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<option>Zero Order Expansion</option>
<option>Operation Sequence</option>
<option>---..Index</option>
<option>---..Code</option>
<option>---..Operation</option>
<option>---..Zero Order</option>
<option>---..Sweep</option>
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<center><b><big><big>exp_2: Operation Sequence and Zero Order Forward Mode</big></big></b></center>
<br/>
<b><big><a name="Mathematical Form" id="Mathematical Form">Mathematical Form</a></big></b>
<br/>
The operation sequence (see below) corresponding to
the algorithm <a href="exp_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a>
 is the same for all values of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.
The mathematical form for the corresponding function 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>
<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>

An algorithmic differentiation package
does not operate on the mathematical function 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>


but rather on the particular algorithm used to compute the function 
(in this case <a href="exp_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a>
).

<br/>
<br/>
<b><big><a name="Zero Order Expansion" id="Zero Order Expansion">Zero Order Expansion</a></big></b>



<br/>
In general, a zero order forward sweep is given a vector

<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">&#x02208;</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>

 and it returns the corresponding vector

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">&#x02208;</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>

 given by

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

The superscript 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>

 denotes zero order derivative; 
i.e., it is equal to the value
of the corresponding variable.
For the example we are considering here, 
both 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>

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

 are equal to one.


<br/>
<br/>
<b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b>
<br/>
An atomic 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 operation is an operation
that has a 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 result and is not made up of 
other more basic operations.
A sequence of atomic 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 operations is called a

<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 operation sequence.
Given an C++ algorithm and its inputs,
there is a corresponding 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 operation sequence for each type. 
If 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 is clear from the context, 
we drop it and just refer to the operation sequence.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>We consider the case where <a href="exp_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a>
 is executed with

<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>.5</mn>
</mrow></math>

.
The table below contains the corresponding operation sequence
and the results of a zero order sweep.

<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 zero order derivative 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 in the operation sequence.

<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">Type v1  = 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">Type v2  = Type(1) + v1;</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>1.5</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">Type v3  = v1 * v1;</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>1</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>3</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'>

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">Type v4  = v3 / Type(2);</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>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</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>4</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'>

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"> Type v5  = v2 + v4;</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>2</mn>
</msub>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</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>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>5</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>
</mrow></math>

<br/>
<b><big><a name="Verification" id="Verification">Verification</a></big></b>
<br/>
The file <a href="exp_2_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_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>.2</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="block"><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">&#x022EF;</mo>
<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></math>

</li><li>
Create a modified version of 
<a href="exp_2_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_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_2_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for0.cpp</span></a>
 in the previous exercise.
</li></ol>




<hr/>Input File: introduction/exp_apx/exp_2.omh

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