<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>exp_eps: Second Order Reverse Sweep</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="exp_eps: Second Order Reverse Sweep"/>
<meta name="keywords" id="keywords" content=" exp_eps: second order reverse sweep exp_eps mode example purpose mathematical form epsilon f_7 index 7: f_6 6: f_5 5: f_4 4: f_3 3: f_2 2: f_1 verification exercises "/>
<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='_exp_eps_rev2_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="exp_eps_for2.cpp.xml" target="_top">Prev</a>
</td><td><a href="exp_eps_rev2.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>Introduction</option>
<option>exp_eps</option>
<option>exp_eps_rev2</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>preprocessor</option>
<option>multi_thread</option>
<option>library</option>
<option>ipopt_solve</option>
<option>Example</option>
<option>speed</option>
<option>Appendix</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>Introduction-></option>
<option>get_started.cpp</option>
<option>exp_2</option>
<option>exp_eps</option>
<option>exp_apx_main.cpp</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>exp_eps-></option>
<option>exp_eps.hpp</option>
<option>exp_eps.cpp</option>
<option>exp_eps_for0</option>
<option>exp_eps_for1</option>
<option>exp_eps_rev1</option>
<option>exp_eps_for2</option>
<option>exp_eps_rev2</option>
<option>exp_eps_cppad</option>
</select>
</td>
<td>
<select onchange='choose_down0(this)'>
<option>exp_eps_rev2-></option>
<option>exp_eps_rev2.cpp</option>
</select>
</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-></option>
<option>Purpose</option>
<option>Mathematical Form</option>
<option>epsilon</option>
<option>f_7</option>
<option>Index 7: f_6</option>
<option>Index 6: f_5</option>
<option>Index 5: f_4</option>
<option>Index 4: f_3</option>
<option>Index 3: f_2</option>
<option>Index 2: f_1</option>
<option>Verification</option>
<option>Exercises</option>
</select>
</td>
</tr></table><br/>
<center><b><big><big>exp_eps: Second Order Reverse Sweep</big></big></b></center>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
In general, a second order reverse sweep is given the
<a href="exp_eps_for1.xml#First Order Expansion" target="_top"><span style='white-space: nowrap'>first order expansion</span></a>
for all of the variables in an operation sequence.
Given a choice of a particular variable,
it computes the derivative,
of that variables first order expansion coefficient,
with respect to all of the independent variables.
<br/>
<br/>
<b><big><a name="Mathematical Form" id="Mathematical Form">Mathematical Form</a></big></b>
<br/>
Suppose that we use the algorithm <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>
to compute
<code><font color="blue"><span style='white-space: nowrap'>exp_eps(</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'>epsilon</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
with
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
is equal to .5
and
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
is equal to .2.
For this case, the mathematical function for the operation sequence
corresponding to <code><font color="blue">exp_eps</font></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'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>ε</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow></math>
The corresponding derivative of the
partial derivative with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>
is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>x</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>ε</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>1</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='normal'>ε</mi>
</msub>
<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='normal'>ε</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>0</mn>
</mtd></mtr></mtable>
</mrow></math>
<br/>
<b><big><a name="epsilon" id="epsilon">epsilon</a></big></b>
<br/>
Since
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>ε</mi>
</mrow></math>
is an independent variable,
it could included as an argument to all of the
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
functions below.
The result would be that all the partials with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>ε</mi>
</mrow></math>
would be zero and hence we drop it to simplify
the presentation.
<br/>
<br/>
<b><big><a name="f_7" id="f_7">f_7</a></big></b>
<br/>
In reverse mode we choose one dependent variable and
compute its derivative with respect to all the independent variables.
For our example, we chose the value returned by <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>
which is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow></math>
.
We begin with the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow></math>
is both an argument and the value of the function; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>1</mn>
</mtd></mtr></mtable>
</mrow></math>
All the other partial derivatives of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow></math>
are zero.
<br/>
<br/>
<b><big><a name="Index 7: f_6" id="Index 7: f_6">Index 7: f_6</a></big></b>
<br/>
The last operation has index 7,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr></mtable>
</mrow></math>
We define the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
as equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow></math>
except that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
are eliminated using
this operation; i.e.
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
<mrow><mo stretchy="true">[</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr></mtable>
</mrow></math>
All the other partial derivatives of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow></math>
are zero.
<br/>
<br/>
<b><big><a name="Index 6: f_5" id="Index 6: f_5">Index 6: f_5</a></big></b>
<br/>
The previous operation has index 6,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mtd></mtr></mtable>
</mrow></math>
We define the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
as equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow></math>
except that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
are eliminated using
this operation; i.e.
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
<mrow><mo stretchy="true">[</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>6</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mtd></mtr></mtable>
</mrow></math>
All the other partial derivatives of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow></math>
are zero.
<br/>
<br/>
<b><big><a name="Index 5: f_4" id="Index 5: f_4">Index 5: f_4</a></big></b>
<br/>
The previous operation has index 5,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">*</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">*</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">*</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr></mtable>
</mrow></math>
We define the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
as equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow></math>
except that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
are eliminated using
this operation; i.e.
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
<mrow><mo stretchy="true">[</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">,</mo>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>
Given the information from the forward sweep, we have
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
,
and the fact that the partial of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow></math>
with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
is zero, we have
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>5</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr></mtable>
</mrow></math>
All the other partial derivatives of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow></math>
are zero.
<br/>
<br/>
<b><big><a name="Index 4: f_3" id="Index 4: f_3">Index 4: f_3</a></big></b>
<br/>
The previous operation has index 4,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr></mtable>
</mrow></math>
We define the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
as equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow></math>
except that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
are eliminated using
this operation; i.e.
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
<mrow><mo stretchy="true">[</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>4</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1.25</mn>
</mtd></mtr></mtable>
</mrow></math>
<br/>
<b><big><a name="Index 3: f_2" id="Index 3: f_2">Index 3: f_2</a></big></b>
<br/>
The previous operation has index 3,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">/</mo>
<mn>1</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">/</mo>
<mn>1</mn>
</mtd></mtr></mtable>
</mrow></math>
We define the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
as equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow></math>
except that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
are eliminated using
this operation; i.e.
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
<mrow><mo stretchy="true">[</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>3</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1.25</mn>
</mtd></mtr></mtable>
</mrow></math>
<br/>
<b><big><a name="Index 2: f_1" id="Index 2: f_1">Index 2: f_1</a></big></b>
<br/>
The previous operation has index 1,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>1</mn>
<mo stretchy="false">*</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>1</mn>
<mo stretchy="false">*</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr></mtable>
</mrow></math>
We define the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
as equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow></math>
except that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
are eliminated using
this operation; i.e.
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
<mrow><mo stretchy="true">[</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow><mo stretchy="true">(</mo><mrow><msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>2</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1.5</mn>
</mtd></mtr></mtable>
</mrow></math>
Note that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow></math>
is equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>
,
so the second partial derivative of
<code><font color="blue"><span style='white-space: nowrap'>exp_eps(</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'>epsilon</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
at
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
equal to .5 and
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
equal .2 is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>x</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>7</mn>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mn>1</mn>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
There is a theorem about algorithmic differentiation that explains why
the other partial of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow></math>
is equal to the first partial of
<code><font color="blue"><span style='white-space: nowrap'>exp_eps(</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'>epsilon</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="Verification" id="Verification">Verification</a></big></b>
<br/>
The file <a href="exp_eps_rev2.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_rev2.cpp</span></a>
contains a routine
that verifies the values computed above.
It returns true for success and false for failure.
It only tests the partial derivatives of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
that might not be equal to the corresponding
partials of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow></math>
; i.e., the
other partials of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
must be equal to the corresponding
partials of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="Exercises" id="Exercises">Exercises</a></big></b>
<ol type="1"><li>
Consider the case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">=</mo>
<mn>.1</mn>
</mrow></math>
and we first preform a zero order forward mode sweep
for the operation sequence used above (in reverse order).
What are the results of a
first order reverse mode sweep; i.e.,
what are the corresponding values for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>k</mi>
</msub>
</mrow>
</mfrac>
</mrow></math>
for all
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>
such that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>k</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">≠</mo>
<mn>0</mn>
</mrow></math>
.
</li><li>
Create a modified version of
<a href="exp_eps_rev2.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_rev2.cpp</span></a>
that verifies the values you obtained for the previous exercise.
Also create and run a main program that reports the result
of calling the modified version of
<a href="exp_eps_rev2.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_rev2.cpp</span></a>
.
</li></ol>
<hr/>Input File: introduction/exp_apx/exp_eps.omh
</body>
</html>