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<option>Second Order Expansion</option>
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<option>---..Operation</option>
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<center><b><big><big>exp_2: Second Order Forward Mode</big></big></b></center>
<br/>
<b><big><a name="Second Order Expansion" id="Second Order Expansion">Second Order Expansion</a></big></b>
<br/>
We define
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
near
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
by the second order expansion
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<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='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow></math>
It follows that for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mn>2</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mfrac><mrow><msup><mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>d</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
<mrow><mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>d</mi>
</mstyle></mrow>
<mspace width='.18em'/>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
In general, a second order forward sweep is given the
<a href="exp_2_for1.xml#First Order Expansion" target="_top"><span style='white-space: nowrap'>first order expansion</span></a>
for all of the variables in an operation sequence,
and the second order derivatives for the independent variables.
It uses these to compute the second order derivative,
and thereby obtain the second order expansion,
for all the variables in the operation sequence.
<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_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a>
to compute
<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>
The corresponding second derivative function is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>x</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<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>
</mrow></math>
<br/>
<b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b>
<br/>
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>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">=</mo>
<mn>.5</mn>
</mrow></math>
.
The corresponding operation sequence,
zero order forward sweep values,
and first order forward sweep values
are inputs and are used by a second order forward 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.
A Forward sweep starts with the first operation
and ends with the last.
<br/>
<br/>
<b><a name="Operation Sequence.Zero" id="Operation Sequence.Zero">Zero</a></b>
<br/>
The Zero column contains the zero order sweep results
for the corresponding variable in the operation sequence
(see <a href="exp_2_for0.xml#Operation Sequence.Sweep" target="_top"><span style='white-space: nowrap'>zero order sweep</span></a>
).
<br/>
<br/>
<b><a name="Operation Sequence.Operation" id="Operation Sequence.Operation">Operation</a></b>
<br/>
The Operation column contains the
first order sweep operation for this variable.
<br/>
<br/>
<b><a name="Operation Sequence.First" id="Operation Sequence.First">First</a></b>
<br/>
The First column contains the first order sweep results
for the corresponding variable in the operation sequence
(see <a href="exp_2_for1.xml#Operation Sequence.Sweep" target="_top"><span style='white-space: nowrap'>first order sweep</span></a>
).
<br/>
<br/>
<b><a name="Operation Sequence.Derivative" id="Operation Sequence.Derivative">Derivative</a></b>
<br/>
The Derivative column contains the
mathematical function corresponding to the second derivative
with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
,
at
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
, for each variable in the sequence.
<br/>
<br/>
<b><a name="Operation Sequence.Second" id="Operation Sequence.Second">Second</a></b>
<br/>
The Second column contains the second order derivatives
for the corresponding variable in the operation sequence; i.e.,
the second order expansion for the <i>i</i>-th variable is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow></math>
We use
<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>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
so that second order differentiation
with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
, at
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
,
is the same as the second partial differentiation
with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>
at
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</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>
</mrow></math>
.
<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'>  </span></code> </td><td align='left' valign='top'>
<b>Zero</b>
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'>
<b>Operation</b>
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'>
<b>First</b>
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'>
<b>Derivative</b>
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'>
<b>Second</b>
</td></tr><tr><td align='left' valign='top'>
1
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
0.5
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </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>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
1
</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
</td></tr><tr><td align='left' valign='top'>
2
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
1.5
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </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>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>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
1
</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>2</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
</td></tr><tr><td align='left' valign='top'>
3
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
0.25
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </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>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>2</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>
<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></math>
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
1
</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>2</mn>
<mo stretchy="false">*</mo>
<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>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>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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>2</mn>
</mrow></math>
</td></tr><tr><td align='left' valign='top'>
4
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
0.125
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'>  </span></code> </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>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>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow></math>
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
.5
</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>2</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
</td></tr><tr><td align='left' valign='top'>
5
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
1.625
</td><td align='left' valign='top'>
<code><span style='white-space: nowrap'> </span></code> </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>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>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>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
</td><td align='left' valign='top'>
</td><td align='left' valign='top'>
1.5
</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>2</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>2</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
</td></tr>
</table>
</center><b><big><a name="Return Value" id="Return Value">Return Value</a></big></b>
<br/>
The second derivative of the return value for this case is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mn>1</mn>
</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>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
<msub><mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow><mo stretchy="true">]</mo></mrow>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">=</mo>
<msub><mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<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='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>x</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>x</mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>
(We have used the fact 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>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
.)
<br/>
<br/>
<b><big><a name="Verification" id="Verification">Verification</a></big></b>
<br/>
The file <a href="exp_2_for2.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for2.cpp</span></a>
contains a routine
which 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>
Which statement in the routine defined by <a href="exp_2_for2.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for2.cpp</span></a>
uses
the values that are calculated by the routine
defined by <a href="exp_2_for1.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for1.cpp</span></a>
?
</li><li>
Suppose that
<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>
,
what are the results of a zero, first, and second 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>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mn>5</mn>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mn>2</mn>
</mrow></math>
.
</li><li>
Create a modified version of <a href="exp_2_for2.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for2.cpp</span></a>
that verifies
the derivative values from the previous exercise.
Also create and run a main program that reports the result
of calling the modified version of
<a href="exp_2_for2.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for2.cpp</span></a>
.
</li></ol>
<hr/>Input File: introduction/exp_apx/exp_2.omh
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