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<center><b><big><big>An Arbitrary Order Gear Method</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<br/>
<code><font color="blue"><span style='white-space: nowrap'># include <cppad/ode_gear.hpp><br/>
</span></font></code>
<code><font color="blue"><span style='white-space: nowrap'>OdeGear(</span></font><i><font color="black"><span style='white-space: nowrap'>F</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'>, </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'>T</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>X</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>e</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/>
This routine applies
<a href="odegear.xml#Gear's Method" target="_top"><span style='white-space: nowrap'>Gear's Method</span></a>
to solve an explicit set of ordinary differential equations.
We are given
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">:</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mo stretchy="false">×</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">→</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
be a smooth function.
This routine solves the following initial value problem
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>x</mi>
<mo stretchy="false">′</mo>
</msup>
<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" >
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>t</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></mtable>
</mrow></math>
for the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow></math>
.
If your set of ordinary differential equations are not stiff
an explicit method may be better (perhaps <a href="runge45.xml" target="_top"><span style='white-space: nowrap'>Runge45</span></a>
.)
<br/>
<br/>
<b><big><a name="Include" id="Include">Include</a></big></b>
<br/>
The file <code><font color="blue">cppad/ode_gear.hpp</font></code> is included by <code><font color="blue">cppad/cppad.hpp</font></code>
but it can also be included separately with out the rest of
the <code><font color="blue">CppAD</font></code> routines.
<br/>
<br/>
<b><big><a name="Fun" id="Fun">Fun</a></big></b>
<br/>
The class
<code><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i></code>
and the object
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i></code>
satisfy the prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>Fun</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>
This must support the following set of calls
<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'>.Ode(</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </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><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>.Ode_dep(</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>f_x</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><a name="Fun.t" id="Fun.t">t</a></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>t</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see description of <a href="odegear.xml#Scalar" target="_top"><span style='white-space: nowrap'>Scalar</span></a>
below).
<br/>
<br/>
<b><a name="Fun.x" id="Fun.x">x</a></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     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'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and has size
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
(see description of <a href="odegear.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below).
<br/>
<br/>
<b><a name="Fun.f" id="Fun.f">f</a></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
to
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>.Ode</span></font></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'>f</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
On input and output,
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
is a vector of size
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
and the input values of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
do not matter.
On output,
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
is set equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
(see
<code><i><font color="black"><span style='white-space: nowrap'>f(t, x)</span></font></i></code>
in <a href="odegear.xml#Purpose" target="_top"><span style='white-space: nowrap'>Purpose</span></a>
).
<br/>
<br/>
<b><a name="Fun.f_x" id="Fun.f_x">f_x</a></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>f_x</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'>f_x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
On input and output,
<code><i><font color="black"><span style='white-space: nowrap'>f_x</span></font></i></code>
is a vector of size
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
and the input values of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>f_x</span></font></i></code>
do not matter.
On output,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">_</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<msub><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>x</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<msub><mi mathvariant='italic'>f</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<b><a name="Fun.Warning" id="Fun.Warning">Warning</a></b>
<br/>
The arguments
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
, and
<code><i><font color="black"><span style='white-space: nowrap'>f_x</span></font></i></code>
must have a call by reference in their prototypes; i.e.,
do not forget the <code><font color="blue">&</font></code> in the prototype for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>f_x</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="m" id="m">m</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     size_t </span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the order (highest power of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
)
used to represent 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'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
in the multi-step method.
Upon return from <code><font color="blue">OdeGear</font></code>,
the <code><i>i</i></code>-th component of the polynomial is defined by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>p</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
</mrow></math>
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>m</mi>
</mrow></math>
(where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mn>0</mn>
<mo stretchy="false">≤</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false"><</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
).
The value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
</mrow></math>
must be greater than or equal one.
<br/>
<br/>
<b><big><a name="n" id="n">n</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     size_t </span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the range space dimension of the
vector valued 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'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="T" id="T">T</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>T</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     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'>T</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and size greater than or equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow></math>
.
For
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mi mathvariant='italic'>m</mi>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow></math>
is the time
corresponding to time corresponding
to a previous point in the multi-step method.
The value
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">]</mo>
</mrow></math>
is the time
of the next point in the multi-step method.
The array
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>T</mi>
</mrow></math>
must be monotone increasing; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false"><</mo>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mrow></math>
.
Above and below we often use the shorthand
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>T</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="X" id="X">X</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>X</span></font></i></code>
has the 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'>X</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and size greater than or equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
.
On input to <code><font color="blue">OdeGear</font></code>,
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow></math>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<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'>n</mi>
<mn>-1</mn>
</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'>j</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow></math>
Upon return from <code><font color="blue">OdeGear</font></code>,
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">≈</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<b><big><a name="e" id="e">e</a></big></b>
<br/>
The vector
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
is an approximate error bound for the result; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>e</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">≥</mo>
<mo stretchy="false">|</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">|</mo>
</mrow></math>
The order of this approximation is one less than the order of
the solution; i.e.,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>e</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>O</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>h</mi>
<mi mathvariant='italic'>m</mi>
</msup>
<mo stretchy="false">)</mo>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>h</mi>
</mrow></math>
is the maximum of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="Scalar" id="Scalar">Scalar</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i></code>
must satisfy the conditions
for a <a href="numerictype.xml" target="_top"><span style='white-space: nowrap'>NumericType</span></a>
type.
The routine <a href="checknumerictype.xml" target="_top"><span style='white-space: nowrap'>CheckNumericType</span></a>
will generate an error message
if this is not the case.
In addition, the following operations must be defined for
<code><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i></code>
objects
<code><i><font color="black"><span style='white-space: nowrap'>a</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>b</span></font></i></code>
:
<table><tr><td align='left' valign='top'>
<b>Operation</b> </td><td align='left' valign='top'>
<b>Description</b> </td></tr><tr><td align='left' valign='top'>
<code><i><font color="black"><span style='white-space: nowrap'>a</span></font></i><font color="blue"><span style='white-space: nowrap'> < </span></font><i><font color="black"><span style='white-space: nowrap'>b</span></font></i></code>
</td><td align='left' valign='top'>
less than operator (returns a <code><font color="blue">bool</font></code> object)
</td></tr>
</table>
<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 Scalar</span></a>
.
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="ode_gear.cpp.xml" target="_top"><span style='white-space: nowrap'>ode_gear.cpp</span></a>
contains an example and test a test of using this routine.
It returns true if it succeeds and false otherwise.
<br/>
<br/>
<b><big><a name="Source Code" id="Source Code">Source Code</a></big></b>
<br/>
The source code for this routine is in the file
<code><font color="blue">cppad/ode_gear.hpp</font></code>.
<br/>
<br/>
<b><big><a name="Theory" id="Theory">Theory</a></big></b>
<br/>
For this discussion we use the shorthand
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>
for the value
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
which is not to be confused
with
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">∈</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
</mrow></math>
in the notation above.
The interpolating polynomial
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>m</mi>
</munderover>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow><munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow></math>
The derivative
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>p</mi>
<mo stretchy="false">′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>
is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>p</mi>
<mo stretchy="false">′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>m</mi>
</munderover>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow><munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow></math>
Evaluating the derivative at the point
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
</mrow></math>
we have
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>p</mi>
<mo stretchy="false">′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow><munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow><munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow><munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>x</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
</mrow>
</munder>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
<munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mo stretchy="false">ℓ</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">ℓ</mo>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>
We define the vector
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>α</mi>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow></math>
by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='normal'>α</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>m</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>m</mi>
</mtd></mtr><mtr><mtd columnalign="left" >
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
<munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>otherwise</mi>
</mstyle></mrow>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>p</mi>
<mo stretchy="false">′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='normal'>α</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>α</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</msub>
</mrow></math>
Gear's method determines
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</msub>
</mrow></math>
by solving the following
nonlinear equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='normal'>α</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>α</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</msub>
</mrow></math>
Newton's method for solving this equation determines iterates,
which we denote by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mi mathvariant='italic'>k</mi>
</msubsup>
</mrow></math>
, by solving the following affine
approximation of the equation above
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mi mathvariant='italic'>k</mi>
</msubsup>
<mo stretchy="false">-</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='normal'>α</mi>
<mn>0</mn>
</msub>
<msubsup><mi mathvariant='italic'>x</mi>
<mn>0</mn>
<mi mathvariant='italic'>k</mi>
</msubsup>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>α</mi>
<mn>1</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>α</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
<mrow><mo stretchy="true">[</mo><mrow><msub><mi mathvariant='normal'>α</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mi mathvariant='italic'>I</mi>
<mo stretchy="false">-</mo>
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</msub>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mrow><mo stretchy="true">[</mo><mrow><mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<mo stretchy="false">-</mo>
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>x</mi>
</msub>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mo stretchy="false">,</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msubsup>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='normal'>α</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">-</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='normal'>α</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
</mrow><mo stretchy="true">]</mo></mrow>
</mtd></mtr></mtable>
</mrow></math>
In order to initialize Newton's method; i.e. choose
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mn>0</mn>
</msubsup>
</mrow></math>
we define the vector
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>β</mi>
<mo stretchy="false">∈</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mrow><mi mathvariant='italic'>m</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow></math>
by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='normal'>β</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<munder><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mtd></mtr><mtr><mtd columnalign="left" >
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
<munder><mo displaystyle='true' largeop='true'>∏</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">≠</mo>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>otherwise</mi>
</mstyle></mrow>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>p</mi>
<mo stretchy="false">′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='normal'>β</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>β</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</msub>
</mrow></math>
We solve the following approximation of the equation above to determine
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mn>0</mn>
</msubsup>
</mrow></math>
:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">,</mo>
<msub><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='normal'>β</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>β</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>β</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<msubsup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
<mn>0</mn>
</msubsup>
</mrow></math>
<br/>
<b><big><a name="Gear's Method" id="Gear's Method">Gear's Method</a></big></b>
<br/>
C. W. Gear,
``Simultaneous Numerical Solution of Differential-Algebraic Equations,''
IEEE Transactions on Circuit Theory,
vol. 18, no. 1, pp. 89-95, Jan. 1971.
<hr/>Input File: cppad/ode_gear.hpp
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