>
).
<br/>
<br/>
<b><big><a name="Directional Derivative" id="Directional Derivative">Directional Derivative</a></big></b>
<br/>
The derivative of the absolute value function is one for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">></mo>
<mn>0</mn>
</mrow></math>
and minus one for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false"><</mo>
<mn>0</mn>
</mrow></math>
.
The subtitle issue is
how to compute its directional derivative
what
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>The function corresponding to the argument <i>x</i>
and the result <i>y</i> are represented
by their Taylor coefficients; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
</mtd></mtr></mtable>
</mrow></math>
Note that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>
is the value of <i>x</i> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>
is the value of <i>y</i>.
In the equations above, the order
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>
is specified
by a call to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
or <a href="reverse.xml" target="_top"><span style='white-space: nowrap'>Reverse</span></a>
as follows:
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font></code><i><span style='white-space: nowrap'>p</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>dx</span></i><code><font color="blue"><span style='white-space: nowrap'>)<br/>
     </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><font color="blue"><span style='white-space: nowrap'>.Reverse(</span></font></code><i><span style='white-space: nowrap'>p</span></i><code><font color="blue"><span style='white-space: nowrap'>+1, </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>If all of the Taylor coefficients of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
are zero,
we define
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>p</mi>
</mrow></math>
.
Otherwise, we define
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
</mrow></math>
to be the minimal index such that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">≠</mo>
<mn>0</mn>
</mrow></math>
.
Note that if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">≠</mo>
<mn>0</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
.
The Taylor coefficient representation of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
(and hence it's derivatives) are computed as
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mo stretchy="false">ℓ</mo>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mo stretchy="false">ℓ</mo>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">></mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="left" >
<mn>0</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="left" >
<mo stretchy="false">-</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mo stretchy="false">ℓ</mo>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false"><</mo>
<mn>0</mn>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mrow></math>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="abs.cpp.xml" target="_top"><span style='white-space: nowrap'>Abs.cpp</span></a>
contains an example and test of this function.
It returns true if it succeeds and false otherwise.
<hr/>Input File: cppad/local/abs.hpp
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