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<a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a>
</td>
<td><a href="forwardone.xml" target="_top">Prev</a>
</td><td><a href="size_taylor.xml" target="_top">Next</a>
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<option>index</option>
<option>search</option>
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<option>Up-></option>
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<option>ADFun</option>
<option>FunEval</option>
<option>Forward</option>
<option>ForwardAny</option>
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<td>
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<option>ADFun-></option>
<option>Independent</option>
<option>FunConstruct</option>
<option>Dependent</option>
<option>abort_recording</option>
<option>seq_property</option>
<option>FunEval</option>
<option>Drivers</option>
<option>FunCheck</option>
<option>omp_max_thread</option>
<option>optimize</option>
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<option>FunEval-></option>
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<option>Reverse</option>
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<option>Forward-></option>
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<option>ForwardAny</option>
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<option>Forward.cpp</option>
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<td>ForwardAny</td>
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<option>Headings-></option>
<option>Syntax</option>
<option>Purpose</option>
<option>---..Function Values</option>
<option>---..Derivative Values</option>
<option>X(t)</option>
<option>Y(t)</option>
<option>f</option>
<option>p</option>
<option>x_p</option>
<option>y_p</option>
<option>Vector</option>
<option>Zero Order</option>
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<center><b><big><big>Any Order Forward Mode</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<br/>
<code><font color="blue"></font></code><i><span style='white-space: nowrap'>y_p</span></i><code><font color="blue"><span style='white-space: nowrap'> = </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font></code><i><span style='white-space: nowrap'>p</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>x_p</span></i><code><font color="blue"><span style='white-space: nowrap'> )</span></font></code>
<code><span style='white-space: nowrap'><br/>
</span></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 <i>f</i>.
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/>
<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="forwardzero.xml" target="_top"><span style='white-space: nowrap'>ForwardZero</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="forwardone.xml" target="_top"><span style='white-space: nowrap'>ForwardOne</span></a>
is simpler to understand
than this explanation of the general case.
<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
a sequence of 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'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
</mrow></math>
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'>p</mi>
</mrow></math>
,
the vector
<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>
above is defined as the value of
<i>x_k</i> in the previous call (counting this call) of the form
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font></code><i><span style='white-space: nowrap'>k</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>x_k</span></i><code><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>If there is no previous call with
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is the value of the independent variables when the
corresponding
AD of <i>Base</i>
<a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation sequence</span></a>
was recorded.
Note that
<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 related to the <i>k</i>-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>
by
<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 <i>k</i>-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'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<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'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</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'>p</mi>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mo stretchy="false">-</mo>
<mi mathvariant='italic'>p</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 <i>k</i>-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
<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="f" id="f">f</a></big></b>
<br/>
The <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
object <i>f</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     ADFun<</span></font></code><i><span style='white-space: nowrap'>Base</span></i><code><font color="blue"><span style='white-space: nowrap'>> </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><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 <i>f</i> is not <code><font color="blue">const</font></code>.
Before this call to <code><font color="blue">Forward</font></code>, the value returned by
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>f</span></i><code><font color="blue"><span style='white-space: nowrap'>.size_taylor()<br/>
</span></font></code>must be greater than or equal
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>
.
After this call it will be will be
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow></math>
(see <a href="size_taylor.xml" target="_top"><span style='white-space: nowrap'>size_taylor</span></a>
).
<br/>
<br/>
<b><big><a name="p" id="p">p</a></big></b>
<br/>
The argument <i>p</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     size_t </span></font></code><i><span style='white-space: nowrap'>p</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>and specifies the order of the Taylor coefficients to be calculated.
<br/>
<br/>
<b><big><a name="x_p" id="x_p">x_p</a></big></b>
<br/>
The argument <i>x_p</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>Vector</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>x_p</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>(see <a href="forwardany.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below)
and its size
must be equal to <i>n</i>, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
space for <i>f</i>.
The <i>p</i>-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 this value; i.e.,
<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'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">_</mo>
<mi mathvariant='italic'>p</mi>
</mrow></math>
.
(The lower order 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>
are
defined by previous calls to <code><font color="blue">Forward</font></code>.)
<br/>
<br/>
<b><big><a name="y_p" id="y_p">y_p</a></big></b>
<br/>
The return value <i>y_p</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>Vector</span></i><code><font color="blue"><span style='white-space: nowrap'> </span></font></code><i><span style='white-space: nowrap'>y_p</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>(see <a href="forwardany.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below)
and its value is
The <i>p</i>-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>
; i.e.,
<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'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>y</mi>
<mo stretchy="false">_</mo>
<mi mathvariant='italic'>p</mi>
</mrow></math>
.
The size of <i>y_p</i>
is equal to <i>m</i>, the dimension of the
<a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>
space for <i>f</i>.
<br/>
<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
The type <i>Vector</i> 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>
<i>Base</i>.
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/>
In the case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
,
the result <i>y_p</i> is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">)</mo>
<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" >
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">[</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">]</mo>
</mtd></mtr></mtable>
</mrow></math>
The agrees with the simplification where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
above are replaced by
<code><font color="blue">0</font></code>,
<i>x</i>, and
<i>y</i>
in <a href="forwardzero.xml" target="_top"><span style='white-space: nowrap'>ForwardZero</span></a>
.
<br/>
<br/>
<b><big><a name="First Order" id="First Order">First Order</a></big></b>
<br/>
In the case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow></math>
,
the result <i>y_p</i> is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>X</mi>
<msup><mo stretchy="false">)</mo>
<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>
</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>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>X</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<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" >
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>
The agrees with the simplification where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
above are replaced by
<code><font color="blue">1</font></code>,
<i>x</i>,
<i>dx</i>, and
<i>dy</i>
in <a href="forwardone.xml" target="_top"><span style='white-space: nowrap'>ForwardOne</span></a>
.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>Note that if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is the <i>j</i>-th
<a href="glossary.xml#Elementary Vector" target="_top"><span style='white-space: nowrap'>elementary vector</span></a>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>F</mi>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<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>
</mrow></math>
<br/>
<b><big><a name="Second Order" id="Second Order">Second Order</a></big></b>
<br/>
In the case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">=</mo>
<mn>2</mn>
</mrow></math>
,
the <i>i</i>-th element of
the result <i>y_p</i> is given by
<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>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</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>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>X</mi>
<msup><mo stretchy="false">)</mo>
<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>
<mrow><mo stretchy="true">[</mo><mrow><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>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<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>
<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>
<msup><mo stretchy="false">)</mo>
<mi mathvariant='italic'>T</mi>
</msup>
<mo stretchy="false">*</mo>
<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>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>X</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
</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>
<mrow><mo stretchy="true">[</mo><mrow><mn>2</mn>
<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>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mo stretchy="false">)</mo>
<mi mathvariant='italic'>T</mi>
</msup>
<mo stretchy="false">*</mo>
<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>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow><mo stretchy="true">]</mo></mrow>
</mtd></mtr></mtable>
</mrow></math>
Note that if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is the <i>j</i>-th
<a href="glossary.xml#Elementary Vector" target="_top"><span style='white-space: nowrap'>elementary vector</span></a>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is zero,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>2</mn>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd></mtr></mtable>
</mrow></math>
If
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is the sum of the <i>j</i>-th and <i>l</i>-th
<a href="glossary.xml#Elementary Vector" target="_top"><span style='white-space: nowrap'>elementary vectors</span></a>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is zero,
<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>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
</mrow>
</mfrac>
<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>
</mrow><mo stretchy="true">]</mo></mrow>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo stretchy="false">-</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">-</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mn>2</mn>
</msup>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>
<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>
contains an example and test of this operation.
It returns true if it succeeds and false otherwise.
<hr/>Input File: omh/forward.omh
</body>
</html>
