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<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
      xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>Sign Function</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="Sign Function"/>
<meta name="keywords" id="keywords" content=" sign Ad function syntax purpose x y operation sequence complex types derivative example "/>
<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='_sign_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="abs.cpp.xml" target="_top">Prev</a>
</td><td><a href="sign.cpp.xml" target="_top">Next</a>
</td><td>
<select onchange='choose_across0(this)'>
<option>Index-&gt;</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-&gt;</option>
<option>CppAD</option>
<option>AD</option>
<option>ADValued</option>
<option>MathOther</option>
<option>sign</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>AD-&gt;</option>
<option>ad_ctor</option>
<option>ad_assign</option>
<option>Convert</option>
<option>ADValued</option>
<option>BoolValued</option>
<option>VecAD</option>
<option>base_require</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>ADValued-&gt;</option>
<option>Arithmetic</option>
<option>std_math_ad</option>
<option>MathOther</option>
<option>CondExp</option>
<option>Discrete</option>
<option>atomic</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>MathOther-&gt;</option>
<option>abs</option>
<option>sign</option>
<option>atan2</option>
<option>erf</option>
<option>pow</option>
<option>limits</option>
</select>
</td>
<td>
<select onchange='choose_down0(this)'>
<option>sign-&gt;</option>
<option>sign.cpp</option>
</select>
</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-&gt;</option>
<option>Syntax</option>
<option>Purpose</option>
<option>x</option>
<option>y</option>
<option>Operation Sequence</option>
<option>Complex Types</option>
<option>Derivative</option>
<option>Example</option>
</select>
</td>
</tr></table><br/>





<center><b><big><big>Sign Function</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'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;sign(</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>



<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
Evaluates the <code><font color="blue">sign</font></code> function which is defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>sign</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mo stretchy="false">+</mo>
<mn>1</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">&gt;</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mn>0</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mn>-1</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">&lt;</mo>
<mn>0</mn>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mrow></math>

<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 one of the following prototypes

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;VecAD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;::reference&#xA0;&amp;</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>
<br/>
<b><big><a name="y" id="y">y</a></big></b>
<br/>
The result 
<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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;</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>
<br/>
<b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b>
<br/>
This is an AD of 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>

<a href="glossary.xml#Operation.Atomic" target="_top"><span style='white-space: nowrap'>atomic&#xA0;operation</span></a>

and hence is part of the current
AD of 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>

<a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation&#xA0;sequence</span></a>
.

<br/>
<br/>
<b><big><a name="Complex Types" id="Complex Types">Complex Types</a></big></b>
<br/>
The <code><font color="blue">sign</font></code> function is not defined for the AD type sequences
above <code><font color="blue">std::complex&lt;float&gt;</font></code> or <code><font color="blue">std::complex&lt;double&gt;</font></code>
because these are not <a href="base_ordered.xml#Ordered Type" target="_top"><span style='white-space: nowrap'>ordered&#xA0;types</span></a>
. 

<br/>
<br/>
<b><big><a name="Derivative" id="Derivative">Derivative</a></big></b>
<br/>
CppAD computes the derivative of the <code><font color="blue">sign</font></code> function as zero for all
argument values 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.
The correct mathematical derivative is different and
is given by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>sign</mi>
</mstyle></mrow>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>2</mn>
<mi mathvariant='normal'>&#x003B4;</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

where 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B4;</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 is the Dirac Delta function.

<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>

<br/>
The file
<a href="sign.cpp.xml" target="_top"><span style='white-space: nowrap'>sign.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/sign.hpp

</body>
</html>