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<center><b><big><big>exp_eps: First Order Reverse Sweep</big></big></b></center>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
First order reverse mode uses the
<a href="exp_eps_for0.xml#Operation Sequence" target="_top"><span style='white-space: nowrap'>operation sequence</span></a>
,
and zero order forward sweep values,
to compute the first order derivative
of one dependent variable with respect to all the independent variables.
The computations are done in reverse
of the order of the computations in the original algorithm.
<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></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>epsilon</span></i><code><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
with <i>x</i> is equal to .5
and <i>epsilon</i> is equal to .2.
For this case, the mathematical function for the operation sequence
corresponding to <code><font color="blue">exp_eps</font></code> is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>ε</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow></math>
The corresponding partial derivatives,
and the value of the derivatives, are
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>ε</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>1</mn>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">=</mo>
<mn>1.5</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='normal'>ε</mi>
</msub>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>ε</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mn>0</mn>
</mtd></mtr></mtable>
</mrow></math>
<br/>
<b><big><a name="epsilon" id="epsilon">epsilon</a></big></b>
<br/>
Since
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>ε</mi>
</mrow></math>
is an independent variable,
it could included as an argument to all of the
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
functions below.
The result would be that all the partials with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>ε</mi>
</mrow></math>
would be zero and hence we drop it to simplify
the presentation.
<br/>
<br/>
<b><big><a name="f_7" id="f_7">f_7</a></big></b>
<br/>
In reverse mode we choose one dependent variable and
compute its derivative with respect to all the independent variables.
For our example, we chose the value returned by <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>
which is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow></math>
.
We begin with the function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow></math>
is both an argument and the value of the function; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</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>
<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>
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>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
<mo stretchy="false">)</mo>
</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>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow></math>
is 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>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
<mo stretchy="false">,</mo>
<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>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>7</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
</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>
<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>
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>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">)</mo>
</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>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
</mrow></math>
is 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>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
<mo stretchy="false">,</mo>
<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>
<mo stretchy="false">]</mo>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>6</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</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>
<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>
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>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">)</mo>
</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>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow></math>
is 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>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">,</mo>
<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>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
Given the information from the forward sweep, we have
<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>
<mn>0.5</mn>
</mrow></math>
and
<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>
<mn>0.5</mn>
</mrow></math>
.
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>5</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>1</mn>
</mtd></mtr></mtable>
</mrow></math>
<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>
<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>
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>
<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>2</mn>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
<mo stretchy="false">)</mo>
</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>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow></math>
is 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>
<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>2</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>4</mn>
</msub>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>4</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</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>
<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>
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>
<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>2</mn>
</msub>
<mo stretchy="false">)</mo>
</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>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow></math>
is 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>4</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>2</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>2</mn>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mo stretchy="false">=</mo>
<mn>0.25</mn>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</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>
<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>
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>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">)</mo>
</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>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow></math>
is 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>
<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>2</mn>
</msub>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">*</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mn>1</mn>
</msub>
</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 derivative of <code><font color="blue"><span style='white-space: nowrap'>exp_eps(</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>epsilon</span></i><code><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
at <i>x</i> equal to .5 and <i>epsilon</i> equal .2 is
1.5 in the <i>x</i> direction and zero in the <i>epsilon</i> direction.
We also note that
<a href="exp_eps_for1.xml" target="_top"><span style='white-space: nowrap'>forward</span></a>
forward mode gave the
same result for the partial in the <i>x</i> direction.
<br/>
<br/>
<b><big><a name="Verification" id="Verification">Verification</a></big></b>
<br/>
The file <a href="exp_eps_rev1.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_rev1.cpp</span></a>
contains a routine
that verifies the values computed above.
It returns true for success and false for failure.
It only tests the partial derivatives of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
that might not be equal to the corresponding
partials of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow></math>
; i.e., the
other partials of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
must be equal to the corresponding
partials of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>f</mi>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="Exercises" id="Exercises">Exercises</a></big></b>
<ol type="1"><li>
Consider the case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">=</mo>
<mn>.1</mn>
</mrow></math>
and we first preform a zero order forward mode sweep
for the operation sequence used above (in reverse order).
What are the results of a
first order reverse mode sweep; i.e.,
what are the corresponding values for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>k</mi>
</msub>
</mrow>
</mfrac>
</mrow></math>
for all
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>
such that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>v</mi>
<mi mathvariant='italic'>k</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">≠</mo>
<mn>0</mn>
</mrow></math>
.
</li><li>
Create a modified version of
<a href="exp_eps_rev1.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_rev1.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_rev1.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps_rev1.cpp</span></a>
.
</li></ol>
<hr/>Input File: introduction/exp_apx/exp_eps.omh
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