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<option>Headings-&gt;</option>
<option>Syntax</option>
<option>---..Define Function</option>
<option>---..Use Function</option>
<option>---..Callback Routines</option>
<option>---..Free Static Memory</option>
<option>Purpose</option>
<option>Partial Implementation</option>
<option>CPPAD_USER_ATOMIC</option>
<option>---..Tvector</option>
<option>---..Base</option>
<option>ok</option>
<option>id</option>
<option>k</option>
<option>n</option>
<option>m</option>
<option>tx</option>
<option>ty</option>
<option>---..forward</option>
<option>---..reverse</option>
<option>afun</option>
<option>---..ax</option>
<option>---..ay</option>
<option>---..Parallel Mode</option>
<option>forward</option>
<option>---..Usage</option>
<option>---..vx</option>
<option>---..vy</option>
<option>reverse</option>
<option>---..Usage</option>
<option>---..py</option>
<option>---..px</option>
<option>for_jac_sparse</option>
<option>---..Usage</option>
<option>---..q</option>
<option>---..r</option>
<option>---..s</option>
<option>rev_jac_sparse</option>
<option>---..Usage</option>
<option>---..q</option>
<option>---..s</option>
<option>---..r</option>
<option>rev_hes_sparse</option>
<option>---..Usage</option>
<option>---..q</option>
<option>---..r</option>
<option>---..s</option>
<option>---..t</option>
<option>---..u</option>
<option>---..v</option>
<option>clear</option>
<option>---..Restriction</option>
<option>Example</option>
<option>---..Simple</option>
<option>---..Use AD</option>
<option>---..Tangent Function</option>
<option>---..Matrix Multiplication</option>
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<center><b><big><big>User Defined Atomic AD Functions</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<br/>
Using <code><font color="blue">CPPAD_USER_ATOMIC</font></code> has been deprecated. 
Use <a href="atomic_base.xml" target="_top"><span style='white-space: nowrap'>atomic_base</span></a>
 instead.

<br/>
<br/>
<b><a name="Syntax.Define Function" id="Syntax.Define Function">Define Function</a></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>CPPAD_USER_ATOMIC(</span></font><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Tvector</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#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;<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>reverse</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>for_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_hes_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
)<br/>
</span></font></code>
<br/>
<b><a name="Syntax.Use Function" id="Syntax.Use Function">Use Function</a></b>

<br/>

<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><a name="Syntax.Callback Routines" id="Syntax.Callback Routines">Callback Routines</a></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;</span></font><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ty</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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>reverse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>px</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>py</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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>for_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</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'>s</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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</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'>s</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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_hes_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</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'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><a name="Syntax.Free Static Memory" id="Syntax.Free Static Memory">Free Static Memory</a></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>user_atomic&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;::clear()</span></font></code>


<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
In some cases, the user knows how to compute the derivative
of a function 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mspace width='.3em'/>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>where</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<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>

more efficiently than by coding it using 
<code><font color="blue"><span style='white-space: nowrap'>AD&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;</span></font></code>
 
<a href="glossary.xml#Operation.Atomic" target="_top"><span style='white-space: nowrap'>atomic</span></a>
 operations
and letting CppAD do the rest.
In this case, <code><font color="blue">CPPAD_USER_ATOMIC</font></code> can be used
add the user code for 
<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 its derivatives,
to the set of 
<code><font color="blue"><span style='white-space: nowrap'>AD&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;</span></font></code>
 atomic operations. 
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>Another possible purpose is to reduce the size of the tape; 
see <a href="old_atomic.xml#Example.Use AD" target="_top"><span style='white-space: nowrap'>use&#xA0;AD</span></a>


<br/>
<br/>
<b><big><a name="Partial Implementation" id="Partial Implementation">Partial Implementation</a></big></b>
<br/>
The routines 
<a href="old_atomic.xml#forward" target="_top"><span style='white-space: nowrap'>forward</span></a>
,
<a href="old_atomic.xml#reverse" target="_top"><span style='white-space: nowrap'>reverse</span></a>
,
<a href="old_atomic.xml#for_jac_sparse" target="_top"><span style='white-space: nowrap'>for_jac_sparse</span></a>
,
<a href="old_atomic.xml#rev_jac_sparse" target="_top"><span style='white-space: nowrap'>rev_jac_sparse</span></a>
, and
<a href="old_atomic.xml#rev_hes_sparse" target="_top"><span style='white-space: nowrap'>rev_hes_sparse</span></a>
,
must be defined by the user.
The 
<code><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i></code>
 the routine, 
for the case 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;0</span></font></code>
, must be implemented.
Functions with the correct prototype,
that just return <code><font color="blue">false</font></code>, 
can be used for the other cases 
(unless they are required by your calculations). 
For example, you need not implement

<code><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i></code>
 for the case 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;==&#xA0;2</span></font></code>
 until you require
forward mode calculation of second derivatives.

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

<br/>
The macro 

<code><font color="blue"><span style='white-space: nowrap'><br/>
CPPAD_USER_ATOMIC(</span></font><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Tvector</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#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;<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>reverse</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>for_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_hes_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
)<br/>
</span></font></code>
defines the 
<code><font color="blue"><span style='white-space: nowrap'>AD&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;</span></font></code>
 routine 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
.
This macro can be placed within a namespace 
(not the <code><font color="blue">CppAD</font></code> namespace) 
but must be outside of any routine.

<br/>
<br/>
<b><a name="CPPAD_USER_ATOMIC.Tvector" id="CPPAD_USER_ATOMIC.Tvector">Tvector</a></b>
<br/>
The macro argument 
<code><i><font color="black"><span style='white-space: nowrap'>Tvector</span></font></i></code>
 must be a
<a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>simple&#xA0;vector&#xA0;template&#xA0;class</span></a>
.
It determines the type of vectors used as arguments to the routine 

<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
.

<br/>
<br/>
<b><a name="CPPAD_USER_ATOMIC.Base" id="CPPAD_USER_ATOMIC.Base">Base</a></b>
<br/>
The macro argument 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
 specifies the 
<a href="base_require.xml" target="_top"><span style='white-space: nowrap'>base&#xA0;type</span></a>

corresponding to 
<code><font color="blue"><span style='white-space: nowrap'>AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base&gt;</span></font></i></code>
 operation sequences.
Calling the routine 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
 will add the operator defined
by this macro to an 
<code><font color="blue"><span style='white-space: nowrap'>AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base&gt;</span></font></i></code>
 operation sequence.

<br/>
<br/>
<b><big><a name="ok" id="ok">ok</a></big></b>
<br/>
For all routines documented below,
the return value 
<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>
If it is <code><font color="blue">true</font></code>, the corresponding evaluation succeeded,
otherwise it failed.

<br/>
<br/>
<b><big><a name="id" id="id">id</a></big></b>
<br/>
For all routines documented below,
the argument 
<code><i><font color="black"><span style='white-space: nowrap'>id</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
Its value in all other calls is the same as in the corresponding
call to 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
.
It can be used to store and retrieve extra information about
a specific call to 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
. 

<br/>
<br/>
<b><big><a name="k" id="k">k</a></big></b>
<br/>
For all routines documented below, the argument 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The value 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
 is the order of the Taylor coefficient that
we are evaluating (<a href="old_atomic.xml#forward" target="_top"><span style='white-space: nowrap'>forward</span></a>
)
or taking the derivative of (<a href="old_atomic.xml#reverse" target="_top"><span style='white-space: nowrap'>reverse</span></a>
).

<br/>
<br/>
<b><big><a name="n" id="n">n</a></big></b>
<br/>
For all routines documented below, 
the argument 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It is the size of the vector 
<code><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i></code>
 in the corresponding call to

<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
; i.e.,
the dimension of the domain space for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="m" id="m">m</a></big></b>
<br/>
For all routines documented below, the argument 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It is the size of the vector 
<code><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i></code>
 in the corresponding call to

<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
; i.e.,
the dimension of the range space for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="tx" id="tx">tx</a></big></b>
<br/>
For all routines documented below, 
the argument 
<code><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&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;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
For 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>

,
we use the Taylor coefficient notation

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>tx</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">*</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>X</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mn>0</mn>
</msubsup>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mn>1</mn>
</msubsup>
<msup><mi mathvariant='italic'>t</mi>
<mn>1</mn>
</msup>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mi mathvariant='italic'>k</mi>
</msubsup>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msup>
</mtd></mtr></mtable>
</mrow></math>

If 
<code><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
the other components of 
<code><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i></code>
 are not specified and should not be used.
Note that superscripts represent an index for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
</mrow></math>


and an exponent for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02113;</mo>
</msup>
</mrow></math>

.
Also note that the Taylor coefficients for 
<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>

 correspond
to the derivatives of 
<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>

 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>

 in the following way:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msubsup><mi mathvariant='italic'>X</mi>
<mi mathvariant='italic'>j</mi>
<mrow><mo stretchy="false">(</mo>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>

<br/>
<b><big><a name="ty" id="ty">ty</a></big></b>
<br/>
In calls to <a href="old_atomic.xml#forward" target="_top"><span style='white-space: nowrap'>forward</span></a>
, 
the argument 
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&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;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
while in calls to <a href="old_atomic.xml#reverse" target="_top"><span style='white-space: nowrap'>reverse</span></a>
 it has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&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;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
For all calls, 
<code><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>

,
we use the Taylor coefficient notation

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>ty</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>Y</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mn>0</mn>
</msubsup>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mn>1</mn>
</msubsup>
<msup><mi mathvariant='italic'>t</mi>
<mn>1</mn>
</msup>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mi mathvariant='italic'>k</mi>
</msubsup>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

where 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msup>
<mo stretchy="false">&#x02192;</mo>
<mn>0</mn>
</mrow></math>

 as 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02192;</mo>
<mn>0</mn>
</mrow></math>

.
If 
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
,
the other components of 
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i></code>
 are not specified and should not be used.
Note that superscripts represent an index for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
</mrow></math>


and an exponent for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02113;</mo>
</msup>
</mrow></math>

.
Also note that the Taylor coefficients for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 correspond
to the derivatives of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</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>

 in the following way:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msubsup><mi mathvariant='italic'>Y</mi>
<mi mathvariant='italic'>j</mi>
<mrow><mo stretchy="false">(</mo>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>

<br/>
<b><a name="ty.forward" id="ty.forward">forward</a></b>
<br/>
In the case of 
<code><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i></code>
,
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>

, 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>ty</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
</mrow></math>

 is an output
and all the other components of 
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i></code>
 are inputs.

<br/>
<br/>
<b><a name="ty.reverse" id="ty.reverse">reverse</a></b>
<br/>
In the case of 
<code><i><font color="black"><span style='white-space: nowrap'>reverse</span></font></i></code>
,
all the components of 
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i></code>
 are inputs.

<br/>
<br/>
<b><big><a name="afun" id="afun">afun</a></big></b>
<br/>
The macro argument 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
,
is the name of the AD function corresponding to this atomic
operation (as it is used in the source code).
CppAD uses the other functions,
where the arguments are vectors with elements of type 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
,
to implement the function 

<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'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
where the argument are vectors with elements of type 
<code><font color="blue"><span style='white-space: nowrap'>AD&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;</span></font></code>
.

<br/>
<br/>
<b><a name="afun.ax" id="afun.ax">ax</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
 argument 
<code><i><font color="black"><span style='white-space: nowrap'>ax</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'>Tvector</span></font></i><font color="blue"><span style='white-space: nowrap'>&lt;&#xA0;AD&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;&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It is the argument vector 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>

 
at which the 
<code><font color="blue"><span style='white-space: nowrap'>AD&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;</span></font></code>
 version of 

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

 is to be evaluated.
The dimension of the domain space for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>


is specified by <a href="old_atomic.xml#n" target="_top"><span style='white-space: nowrap'>n</span></a>
 
<code><font color="blue"><span style='white-space: nowrap'>=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()</span></font></code>
,
which must be greater than zero.

<br/>
<br/>
<b><a name="afun.ay" id="afun.ay">ay</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
 result 
<code><i><font color="black"><span style='white-space: nowrap'>ay</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'>Tvector</span></font></i><font color="blue"><span style='white-space: nowrap'>&lt;&#xA0;AD&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;&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The input values of its elements 
are not specified (must not matter).
Upon return, it is the 
<code><font color="blue"><span style='white-space: nowrap'>AD&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;</span></font></code>
 version of the 
result vector 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.
The dimension of the range space for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>


is specified by <a href="old_atomic.xml#m" target="_top"><span style='white-space: nowrap'>m</span></a>
 
<code><font color="blue"><span style='white-space: nowrap'>=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()</span></font></code>
,
which must be greater than zero.

<br/>
<br/>
<b><a name="afun.Parallel Mode" id="afun.Parallel Mode">Parallel Mode</a></b>


<br/>
The first call to 

<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'>afun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
must not be in <a href="ta_in_parallel.xml" target="_top"><span style='white-space: nowrap'>parallel</span></a>
 mode.
In addition, the
<a href="old_atomic.xml#clear" target="_top"><span style='white-space: nowrap'>old_atomic&#xA0;clear</span></a>

routine cannot be called while in parallel mode.

<br/>
<br/>
<b><big><a name="forward" id="forward">forward</a></big></b>
<br/>
The macro argument 
<code><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i></code>
 is a
user defined function

<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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
that computes results during a <a href="forward.xml" target="_top"><span style='white-space: nowrap'>forward</span></a>
 mode sweep.
For this call, we are given the Taylor coefficients in 
<code><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i></code>
 
form order zero through 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
,
and the Taylor coefficients in  
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i></code>
 with order less than 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
.
The 
<code><i><font color="black"><span style='white-space: nowrap'>forward</span></font></i></code>
 routine computes the 

<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
 order Taylor coefficients for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
</mrow></math>

 using the definition

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>

. 
For example, 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>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mn>0</mn>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mn>1</mn>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>Y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</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>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msup>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mn>2</mn>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>Y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<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>0</mn>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<msubsup><mi mathvariant='italic'>f</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'>x</mi>
<mn>0</mn>
</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>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msup>
<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>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msup>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<msubsup><mi mathvariant='italic'>f</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'>x</mi>
<mn>0</mn>
</msup>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msup>
<mo stretchy="false">+</mo>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msup>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>2</mn>
</msup>
</mtd></mtr></mtable>
</mrow></math>

Then, 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>

, it sets

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>ty</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mi mathvariant='italic'>k</mi>
</msubsup>
</mrow></math>

The other components of 
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i></code>
 must be left unchanged.

<br/>
<br/>
<b><a name="forward.Usage" id="forward.Usage">Usage</a></b>
<br/>
This routine is used,
with 
<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;&#xA0;0</span></font></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;==&#xA0;0</span></font></code>
,
by calls to 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
.
It is used,
with 
<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;=&#xA0;0</span></font></code>
 and

<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
 equal to the order of the derivative begin computed,
by calls to <a href="forward_order.xml" target="_top"><span style='white-space: nowrap'>forward</span></a>
.

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&lt;bool&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The case 
<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;&#xA0;0</span></font></code>
 occurs 
once for each call to 
<code><i><font color="black"><span style='white-space: nowrap'>afun</span></font></i></code>
,
during the call,
and before any of the other callbacks corresponding to that call.
Hence such a call can be used to cache information attached to 
the corresponding 
<code><i><font color="black"><span style='white-space: nowrap'>id</span></font></i></code>

(such as the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i></code>
).
If 
<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;&#xA0;0</span></font></code>
 then

<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;==&#xA0;0</span></font></code>
, 

<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
, and
for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,

<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 is true if and only if

<code><i><font color="black"><span style='white-space: nowrap'>ax</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 is a <a href="glossary.xml#Variable" target="_top"><span style='white-space: nowrap'>variable</span></a>
.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>If 
<code><i><font color="black"><span style='white-space: nowrap'>vx</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;==&#xA0;0</span></font></code>
, 
then 
<code><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;==&#xA0;0</span></font></code>
 and neither of these vectors
should be used.

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;bool&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
If 
<code><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;==&#xA0;0</span></font></code>
, it should not be used.
Otherwise, 

<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;==&#xA0;0</span></font></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
The input values of the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i></code>
 
are not specified (must not matter).
Upon return, for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</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>

,

<code><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 is true if and only if

<code><i><font color="black"><span style='white-space: nowrap'>ay</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 is a variable.
(CppAD uses 
<code><i><font color="black"><span style='white-space: nowrap'>vy</span></font></i></code>
 to reduce the necessary computations.)

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

is a user defined function

<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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>reverse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>px</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>py</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
that computes results during a <a href="reverse.xml" target="_top"><span style='white-space: nowrap'>reverse</span></a>
 mode sweep. 
The input value of the vectors 
<code><i><font color="black"><span style='white-space: nowrap'>tx</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>ty</span></font></i></code>

contain Taylor coefficient, up to order 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
,
for 
<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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 respectively.
We use the 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
</mrow></math>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
</mrow></math>


to denote these Taylor coefficients where the implicit range indices are

<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>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>

.
Using the calculations done by <a href="old_atomic.xml#forward" target="_top"><span style='white-space: nowrap'>forward</span></a>
,
the Taylor coefficients 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
</mrow></math>

 are a function of the Taylor
coefficients for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
</mrow></math>

; i.e., given 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>


we define the function

<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>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">&#x000D7;</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

 by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">=</mo>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">(</mo>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
<mo stretchy="false">)</mo>
</mrow></math>

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>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">&#x000D7;</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">&#x02192;</mo>
<mi mathvariant='italic'>B</mi>
</mrow></math>


to denote an arbitrary scalar valued function of the Taylor coefficients for 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 and write  
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
<mo stretchy="false">)</mo>
</mrow></math>

.
The <code><font color="blue">reverse</font></code> routine
is given the derivative of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
</mrow></math>

 with respect to

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
</mrow></math>

 and computes its derivative with respect
to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
</mrow></math>

.

<br/>
<br/>
<b><a name="reverse.Usage" id="reverse.Usage">Usage</a></b>
<br/>
This routine is used,
with 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1</span></font></code>
 equal to the order of the derivative being calculated,
by calls to <a href="reverse_any.xml" target="_top"><span style='white-space: nowrap'>reverse</span></a>
.

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&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;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>py</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>py</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>py</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
</mrow></math>

If 
<code><i><font color="black"><span style='white-space: nowrap'>py</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
,
the other components of 
<code><i><font color="black"><span style='white-space: nowrap'>py</span></font></i></code>
 are not specified and should not be used.

<br/>
<br/>
<b><a name="reverse.px" id="reverse.px">px</a></b>
<br/>
We define the function 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>

The 
<code><i><font color="black"><span style='white-space: nowrap'>reverse</span></font></i></code>
 argument 
<code><i><font color="black"><span style='white-space: nowrap'>px</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&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;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>px</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>px</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
The input values of the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>px</span></font></i></code>
 
are not specified (must not matter).
Upon return,
for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>px</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">*</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">]</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mi mathvariant='italic'>p</mi>
</msubsup>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">(</mo>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">{</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">}</mo>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mi mathvariant='italic'>p</mi>
</msubsup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</munderover>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>k</mi>
</munderover>
<mo stretchy="false">(</mo>
<mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mi mathvariant='italic'>p</mi>
</msubsup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</munderover>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
<mi mathvariant='italic'>k</mi>
</munderover>
<mi mathvariant='italic'>py</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">(</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mi mathvariant='italic'>p</mi>
</msubsup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

Note that we have used the fact that for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">&#x02113;</mo>
<mo stretchy="false">&lt;</mo>
<mi mathvariant='italic'>p</mi>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02113;</mo>
</msubsup>
<mo stretchy="false">/</mo>
<mo stretchy="false">&#x02202;</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mi mathvariant='italic'>p</mi>
</msubsup>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>

.
If 
<code><i><font color="black"><span style='white-space: nowrap'>px</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;&#xA0;(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;1)&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
the other components of 
<code><i><font color="black"><span style='white-space: nowrap'>px</span></font></i></code>
 are not specified and should not be used.

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

is a user defined function

<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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>for_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</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'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
that is used to compute results during a forward Jacobian sparsity sweep.
For a fixed 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow></math>

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

,
the Jacobian 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>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>q</mi>
</msup>
</mrow></math>

 is

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>R</mi>
</mrow></math>

Given a <a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity&#xA0;pattern</span></a>
 for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
</mrow></math>

,

<code><i><font color="black"><span style='white-space: nowrap'>for_jac_sparse</span></font></i></code>
 computes a sparsity pattern for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><a name="for_jac_sparse.Usage" id="for_jac_sparse.Usage">Usage</a></b>
<br/>
This routine is used by calls to <a href="forsparsejac.xml" target="_top"><span style='white-space: nowrap'>ForSparseJac</span></a>
.

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the number of columns in 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>

 and the Jacobian 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>

. 

<br/>
<br/>
<b><a name="for_jac_sparse.r" id="for_jac_sparse.r">r</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>for_jac_sparse</span></font></i></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;CppAD::vector&lt;&#xA0;std::set&lt;size_t&gt;&#xA0;&gt;&amp;&#xA0;</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>
and 
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
For 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,
all the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 are between
zero and 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
 inclusive.
This specifies a sparsity pattern for the matrix 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
</mrow></math>

.

<br/>
<br/>
<b><a name="for_jac_sparse.s" id="for_jac_sparse.s">s</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>for_jac_sparse</span></font></i></code>
 return value 
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;&#xA0;std::set&lt;size_t&gt;&#xA0;&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"></font></code>
.
The input values of its sets 
are not specified (must not matter). Upon return 
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>

,
all the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 are between
zero and 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
 inclusive.
This represents a sparsity pattern for the matrix 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

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

is a user defined function

<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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_jac_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</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'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
that is used to compute results during a reverse Jacobian sparsity sweep.
For a fixed 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>m</mi>
</mrow></math>

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

,
the Jacobian of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>

 is

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

Given a <a href="glossary.xml#Sparsity Pattern" target="_top"><span style='white-space: nowrap'>sparsity&#xA0;pattern</span></a>
 for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
</mrow></math>

,

<code><i><font color="black"><span style='white-space: nowrap'>rev_jac_sparse</span></font></i></code>
 computes a sparsity pattern for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><a name="rev_jac_sparse.Usage" id="rev_jac_sparse.Usage">Usage</a></b>
<br/>
This routine is used by calls to <a href="revsparsejac.xml" target="_top"><span style='white-space: nowrap'>RevSparseJac</span></a>

and to <a href="optimize.xml" target="_top"><span style='white-space: nowrap'>optimize</span></a>
.


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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the number of rows in 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>q</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>m</mi>
</mrow>
</msup>
</mrow></math>

 and the Jacobian 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>q</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>

. 

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&lt;&#xA0;std::set&lt;size_t&gt;&#xA0;&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
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>

, 
all the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>

are between zero and 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
 inclusive.
This specifies a sparsity pattern for the matrix 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>S</mi>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
</mrow></math>

.

<br/>
<br/>
<b><a name="rev_jac_sparse.r" id="rev_jac_sparse.r">r</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>rev_jac_sparse</span></font></i></code>
 return value 
<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;CppAD::vector&lt;&#xA0;std::set&lt;size_t&gt;&#xA0;&gt;&amp;&#xA0;</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>
and 
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
The input values of its sets 
are not specified (must not matter).
Upon return for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,
all the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>

are between zero and 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
 inclusive.
This represents a sparsity pattern for the matrix 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
</mrow></math>

.

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

is a user defined function

<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'>ok</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>rev_hes_sparse</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>id</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</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'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
There is an unspecified scalar valued function 

<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'>m</mi>
</msup>
<mo stretchy="false">&#x02192;</mo>
<mi mathvariant='italic'>B</mi>
</mrow></math>

.
Given a sparsity pattern for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
</mrow></math>


and information about the function 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
</mrow></math>

,
this routine computes the sparsity pattern for

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>V</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mrow></math>

<br/>
<b><a name="rev_hes_sparse.Usage" id="rev_hes_sparse.Usage">Usage</a></b>
<br/>
This routine is used by calls to <a href="revsparsehes.xml" target="_top"><span style='white-space: nowrap'>RevSparseHes</span></a>
.

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the number of columns in the sparsity patterns.

<br/>
<br/>
<b><a name="rev_hes_sparse.r" id="rev_hes_sparse.r">r</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>rev_hes_sparse</span></font></i></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;CppAD::vector&lt;&#xA0;std::set&lt;size_t&gt;&#xA0;&gt;&amp;&#xA0;</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>
and 
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
For 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,
all the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>r</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>
 are between
zero and 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
 inclusive.
This specifies a sparsity pattern for the matrix 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>

.

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&lt;bool&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
This specifies a sparsity pattern for the matrix 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mn>1</mn>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>m</mi>
</mrow>
</msup>
</mrow></math>

.

<br/>
<br/>
<b><a name="rev_hes_sparse.t" id="rev_hes_sparse.t">t</a></b>
<br/>
The 
<code><i><font color="black"><span style='white-space: nowrap'>rev_hes_sparse</span></font></i></code>
 argument 
<code><i><font color="black"><span style='white-space: nowrap'>t</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;bool&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
The input values of its elements 
are not specified (must not matter).
Upon return it represents a sparsity pattern for the matrix 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mn>1</mn>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>n</mi>
</mrow>
</msup>
</mrow></math>

 defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;CppAD::vector&lt;&#xA0;std::set&lt;size_t&gt;&#xA0;&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
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>

,
all the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>

are between zero and 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
 inclusive.
This specifies a sparsity pattern
for the matrix 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>U</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>

 defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>U</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">{</mo>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>y</mi>
</msub>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">]</mo>
<msub><mo stretchy="false">}</mo>
<mrow><mi mathvariant='italic'>u</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" >
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">{</mo>
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">]</mo>
<msub><mo stretchy="false">}</mo>
<mrow><mi mathvariant='italic'>u</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'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mtd></mtr></mtable>
</mrow></math>

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;CppAD::vector&lt;&#xA0;std::set&lt;size_t&gt;&#xA0;&gt;&amp;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and 
<code><i><font color="black"><span style='white-space: nowrap'>v</span></font></i><font color="blue"><span style='white-space: nowrap'>.size()&#xA0;&gt;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
The input values of its elements 
are not specified (must not matter).
Upon return, for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,
all the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>v</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]</span></font></code>

are between zero and 
<code><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
 inclusive.
This represents a sparsity pattern for the matrix 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>V</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x02208;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
</mrow></math>

 defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>V</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">[</mo>
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<msub><mo stretchy="false">]</mo>
<mrow><mi mathvariant='italic'>u</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" >
<msub><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>u</mi>
</msub>
<mo stretchy="false">[</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>R</mi>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<msub><mo stretchy="false">]</mo>
<mrow><mi mathvariant='italic'>u</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" >
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>g</mi>
<mo stretchy="false">&#x02218;</mo>
<mi mathvariant='italic'>f</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
<mo stretchy="false">+</mo>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant='italic'>m</mi>
</munderover>
<mo stretchy="false">[</mo>
<msup><mi mathvariant='italic'>g</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">)</mo>
<msub><mo stretchy="false">]</mo>
<mi mathvariant='italic'>i</mi>
</msub>
<mspace width='.3em'/>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>f</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>T</mi>
</mstyle></mrow>
</msup>
<mi mathvariant='italic'>U</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant='italic'>m</mi>
</munderover>
<mi mathvariant='italic'>S</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msub><mo stretchy="false">)</mo>
<mi mathvariant='italic'>i</mi>
</msub>
<mspace width='.3em'/>
<msubsup><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant='italic'>R</mi>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><big><a name="clear" id="clear">clear</a></big></b>
<br/>
User atomic functions hold onto static work space in order to
increase speed by avoiding system memory allocation calls.
The function call 
<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;user_atomic&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;::clear()<br/>
</span></font></code>
makes to work space <a href="ta_available.xml" target="_top"><span style='white-space: nowrap'>available</span></a>
 to
for other uses by the same thread.
This should be called when you are done using the 
user atomic functions for a specific value of 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
.

<br/>
<br/>
<b><a name="clear.Restriction" id="clear.Restriction">Restriction</a></b>
<br/>
The user atomic <code><font color="blue">clear</font></code> routine cannot be called
while in <a href="ta_in_parallel.xml" target="_top"><span style='white-space: nowrap'>parallel</span></a>
 execution mode.


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


<br/>
<br/>
<b><a name="Example.Simple" id="Example.Simple">Simple</a></b>
<br/>
The file <a href="old_reciprocal.cpp.xml" target="_top"><span style='white-space: nowrap'>old_reciprocal.cpp</span></a>
 contains the simplest example and test
of a user atomic operation.

<br/>
<br/>
<b><a name="Example.Use AD" id="Example.Use AD">Use AD</a></b>
<br/>
The examples 
<a href="old_usead_1.cpp.xml" target="_top"><span style='white-space: nowrap'>old_usead_1.cpp</span></a>
 and <a href="old_usead_2.cpp.xml" target="_top"><span style='white-space: nowrap'>old_usead_2.cpp</span></a>

use AD to compute the derivatives 
inside a user defined atomic function.
This may have the advantage of reducing the size of the tape, because
a repeated section of code would only be taped once.

<br/>
<br/>
<b><a name="Example.Tangent Function" id="Example.Tangent Function">Tangent Function</a></b>
<br/>
The file <a href="old_tan.cpp.xml" target="_top"><span style='white-space: nowrap'>old_tan.cpp</span></a>
 contains an example and test
implementation of the tangent function as a user atomic operation.

<br/>
<br/>
<b><a name="Example.Matrix Multiplication" id="Example.Matrix Multiplication">Matrix Multiplication</a></b>
<br/>
The file  <a href="old_mat_mul.cpp.xml" target="_top"><span style='white-space: nowrap'>old_mat_mul.cpp</span></a>
 contains an example and test
implementation of matrix multiplication a a user atomic operation.


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

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