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<center><b><big><big>Check an ADFun Sequence of Operations</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<br/>

<code><i><font color="black"><span style='white-space: nowrap'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;FunCheck(</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>g</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</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'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>a</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>

<code><span style='white-space: nowrap'><br/>
</span></code><b>See Also</b>
<a href="comparechange.xml" target="_top"><span style='white-space: nowrap'>CompareChange</span></a>



<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
We use 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>

 to denote the
<a href="glossary.xml#AD Function" target="_top"><span style='white-space: nowrap'>AD&#xA0;function</span></a>
 corresponding to 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
We use 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>

 to denote the
function corresponding to the C++ function object 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
.
This routine check if

<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>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

If 
<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>
<mo stretchy="false">&#x02260;</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

, the
<a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation&#xA0;sequence</span></a>

corresponding to 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 does not represents the algorithm used
by 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 to calculate values for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
</mrow></math>


(see <a href="funcheck.xml#Discussion" target="_top"><span style='white-space: nowrap'>Discussion</span></a>
 below).

<br/>
<br/>
<b><big><a name="f" id="f">f</a></big></b>
<br/>
The <code><font color="blue">FunCheck</font></code> argument 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;ADFun&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
Note that the <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
 object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 is not <code><font color="blue">const</font></code>
(see <a href="funcheck.xml#FunCheck Uses Forward" target="_top"><span style='white-space: nowrap'>Forward</span></a>
 below).

<br/>
<br/>
<b><big><a name="g" id="g">g</a></big></b>
<br/>
The <code><font color="blue">FunCheck</font></code> argument 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>g</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(
<code><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i></code>
 is defined the properties of 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
).
The C++ function object 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 supports the syntax

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>g</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
which computes 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><a name="g.x" id="g.x">x</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 argument 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="funcheck.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
 below)
and its size 
must be equal to 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
 space for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.

<br/>
<br/>
<b><big><a name="y" id="y">y</a></big></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 result 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its value is 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.
The size of 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
 
is equal to 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>
 space for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.

<br/>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The <code><font color="blue">FunCheck</font></code> argument 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and its size 
must be equal to 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
 space for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
This specifies that point at which to compare the values
calculated by 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>G</span></font></i></code>
. 

<br/>
<br/>
<b><big><a name="r" id="r">r</a></big></b>
<br/>
The <code><font color="blue">FunCheck</font></code> argument 
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the relative error the element by element
comparison of the value of 
<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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="a" id="a">a</a></big></b>
<br/>
The <code><font color="blue">FunCheck</font></code> argument 
<code><i><font color="black"><span style='white-space: nowrap'>a</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>a</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the absolute error the element by element
comparison of the value of 
<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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="ok" id="ok">ok</a></big></b>
<br/>
The <code><font color="blue">FunCheck</font></code> result 
<code><i><font color="black"><span style='white-space: nowrap'>ok</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;bool&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It is true, if for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow></math>

 
either the relative error bound is satisfied

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mo stretchy="false">|</mo>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>G</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">|</mo>
<mo stretchy="false">&#x02264;</mo>
<mi mathvariant='italic'>r</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">|</mo>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">|</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">|</mo>
<msub><mi mathvariant='italic'>G</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">|</mo>
<mo stretchy="false">)</mo>
</mrow></math>

or the absolute error bound is satisfied

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mo stretchy="false">|</mo>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>G</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">|</mo>
<mo stretchy="false">&#x02264;</mo>
<mi mathvariant='italic'>a</mi>
</mrow></math>

It is false if for some 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 neither
of these bounds is satisfied.

<br/>
<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
The type 
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
 must be a <a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
 class with
<a href="simplevector.xml#Elements of Specified Type" target="_top"><span style='white-space: nowrap'>elements&#xA0;of&#xA0;type</span></a>


<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
.
The routine <a href="checksimplevector.xml" target="_top"><span style='white-space: nowrap'>CheckSimpleVector</span></a>
 will generate an error message
if this is not the case.

<br/>
<br/>
<b><big><a name="FunCheck Uses Forward" id="FunCheck Uses Forward">FunCheck Uses Forward</a></big></b>
<br/>
After each call to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
,
the object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 contains the corresponding 
<a href="glossary.xml#Taylor Coefficient" target="_top"><span style='white-space: nowrap'>Taylor&#xA0;coefficients</span></a>
.
After <code><font color="blue">FunCheck</font></code>,
the previous calls to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
 are undefined.

<br/>
<br/>
<b><big><a name="Discussion" id="Discussion">Discussion</a></big></b>
<br/>
Suppose that the algorithm corresponding to 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 contains

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;if(&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&gt;=&#xA0;0&#xA0;)<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;exp(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;else&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;exp(-</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
where 
<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'>y</span></font></i></code>
 are 
<code><font color="blue"><span style='white-space: nowrap'>AD&lt;double&gt;</span></font></code>
 objects.
It follows that the 
AD of <code><font color="blue">double</font></code> <a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation&#xA0;sequence</span></a>

depends on the value of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.
If the sequence of operations stored in 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 corresponds to 

<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 with 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">&#x02265;</mo>
<mn>0</mn>
</mrow></math>

, 
the function values computed using 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 when 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">&lt;</mo>
<mn>0</mn>
</mrow></math>


will not agree with the function values computed by 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>g</mi>
</mrow></math>

.
This is because the operation sequence corresponding to 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 changed
(and hence the object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 does not represent the function

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

 for this value of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
).
In this case, you probably want to re-tape the calculations
performed by 
<code><i><font color="black"><span style='white-space: nowrap'>g</span></font></i></code>
 with the
<a href="glossary.xml#Tape.Independent Variable" target="_top"><span style='white-space: nowrap'>independent&#xA0;variables</span></a>
 
equal to the values in 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 
(so AD operation sequence properly represents the algorithm
for this value of independent variables).


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

<br/>
The file
<a href="fun_check.cpp.xml" target="_top"><span style='white-space: nowrap'>fun_check.cpp</span></a>

contains an example and test of this function.   
It returns true if it succeeds and false otherwise.


<hr/>Input File: cppad/local/fun_check.hpp

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