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<center><b><big><big>Second Order Forward Mode: Derivative Values</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'>y2</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(1, </span></font><i><font color="black"><span style='white-space: nowrap'>x2</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
We use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">→</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>
.
The result of the syntax above is that for
<code><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'> = 0 , </span></font><i><font color="black"><span style='white-space: nowrap'>...</span></font></i><font color="blue"><span style='white-space: nowrap'> , </span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>-1</span></font></code>
,
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>y2</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>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<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>
<mi mathvariant='italic'>x</mi>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>x</mi>
<mn>2</mn>
<mo stretchy="false">+</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<mi mathvariant='italic'>x</mi>
<msup><mn>1</mn>
<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>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>x</mi>
<mn>1</mn>
</mrow></math>
<code><span style='white-space: nowrap'><br/>
</span></code>where
<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>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>
is the Jacobian of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
</mrow></math>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<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>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow></math>
is the Hessian of th <code><i>i</i></code>-th component of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
</mrow></math>
,
evaluated at
<code><i><font color="black"><span style='white-space: nowrap'>x0</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="f" id="f">f</a></big></b>
<br/>
The 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>.
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><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.size_order()<br/>
</span></font></code>
must be greater than or equal two.
After this call it will be will be three (see <a href="size_order.xml" target="_top"><span style='white-space: nowrap'>size_order</span></a>
).
<br/>
<br/>
<b><big><a name="x0" id="x0">x0</a></big></b>
<br/>
The vector
<code><i><font color="black"><span style='white-space: nowrap'>x0</span></font></i></code>
in the formula for
<code><i><font color="black"><span style='white-space: nowrap'>y2</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>
corresponds to the previous call to <a href="forward_zero.xml" target="_top"><span style='white-space: nowrap'>forward_zero</span></a>
using this ADFun object
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
; i.e.,
<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(0, </span></font><i><font color="black"><span style='white-space: nowrap'>x0</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
If there is no previous call with the first argument zero,
the value of the <a href="independent.xml" target="_top"><span style='white-space: nowrap'>independent</span></a>
variables
during the recording of the AD sequence of operations is used
for
<code><i><font color="black"><span style='white-space: nowrap'>x0</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="x1" id="x1">x1</a></big></b>
<br/>
The vector
<code><i><font color="black"><span style='white-space: nowrap'>x1</span></font></i></code>
in the formula for
<code><i><font color="black"><span style='white-space: nowrap'>y2</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>
corresponds to the previous call to <a href="forward_one.xml" target="_top"><span style='white-space: nowrap'>forward_one</span></a>
using this ADFun object
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
; i.e.,
<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(1, </span></font><i><font color="black"><span style='white-space: nowrap'>x1</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><big><a name="x2" id="x2">x2</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>x2</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'>x2</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="forward_two.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below)
and its size must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
space for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="y2" id="y2">y2</a></big></b>
<br/>
The result
<code><i><font color="black"><span style='white-space: nowrap'>y2</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'>y2</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="forward_two.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below)
The size of
<code><i><font color="black"><span style='white-space: nowrap'>y1</span></font></i></code>
is equal to
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>
space for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
Its value is given element-wise by the formula in the
<a href="forward_two.xml#Purpose" target="_top"><span style='white-space: nowrap'>purpose</span></a>
above.
<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="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.
<br/>
<br/>
<b><big><a name="Special Case" id="Special Case">Special Case</a></big></b>
<br/>
This is special case of <a href="forward_order.xml" target="_top"><span style='white-space: nowrap'>forward_order</span></a>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<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>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>X</mi>
<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" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<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>
<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>
<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>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Y</mi>
<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" >
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<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>
<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>
</mtd></mtr></mtable>
</mrow></math>
and
<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>
.
For this special case,
<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>
,
<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>
<code><font color="blue"><span style='white-space: nowrap'>= </span></font><i><font color="black"><span style='white-space: nowrap'>x0</span></font></i></code>
,
<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>
<code><font color="blue"><span style='white-space: nowrap'>= </span></font><i><font color="black"><span style='white-space: nowrap'>x1</span></font></i></code>
,
<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>
<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>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</msup>
</mrow></math>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<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>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">=</mo>
<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>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">]</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">)</mo>
</mrow></math>
Restricting our attention to the <code><i>i</i></code>-th component, and
taking the derivative with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
, we obtain
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msubsup><mi mathvariant='italic'>y</mi>
<mi mathvariant='italic'>i</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<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>
<mi mathvariant='italic'>t</mi>
<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>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</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>
<mo stretchy="false">+</mo>
<mn>2</mn>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
Taking a second derivative with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
,
and evaluating 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>
, we obtain
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<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>
<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>
<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>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</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>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mn>2</mn>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
which agrees with the specification for
<code><i><font color="black"><span style='white-space: nowrap'>y2</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>
in the
<a href="forward_two.xml#Purpose" target="_top"><span style='white-space: nowrap'>purpose</span></a>
above.
<hr/>Input File: omh/forward/forward_two.omh
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