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<option>Operation Sequence</option>
<option>---..Variable</option>
<option>---..Parameter</option>
<option>---..Index</option>
<option>---..Code</option>
<option>---..Operation</option>
<option>---..Zero Order</option>
<option>---..Sweep</option>
<option>Return Value</option>
<option>Comparisons</option>
<option>Verification</option>
<option>Exercises</option>
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<center><b><big><big>exp_eps: Operation Sequence and Zero Order Forward Sweep</big></big></b></center>
<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 form for the operation sequence 
corresponding to the <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>

Note that, for these particular values of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
,
this is the same as the mathematical form for
<a href="exp_2_for0.xml#Mathematical Form" target="_top"><span style='white-space: nowrap'>exp_2</span></a>
.


<br/>
<br/>
<b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b>
<br/>
We consider the 
<a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation&#xA0;sequence</span></a>

corresponding to the algorithm <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>

with the argument 
<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. 

<br/>
<br/>
<b><a name="Operation Sequence.Variable" id="Operation Sequence.Variable">Variable</a></b>
<br/>
We refer to values that depend on the input variables

<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
 as variables.

<br/>
<br/>
<b><a name="Operation Sequence.Parameter" id="Operation Sequence.Parameter">Parameter</a></b>
<br/>
We refer to values that do not depend on the input variables

<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 or 
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
 as parameters.
Operations where the result is a parameter are not included
in the zero order sweep below.


<br/>
<br/>
<b><a name="Operation Sequence.Index" id="Operation Sequence.Index">Index</a></b>
<br/>
The Index column contains the index in the operation sequence
of the corresponding atomic operation and variable. 
A Forward sweep starts with the first operation 
and ends with the last.

<br/>
<br/>
<b><a name="Operation Sequence.Code" id="Operation Sequence.Code">Code</a></b>
<br/>
The Code column contains the C++ source code corresponding 
to the corresponding atomic operation in the sequence. 

<br/>
<br/>
<b><a name="Operation Sequence.Operation" id="Operation Sequence.Operation">Operation</a></b>
<br/>
The Operation column contains the 
mathematical function corresponding to each atomic operation in the sequence.

<br/>
<br/>
<b><a name="Operation Sequence.Zero Order" id="Operation Sequence.Zero Order">Zero Order</a></b>
<br/>
The Zero Order column contains the 
<a href="exp_2_for0.xml#Zero Order Expansion" target="_top"><span style='white-space: nowrap'>zero&#xA0;order&#xA0;derivative</span></a>

for the corresponding variable in the operation sequence.
Forward mode refers to the fact that
these coefficients are computed in the same order as the original algorithm;
i.e., in order of increasing index.


<br/>
<br/>
<b><a name="Operation Sequence.Sweep" id="Operation Sequence.Sweep">Sweep</a></b>

<center>
<table><tr><td align='left'  valign='top'>

<b>Index</b>
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <b>Code</b> 
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <b>Operation</b>
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <b>Zero Order</b>
</td></tr><tr><td align='left'  valign='top'>

1
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <code><font color="blue">abs_x = x;</font></code>
	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>x</mi>
</mrow></math>


	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<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>


</td></tr><tr><td align='left'  valign='top'>

2
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <code><font color="blue">temp  = term * abs_x;</font></code>
	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">*</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow></math>


	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<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>
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mrow></math>


</td></tr><tr><td align='left'  valign='top'>

3
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <code><font color="blue">term = temp / Type(k);</font></code>
	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
<mo stretchy="false">/</mo>
<mn>1</mn>
</mrow></math>


	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<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>


</td></tr><tr><td align='left'  valign='top'>

4
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <code><font color="blue">sum  = sum + term;</font></code>
	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow></math>


	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<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>
<mo stretchy="false">=</mo>
<mn>1.5</mn>
</mrow></math>


</td></tr><tr><td align='left'  valign='top'>

5
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
  <code><font color="blue">temp  = term * abs_x;</font></code>
	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
<mo stretchy="false">*</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow></math>


	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<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>
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mrow></math>


</td></tr><tr><td align='left'  valign='top'>

6
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <code><font color="blue">term = temp / Type(k);</font></code>
	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow></math>


	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<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>
<mo stretchy="false">=</mo>
<mn>0.125</mn>
</mrow></math>


</td></tr><tr><td align='left'  valign='top'>

7
	</td><td align='left'  valign='top'>
 <code><span style='white-space: nowrap'>&#xA0;&#xA0;</span></code> </td><td align='left'  valign='top'>
 <code><font color="blue">sum  = sum + term;</font></code>
	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
</mrow></math>


	</td><td align='left'  valign='top'>
 </td><td align='left'  valign='top'>
 
<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>
<mo stretchy="false">=</mo>
<mn>1.625</mn>
</mrow></math>


</td></tr>
</table>
</center><b><big><a name="Return Value" id="Return Value">Return Value</a></big></b>
<br/>
The return value for this case is

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mn>1.625</mn>
<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>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">,</mo>
<msup><mi mathvariant='normal'>&#x003B5;</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow></math>

<br/>
<b><big><a name="Comparisons" id="Comparisons">Comparisons</a></big></b>
<br/>
If 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 were negative,
or if 
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
 were a much smaller or much larger value,
the results of the following comparisons could be different:
<code><font color='blue'><pre style='display:inline'> 
	if( Type(0) &gt; x ) 
	while(term &gt; epsilon)
</pre></font></code>

This in turn would result in a different operation sequence.
Thus the operation sequence above only corresponds to 
<a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>

for values of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
 within a certain range.
Note that there is a neighborhood
of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">=</mo>
<mn>0.5</mn>
</mrow></math>

 for which the comparisons would have the 
same result and hence the operation sequence would be the same.



<br/>
<br/>
<b><big><a name="Verification" id="Verification">Verification</a></big></b>
<br/>
The file <a href="exp_eps_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_for0.cpp</span></a>
 contains a routine 
that verifies the values computed above.
It returns true for success and false for failure.

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

<ol type="1"><li>
Suppose that 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mn>.1</mn>
</mrow></math>

,
what is the result of a zero order forward sweep for 
the operation sequence above; 
i.e., what are the corresponding values for

<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>
<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>
<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>
</mrow></math>

.
</li><li>

Create a modified version of 
<a href="exp_eps_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_for0.cpp</span></a>
 that verifies the values you obtained
for the previous exercise.
</li><li>

Create and run a main program that reports the result of calling 
the modified version 
of <a href="exp_eps_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_for0.cpp</span></a>
 in the previous exercise.
</li></ol>



<hr/>Input File: introduction/exp_apx/exp_eps.omh

</body>
</html>
