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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'>#&#xA0;include&#xA0;&lt;cppad/ode_gear.hpp&gt;<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'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>T</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>X</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</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&#xA0;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">&#x000D7;</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">&#x02192;</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">&#x02032;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</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'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>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'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</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,&#xA0;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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</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">&#x02202;</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">&amp;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It 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 <i>i</i>-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">&#x02026;</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">&#x02264;</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">&lt;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It 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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</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">&#x02026;</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">&lt;</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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</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">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow></math>

, and

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

 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="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">&#x02248;</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">&#x02265;</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'>&#xA0;&lt;&#xA0;</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&#xA0;of&#xA0;type&#xA0;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">&#x02208;</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">&#x02208;</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'>&#x02211;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02260;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02260;</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">&#x02032;</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">&#x02032;</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'>&#x02211;</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'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02260;</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<munder><mo displaystyle='true' largeop='true'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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">&#x02113;</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">&#x02032;</mo>
</msup>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02113;</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">&#x02113;</mo>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</mo>
</mrow>
</munder>
<munder><mo displaystyle='true' largeop='true'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02113;</mo>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02113;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</mo>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02113;</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'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</mo>
</mrow>
</munder>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02260;</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</munder>
<munder><mo displaystyle='true' largeop='true'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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">&#x02113;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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">&#x02113;</mo>
</msub>
<munder><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</mo>
</mrow>
</munder>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02113;</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'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</mo>
</mrow>
</munder>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mfrac><mrow><munder><mo displaystyle='true' largeop='true'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</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">&#x02113;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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">&#x02113;</mo>
</msub>
<munder><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</mo>
</mrow>
</munder>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>t</mi>
<mo stretchy="false">&#x02113;</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'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</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">&#x02113;</mo>
</msub>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
</mrow>
</msup>
<munder><mo displaystyle='true' largeop='true'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</mo>
<mo stretchy="false">&#x02113;</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">&#x02113;</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'>&#x003B1;</mi>
<mo stretchy="false">&#x02208;</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'>&#x003B1;</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'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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">&#x02032;</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'>&#x003B1;</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>&#x003B1;</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'>&#x003B1;</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>&#x003B1;</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">&#x02202;</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'>&#x003B1;</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'>&#x003B1;</mi>
<mn>1</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>&#x003B1;</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'>&#x003B1;</mi>
<mi mathvariant='italic'>m</mi>
</msub>
<mi mathvariant='italic'>I</mi>
<mo stretchy="false">-</mo>
<msub><mo stretchy="false">&#x02202;</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">&#x02202;</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'>&#x003B1;</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">-</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='normal'>&#x003B1;</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'>&#x003B2;</mi>
<mo stretchy="false">&#x02208;</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'>&#x003B2;</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'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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'>&#x0220F;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02260;</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">&#x02032;</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'>&#x003B2;</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>&#x003B2;</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'>&#x003B2;</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='normal'>&#x003B2;</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'>&#x003B2;</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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