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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>Exponential Function Reverse Mode Theory</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="Exponential Function Reverse Mode Theory"/>
<meta name="keywords" id="keywords" content=" exp reverse theory exponential function mode "/>
<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='_expreverse_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="reversetheory.xml" target="_top">Prev</a>
</td><td><a href="logreverse.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>
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<option>ReverseTheory</option>
<option>ExpReverse</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>Appendix-&gt;</option>
<option>Faq</option>
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</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>Theory-&gt;</option>
<option>ForwardTheory</option>
<option>ReverseTheory</option>
<option>reverse_identity</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>ReverseTheory-&gt;</option>
<option>ExpReverse</option>
<option>LogReverse</option>
<option>SqrtReverse</option>
<option>SinCosReverse</option>
<option>AtanReverse</option>
<option>AsinReverse</option>
<option>AcosReverse</option>
<option>tan_reverse</option>
<option>erf_reverse</option>
</select>
</td>
<td>ExpReverse</td>
<td>Headings</td>
</tr></table><br/>
<center><b><big><big>Exponential Function Reverse Mode Theory</big></big></b></center>
We use the reverse theory
<a href="reversetheory.xml#Standard Math Functions" target="_top"><span style='white-space: nowrap'>standard&#xA0;math&#xA0;function</span></a>

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

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

.

The forward mode formulas for the
<a href="expforward.xml" target="_top"><span style='white-space: nowrap'>exponential</span></a>

function are

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mi>exp</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow></math>

if 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>

, and

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</mfrac>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</munderover>
<mi mathvariant='italic'>k</mi>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

for the case 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>

, and for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&gt;</mo>
<mn>0</mn>
</mrow></math>

,


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

If 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&gt;</mo>
<mn>0</mn>
</mrow></math>

, then for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow></math>




<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</mfrac>
<mi mathvariant='italic'>k</mi>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</mfrac>
<mi mathvariant='italic'>k</mi>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>


<hr/>Input File: omh/theory/exp_reverse.omh

</body>
</html>