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<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&#xA0;order&#xA0;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'>,&#xA0;</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'>&#x003B5;</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">&#x02202;</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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'>&#x003B5;</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">&#x02202;</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
</msub>
<msub><mo stretchy="false">&#x02202;</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'>&#x003B5;</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'>&#x003B5;</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'>&#x003B5;</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">&#x02026;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02026;</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">&#x02026;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02026;</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">&#x02026;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02026;</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">&#x02026;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02026;</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">&#x02026;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02026;</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">&#x02026;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</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'>,&#xA0;</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">&#x02202;</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</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">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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'>,&#xA0;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</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">&#x02202;</mo>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>k</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">&#x02260;</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

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