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<option>---..Function Values</option>
<option>---..Derivative Values</option>
<option>Notation</option>
<option>---..n</option>
<option>---..m</option>
<option>f</option>
<option>One Order</option>
<option>q</option>
<option>xq</option>
<option>---..One Order</option>
<option>---..Multiple Orders</option>
<option>---..Restrictions</option>
<option>s</option>
<option>X(t)</option>
<option>Y(t)</option>
<option>yq</option>
<option>---..One Order</option>
<option>---..Multiple Orders</option>
<option>Vector</option>
<option>Zero Order</option>
<option>First Order</option>
<option>Second Order</option>
<option>Example</option>
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<center><b><big><big>Multiple Order Forward Mode</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<br/>
<code><i><font color="black"><span style='white-space: nowrap'>yq</span></font></i><font color="blue"><span style='white-space: nowrap'> = </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>xq</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'>yq</span></font></i><font color="blue"><span style='white-space: nowrap'> = </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>, </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>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
We use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
to denote the
<a href="glossary.xml#AD Function" target="_top"><span style='white-space: nowrap'>AD function</span></a>
corresponding to
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
Given a function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
,
defined by its
<a href="glossary.xml#Taylor Coefficient" target="_top"><span style='white-space: nowrap'>Taylor coefficients</span></a>
,
forward mode computes the Taylor coefficients for the 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'>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>
<br/>
<b><a name="Purpose.Function Values" id="Purpose.Function Values">Function Values</a></b>
<br/>
If you are using forward mode to compute values 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>
,
<a href="forward_zero.xml" target="_top"><span style='white-space: nowrap'>forward_zero</span></a>
is simpler to understand
than this explanation of the general case.
<br/>
<br/>
<b><a name="Purpose.Derivative Values" id="Purpose.Derivative Values">Derivative Values</a></b>
<br/>
If you are using forward mode to compute values for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<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'>dx</mi>
</mrow></math>
,
<a href="forward_one.xml" target="_top"><span style='white-space: nowrap'>forward_one</span></a>
is simpler to understand
than this explanation of the general case.
<br/>
<br/>
<b><big><a name="Notation" id="Notation">Notation</a></big></b>
<br/>
<br/>
<b><a name="Notation.n" id="Notation.n">n</a></b>
<br/>
We use
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
to denote the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
space for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
<br/>
<br/>
<b><a name="Notation.m" id="Notation.m">m</a></b>
<br/>
We use
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
to denote the dimension of the
<a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>
space for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="f" id="f">f</a></big></b>
<br/>
The <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
object
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     ADFun<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>> </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
Note that the <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
object
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
is not <code><font color="blue">const</font></code>.
After this call we will have
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_order()     == </span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'> + 1<br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_direction() == 1<br/>
</span></font></code>
<br/>
<b><big><a name="One Order" id="One Order">One Order</a></big></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
then we are only computing one order.
In this case, before this call we must have
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_order()     >= </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><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_direction() == 1<br/>
</span></font></code>
<br/>
<b><big><a name="q" id="q">q</a></big></b>
<br/>
The 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/>
     size_t </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>
and specifies the highest order of the Taylor coefficients to be calculated.
<br/>
<br/>
<b><big><a name="xq" id="xq">xq</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="forward_order.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below).
As above, we use
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
to denote the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
space for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
The size of
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i></code>
must be either
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
or
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>*(</span></font><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>
.
After this call we will have
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_order()     == </span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'> + 1<br/>
</span></font></code>
<br/>
<b><a name="xq.One Order" id="xq.One Order">One Order</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
the <code><i>q</i></code>-th order Taylor coefficient 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>
is defined by
<code><span style='white-space: nowrap'><br/>
     </span></code>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
</mrow></math>
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i></code>
.
For
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mrow></math>
,
the Taylor coefficient
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is defined by
<code><i><font color="black"><span style='white-space: nowrap'>xk</span></font></i></code>
in the previous call to
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>xk</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><a name="xq.Multiple Orders" id="xq.Multiple Orders">Multiple Orders</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>*(</span></font><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>
,
For
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
</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">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>
,
the <code><i>j</i></code>-th component of the <code><i>k</i></code>-th order Taylor coefficient
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>
is defined by
<code><span style='white-space: nowrap'><br/>
     </span></code>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">=</mo>
</mrow></math>
<code><i><font color="black"></font></i><font color="blue"><span style='white-space: nowrap'>xq</span></font><i><font color="black"><span style='white-space: nowrap'>[ (</span></font></i><font color="blue"><span style='white-space: nowrap'>q</span></font><i><font color="black"><span style='white-space: nowrap'>+1) * </span></font></i><font color="blue"><span style='white-space: nowrap'>j</span></font><i><font color="black"><span style='white-space: nowrap'> + </span></font></i><font color="blue"><span style='white-space: nowrap'>k</span></font><i><font color="black"><span style='white-space: nowrap'> ]</span></font></i></code>
<br/>
<br/>
<b><a name="xq.Restrictions" id="xq.Restrictions">Restrictions</a></b>
<br/>
Note if
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
uses <a href="old_atomic.xml" target="_top"><span style='white-space: nowrap'>old_atomic</span></a>
functions,
the size of
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i></code>
must be
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="s" id="s">s</a></big></b>
<br/>
If the argument
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
is not present, <code><font color="blue">std::cout</font></code>
is used in its place.
Otherwise, this argument has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     std::ostream& </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>
If order zero is begin calculated,
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
specifies where the output corresponding to <a href="printfor.xml" target="_top"><span style='white-space: nowrap'>PrintFor</span></a>
will be written.
If order zero is not being calculated,
<code><i><font color="black"><span style='white-space: nowrap'>s</span></font></i></code>
is not used
<br/>
<br/>
<b><big><a name="X(t)" id="X(t)">X(t)</a></big></b>
<br/>
The function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
is defined using
the Taylor coefficients
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</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>
<msup><mi mathvariant='italic'>t</mi>
<mn>1</mn>
</msup>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>q</mi>
</msup>
</mrow></math>
Note that for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
</mrow></math>
,
the <code><i>k</i></code>-th derivative 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>
is related to the
Taylor coefficients by the equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>X</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<b><big><a name="Y(t)" id="Y(t)">Y(t)</a></big></b>
<br/>
The function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
is defined by
<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>
.
We use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
to denote the <code><i>k</i></code>-th order Taylor coefficient 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>
; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mn>0</mn>
</msup>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mn>1</mn>
</msup>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>q</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>q</mi>
</msup>
<mo stretchy="false">)</mo>
</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'>q</mi>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mo stretchy="false">-</mo>
<mi mathvariant='italic'>q</mi>
</mrow>
</msup>
<mo stretchy="false">→</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">→</mo>
<mn>0</mn>
</mrow></math>
.
Note that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is related to
the <code><i>k</i></code>-th derivative 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>
by the equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>Y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<b><big><a name="yq" id="yq">yq</a></big></b>
<br/>
The return value
<code><i><font color="black"><span style='white-space: nowrap'>yq</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'> </span></font><i><font color="black"><span style='white-space: nowrap'>yq</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="forward_order.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below).
<br/>
<br/>
<b><a name="yq.One Order" id="yq.One Order">One Order</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
the vector
<code><i><font color="black"><span style='white-space: nowrap'>yq</span></font></i></code>
has size
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
.
The <code><i>q</i></code>-th order Taylor coefficient 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>
is returned as
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>yq</span></font></i></code>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">=</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
.
<br/>
<br/>
<b><a name="yq.Multiple Orders" id="yq.Multiple Orders">Multiple Orders</a></b>
<br/>
If
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>*(</span></font><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>
,
the vector
<code><i><font color="black"><span style='white-space: nowrap'>yq</span></font></i></code>
has size
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>*(</span></font><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>
.
For
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
</mrow></math>
,
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow></math>
,
the <code><i>i</i></code>-th component of the <code><i>k</i></code>-th order Taylor coefficient
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>
is returned as
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>yq</span></font></i><font color="blue"><span style='white-space: nowrap'>[ (</span></font><i><font color="black"><span style='white-space: nowrap'>q</span></font></i><font color="blue"><span style='white-space: nowrap'>+1) * </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><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'> ]</span></font></code>
<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>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow></math>
<br/>
<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
must be a <a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
class with
<a href="simplevector.xml#Elements of Specified Type" target="_top"><span style='white-space: nowrap'>elements of type</span></a>
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
.
The routine <a href="checksimplevector.xml" target="_top"><span style='white-space: nowrap'>CheckSimpleVector</span></a>
will generate an error message
if this is not the case.
<br/>
<br/>
<b><big><a name="Zero Order" id="Zero Order">Zero Order</a></big></b>
<br/>
The case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
and
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
corresponds to the zero order
<a href="forward_zero.xml#Special Case" target="_top"><span style='white-space: nowrap'>special case</span></a>
.
<br/>
<br/>
<b><big><a name="First Order" id="First Order">First Order</a></big></b>
<br/>
The case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
and
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
corresponds to the first order
<a href="forward_one.xml#Special Case" target="_top"><span style='white-space: nowrap'>special case</span></a>
.
<br/>
<br/>
<b><big><a name="Second Order" id="Second Order">Second Order</a></big></b>
<br/>
The case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">=</mo>
<mn>2</mn>
</mrow></math>
and
<code><i><font color="black"><span style='white-space: nowrap'>xq</span></font></i><font color="blue"><span style='white-space: nowrap'>.size() == </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
corresponds to the second order
<a href="forward_two.xml#Special Case" target="_top"><span style='white-space: nowrap'>special case</span></a>
.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="forward.cpp.xml" target="_top"><span style='white-space: nowrap'>forward.cpp</span></a>
( <a href="forward_order.cpp.xml" target="_top"><span style='white-space: nowrap'>forward_order.cpp</span></a>
)
contains an example and test using one order (multiple orders).
They return true if they succeed and false otherwise.
<hr/>Input File: omh/forward/forward_order.omh
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