<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>Checkpointing an ODE Solver: Example and Test</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="Checkpointing an ODE Solver: Example and Test"/>
<meta name="keywords" id="keywords" content=" checkpointing ode solver: example test see also purpose problem solver solution "/>
<style type='text/css'>
body { color : black }
body { background-color : white }
A:link { color : blue }
A:visited { color : purple }
A:active { color : purple }
</style>
<script type='text/javascript' language='JavaScript' src='_checkpoint_ode.cpp_xml.js'>
</script>
</head>
<body>
<table><tr>
<td>
<a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a>
</td>
<td><a href="atomic_mul_level.cpp.xml" target="_top">Prev</a>
</td><td><a href="checkpoint_extended_ode.cpp.xml" target="_top">Next</a>
</td><td>
<select onchange='choose_across0(this)'>
<option>Index-></option>
<option>contents</option>
<option>reference</option>
<option>index</option>
<option>search</option>
<option>external</option>
</select>
</td>
<td>
<select onchange='choose_up0(this)'>
<option>Up-></option>
<option>CppAD</option>
<option>AD</option>
<option>ADValued</option>
<option>atomic</option>
<option>checkpoint</option>
<option>checkpoint_ode.cpp</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>ADValued-></option>
<option>Arithmetic</option>
<option>unary_standard_math</option>
<option>binary_math</option>
<option>CondExp</option>
<option>Discrete</option>
<option>numeric_limits</option>
<option>atomic</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>atomic-></option>
<option>checkpoint</option>
<option>atomic_base</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>checkpoint-></option>
<option>checkpoint.cpp</option>
<option>atomic_mul_level.cpp</option>
<option>checkpoint_ode.cpp</option>
<option>checkpoint_extended_ode.cpp</option>
</select>
</td>
<td>checkpoint_ode.cpp</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-></option>
<option>See Also</option>
<option>Purpose</option>
<option>Problem</option>
<option>ODE Solver</option>
<option>ODE</option>
<option>Solution</option>
</select>
</td>
</tr></table><br/>
<center><b><big><big>Checkpointing an ODE Solver: Example and Test</big></big></b></center>
<br/>
<b><big><a name="See Also" id="See Also">See Also</a></big></b>
<br/>
<a href="checkpoint_extended_ode.cpp.xml" target="_top"><span style='white-space: nowrap'>checkpoint_extended_ode.cpp</span></a>
,
<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
In this example we <a href="checkpoint.xml" target="_top"><span style='white-space: nowrap'>checkpoint</span></a>
one step of an ODE solver.
<br/>
<br/>
<b><big><a name="Problem" id="Problem">Problem</a></big></b>
<br/>
We consider the initial value problem with parameter
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>
defined by,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>z</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
Note that if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
needs to be in the equation, one can define
the first component of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</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>
to be equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="ODE Solver" id="ODE Solver">ODE Solver</a></big></b>
<br/>
For this example, we consider the Fourth order Runge-Kutta ODE solver.
Given an approximation solution at time
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
</mrow></math>
denoted by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>Δ</mi>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">-</mo>
<msub><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msub>
</mrow></math>
,
it defines the approximation solution
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
at time
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow></math>
by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>h</mi>
<mn>1</mn>
</msub>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>h</mi>
<mn>2</mn>
</msub>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='normal'>Δ</mi>
<mi mathvariant='italic'>t</mi>
<mspace width='.3em'/>
<msub><mi mathvariant='italic'>h</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">/</mo>
<mn>2</mn>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>h</mi>
<mn>3</mn>
</msub>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='normal'>Δ</mi>
<mi mathvariant='italic'>t</mi>
<mspace width='.3em'/>
<msub><mi mathvariant='italic'>h</mi>
<mn>2</mn>
</msub>
<mo stretchy="false">/</mo>
<mn>2</mn>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>h</mi>
<mn>4</mn>
</msub>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='normal'>Δ</mi>
<mi mathvariant='italic'>t</mi>
<mspace width='.3em'/>
<msub><mi mathvariant='italic'>h</mi>
<mn>3</mn>
</msub>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='normal'>Δ</mi>
<mi mathvariant='italic'>t</mi>
<mspace width='.3em'/>
<mo stretchy="false">(</mo>
<msub><mi mathvariant='italic'>h</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">+</mo>
<mn>2</mn>
<msub><mi mathvariant='italic'>h</mi>
<mn>2</mn>
</msub>
<mo stretchy="false">+</mo>
<mn>2</mn>
<msub><mi mathvariant='italic'>h</mi>
<mn>3</mn>
</msub>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>h</mi>
<mn>4</mn>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mn>6</mn>
</mtd></mtr></mtable>
</mrow></math>
If
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>z</mi>
<mi mathvariant='italic'>k</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mover accent='false'><mrow><mi mathvariant='italic'>z</mi>
</mrow>
<mo stretchy='false'>˜</mo></mover>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msub><mi mathvariant='italic'>z</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>O</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='normal'>Δ</mi>
<msup><mi mathvariant='italic'>t</mi>
<mn>5</mn>
</msup>
<mo stretchy="false">)</mo>
</mrow></math>
.
Other ODE solvers can use a similar method to the one used below.
<br/>
<br/>
<b><big><a name="ODE" id="ODE">ODE</a></big></b>
<br/>
For this example the ODE is defined by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">(</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
</mtd></mtr><mtr><mtd columnalign="center" >
<msub><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msub>
<msub><mi mathvariant='italic'>z</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="center" >
<mo stretchy="false">⋮</mo>
</mtd></mtr><mtr><mtd columnalign="center" >
<msub><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow>
</msub>
<msub><mi mathvariant='italic'>z</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">(</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<msub><mi mathvariant='italic'>z</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="center" >
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<msub><mi mathvariant='italic'>z</mi>
<mn>1</mn>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="center" >
<mo stretchy="false">⋮</mo>
</mtd></mtr><mtr><mtd columnalign="center" >
<msub><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>t</mi>
</msub>
<msub><mi mathvariant='italic'>z</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
<br/>
<b><big><a name="Solution" id="Solution">Solution</a></big></b>
<br/>
The solution of the ODE for this example,
which is used to check the results,
can be calculated by
starting with the first row and then using the solution
for the first row to solve the second and so on.
Doing this we obtain
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">(</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<mi mathvariant='italic'>t</mi>
</mtd></mtr><mtr><mtd columnalign="center" >
<msub><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<msup><mi mathvariant='italic'>t</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mtd></mtr><mtr><mtd columnalign="center" >
<mo stretchy="false">⋮</mo>
</mtd></mtr><mtr><mtd columnalign="center" >
<msub><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow>
</msub>
<msub><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mn>-2</mn>
</mrow>
</msub>
<mo stretchy="false">…</mo>
<msub><mi mathvariant='italic'>x</mi>
<mn>0</mn>
</msub>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">!</mo>
</mtd></mtr></mtable>
</mrow><mo stretchy="true">)</mo></mrow>
</mrow></math>
<code><font color="blue"><pre><tt>
<span style="font-weight: bold"><span style="color: #000080"># include</span></span> <span style="color: #FF0000"><cppad/cppad.hpp></span>
<span style="font-weight: bold"><span style="color: #0000FF">namespace</span></span> <span style="color: #FF0000">{</span>
<span style="font-weight: bold"><span style="color: #0000FF">using</span></span> CppAD<span style="color: #990000">::</span>AD<span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">typedef</span></span> <span style="color: #008080">AD<double></span> a1double<span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">typedef</span></span> <span style="color: #008080">AD<a1double></span> a2double<span style="color: #990000">;</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">typedef</span></span> <span style="font-weight: bold"><span style="color: #000000">CPPAD_TESTVECTOR</span></span><span style="color: #990000">(</span> <span style="color: #009900">double</span> <span style="color: #990000">)</span> a0vector<span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">typedef</span></span> <span style="font-weight: bold"><span style="color: #000000">CPPAD_TESTVECTOR</span></span><span style="color: #990000">(</span> a1double <span style="color: #990000">)</span> a1vector<span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">typedef</span></span> <span style="font-weight: bold"><span style="color: #000000">CPPAD_TESTVECTOR</span></span><span style="color: #990000">(</span> a2double <span style="color: #990000">)</span> a2vector<span style="color: #990000">;</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// set once by main and kept that way</span></span>
<span style="color: #009900">double</span> delta_t_ <span style="color: #990000">=</span> std<span style="color: #990000">::</span>numeric_limits<span style="color: #990000"><</span><span style="color: #009900">double</span><span style="color: #990000">>::</span><span style="font-weight: bold"><span style="color: #000000">quiet_NaN</span></span><span style="color: #990000">();</span>
<span style="color: #008080">size_t</span> n_ <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// The function h( x , y)</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">template</span></span> <span style="color: #990000"><</span><span style="font-weight: bold"><span style="color: #0000FF">class</span></span> <span style="color: #008080">FloatVector</span><span style="color: #990000">></span>
<span style="color: #008080">FloatVector</span> <span style="font-weight: bold"><span style="color: #000000">h</span></span><span style="color: #990000">(</span><span style="font-weight: bold"><span style="color: #0000FF">const</span></span> FloatVector<span style="color: #990000">&</span> x<span style="color: #990000">,</span> <span style="font-weight: bold"><span style="color: #0000FF">const</span></span> FloatVector<span style="color: #990000">&</span> y<span style="color: #990000">)</span>
<span style="color: #FF0000">{</span> <span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> x<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> y<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="color: #008080">FloatVector</span> <span style="font-weight: bold"><span style="color: #000000">result</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
result<span style="color: #990000">[</span><span style="color: #993399">0</span><span style="color: #990000">]</span> <span style="color: #990000">=</span> x<span style="color: #990000">[</span><span style="color: #993399">0</span><span style="color: #990000">];</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">1</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
result<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> x<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">*</span> y<span style="color: #990000">[</span>i<span style="color: #990000">-</span><span style="color: #993399">1</span><span style="color: #990000">];</span>
<span style="font-weight: bold"><span style="color: #0000FF">return</span></span> result<span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">// The 4-th Order Runge-Kutta Step</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">template</span></span> <span style="color: #990000"><</span><span style="font-weight: bold"><span style="color: #0000FF">class</span></span> <span style="color: #008080">FloatVector</span><span style="color: #990000">></span>
<span style="color: #008080">FloatVector</span> <span style="font-weight: bold"><span style="color: #000000">Runge4</span></span><span style="color: #990000">(</span><span style="font-weight: bold"><span style="color: #0000FF">const</span></span> FloatVector<span style="color: #990000">&</span> x<span style="color: #990000">,</span> <span style="font-weight: bold"><span style="color: #0000FF">const</span></span> FloatVector<span style="color: #990000">&</span> z0
<span style="color: #990000">)</span>
<span style="color: #FF0000">{</span> <span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> x<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> z0<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">typedef</span></span> <span style="font-weight: bold"><span style="color: #0000FF">typename</span></span> <span style="color: #008080">FloatVector</span><span style="color: #990000">::</span><span style="color: #008080">value_type</span> Float<span style="color: #990000">;</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="color: #008080">Float</span> dt <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">Float</span></span><span style="color: #990000">(</span>delta_t_<span style="color: #990000">);</span>
<span style="color: #008080">size_t</span> m <span style="color: #990000">=</span> z0<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">();</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="color: #008080">FloatVector</span> <span style="font-weight: bold"><span style="color: #000000">h1</span></span><span style="color: #990000">(</span>m<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">h2</span></span><span style="color: #990000">(</span>m<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">h3</span></span><span style="color: #990000">(</span>m<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">h4</span></span><span style="color: #990000">(</span>m<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">result</span></span><span style="color: #990000">(</span>m<span style="color: #990000">);</span>
h1 <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">h</span></span><span style="color: #990000">(</span> x<span style="color: #990000">,</span> z0 <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> m<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
h2<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> z0<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">+</span> dt <span style="color: #990000">*</span> h1<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">/</span> <span style="color: #993399">2.0</span><span style="color: #990000">;</span>
h2 <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">h</span></span><span style="color: #990000">(</span> x<span style="color: #990000">,</span> h2 <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> m<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
h3<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> z0<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">+</span> dt <span style="color: #990000">*</span> h2<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">/</span> <span style="color: #993399">2.0</span><span style="color: #990000">;</span>
h3 <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">h</span></span><span style="color: #990000">(</span> x<span style="color: #990000">,</span> h3 <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> m<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
h4<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> z0<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">+</span> dt <span style="color: #990000">*</span> h3<span style="color: #990000">[</span>i<span style="color: #990000">];</span>
h4 <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">h</span></span><span style="color: #990000">(</span> x<span style="color: #990000">,</span> h4 <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> m<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
<span style="color: #FF0000">{</span> <span style="color: #008080">Float</span> dz <span style="color: #990000">=</span> dt <span style="color: #990000">*</span> <span style="color: #990000">(</span> h1<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">+</span> <span style="color: #993399">2.0</span><span style="color: #990000">*</span>h2<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">+</span> <span style="color: #993399">2.0</span><span style="color: #990000">*</span>h3<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">+</span> h4<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">)</span> <span style="color: #990000">/</span> <span style="color: #993399">6.0</span><span style="color: #990000">;</span>
result<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> z0<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">+</span> dz<span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
<span style="font-weight: bold"><span style="color: #0000FF">return</span></span> result<span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">// pack x and z into an ode_info vector</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">template</span></span> <span style="color: #990000"><</span><span style="font-weight: bold"><span style="color: #0000FF">class</span></span> <span style="color: #008080">FloatVector</span><span style="color: #990000">></span>
<span style="color: #009900">void</span> <span style="font-weight: bold"><span style="color: #000000">pack</span></span><span style="color: #990000">(</span>
FloatVector<span style="color: #990000">&</span> ode_info <span style="color: #990000">,</span>
<span style="font-weight: bold"><span style="color: #0000FF">const</span></span> FloatVector<span style="color: #990000">&</span> x <span style="color: #990000">,</span>
<span style="font-weight: bold"><span style="color: #0000FF">const</span></span> FloatVector<span style="color: #990000">&</span> z <span style="color: #990000">)</span>
<span style="color: #FF0000">{</span> <span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> ode_info<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">+</span> n_ <span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> x<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> z<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="color: #008080">size_t</span> offset <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
ode_info<span style="color: #990000">[</span>offset <span style="color: #990000">+</span> i<span style="color: #990000">]</span> <span style="color: #990000">=</span> x<span style="color: #990000">[</span>i<span style="color: #990000">];</span>
offset <span style="color: #990000">+=</span> n_<span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
ode_info<span style="color: #990000">[</span>offset <span style="color: #990000">+</span> i<span style="color: #990000">]</span> <span style="color: #990000">=</span> z<span style="color: #990000">[</span>i<span style="color: #990000">];</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">// unpack an ode_info vector</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">template</span></span> <span style="color: #990000"><</span><span style="font-weight: bold"><span style="color: #0000FF">class</span></span> <span style="color: #008080">FloatVector</span><span style="color: #990000">></span>
<span style="color: #009900">void</span> <span style="font-weight: bold"><span style="color: #000000">unpack</span></span><span style="color: #990000">(</span>
<span style="font-weight: bold"><span style="color: #0000FF">const</span></span> FloatVector<span style="color: #990000">&</span> ode_info <span style="color: #990000">,</span>
FloatVector<span style="color: #990000">&</span> x <span style="color: #990000">,</span>
FloatVector<span style="color: #990000">&</span> z <span style="color: #990000">)</span>
<span style="color: #FF0000">{</span> <span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> ode_info<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">+</span> n_ <span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> x<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> z<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="color: #008080">size_t</span> offset <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
x<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> ode_info<span style="color: #990000">[</span>offset <span style="color: #990000">+</span> i<span style="color: #990000">];</span>
offset <span style="color: #990000">+=</span> n_<span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
z<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> ode_info<span style="color: #990000">[</span>offset <span style="color: #990000">+</span> i<span style="color: #990000">];</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">// Algorithm that z(t, x)</span></span>
<span style="color: #009900">void</span> <span style="font-weight: bold"><span style="color: #000000">ode_algo</span></span><span style="color: #990000">(</span><span style="font-weight: bold"><span style="color: #0000FF">const</span></span> a1vector<span style="color: #990000">&</span> ode_info_in<span style="color: #990000">,</span> a1vector<span style="color: #990000">&</span> ode_info_out<span style="color: #990000">)</span>
<span style="color: #FF0000">{</span> <span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> ode_info_in<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">+</span> n_ <span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">assert</span></span><span style="color: #990000">(</span> <span style="font-weight: bold"><span style="color: #000000">size_t</span></span><span style="color: #990000">(</span> ode_info_out<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">size</span></span><span style="color: #990000">()</span> <span style="color: #990000">)</span> <span style="color: #990000">==</span> n_ <span style="color: #990000">+</span> n_ <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// initial ode information</span></span>
<span style="color: #008080">a1vector</span> <span style="font-weight: bold"><span style="color: #000000">x</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">z0</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">unpack</span></span><span style="color: #990000">(</span>ode_info_in<span style="color: #990000">,</span> x<span style="color: #990000">,</span> z0<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// advance z(t, x)</span></span>
<span style="color: #008080">a1vector</span> z1 <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">Runge4</span></span><span style="color: #990000">(</span>x<span style="color: #990000">,</span> z0<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// final ode information</span></span>
<span style="font-weight: bold"><span style="color: #000000">pack</span></span><span style="color: #990000">(</span>ode_info_out<span style="color: #990000">,</span> x<span style="color: #990000">,</span> z1<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">return</span></span><span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="color: #009900">bool</span> <span style="font-weight: bold"><span style="color: #000000">ode</span></span><span style="color: #990000">(</span><span style="color: #009900">void</span><span style="color: #990000">)</span>
<span style="color: #FF0000">{</span> <span style="color: #009900">bool</span> ok <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #0000FF">true</span></span><span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">using</span></span> CppAD<span style="color: #990000">::</span>NearEqual<span style="color: #990000">;</span>
<span style="color: #009900">double</span> eps <span style="color: #990000">=</span> std<span style="color: #990000">::</span>numeric_limits<span style="color: #990000"><</span><span style="color: #009900">double</span><span style="color: #990000">>::</span><span style="font-weight: bold"><span style="color: #000000">epsilon</span></span><span style="color: #990000">();</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// number of terms in the differential equation</span></span>
n_ <span style="color: #990000">=</span> <span style="color: #993399">6</span><span style="color: #990000">;</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// step size for the differentiail equation</span></span>
<span style="color: #008080">size_t</span> n_step <span style="color: #990000">=</span> <span style="color: #993399">10</span><span style="color: #990000">;</span>
<span style="color: #009900">double</span> T <span style="color: #990000">=</span> <span style="color: #993399">1.0</span><span style="color: #990000">;</span>
delta_t_ <span style="color: #990000">=</span> T <span style="color: #990000">/</span> <span style="color: #009900">double</span><span style="color: #990000">(</span>n_step<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// set parameter value and initial value of the ode</span></span>
<span style="color: #008080">a1vector</span> <span style="font-weight: bold"><span style="color: #000000">ax</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">az0</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
<span style="color: #FF0000">{</span> ax<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">a1double</span></span><span style="color: #990000">(</span>i <span style="color: #990000">+</span> <span style="color: #993399">1</span><span style="color: #990000">);</span>
az0<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> <span style="font-weight: bold"><span style="color: #000000">a1double</span></span><span style="color: #990000">(</span><span style="color: #993399">0</span><span style="color: #990000">);</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// pack ode information input vector</span></span>
<span style="color: #008080">a1vector</span> <span style="font-weight: bold"><span style="color: #000000">ode_info_in</span></span><span style="color: #990000">(</span><span style="color: #993399">2</span> <span style="color: #990000">*</span> n_<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">pack</span></span><span style="color: #990000">(</span>ode_info_in<span style="color: #990000">,</span> ax<span style="color: #990000">,</span> az0<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// create checkpoint version of the algorithm</span></span>
<span style="color: #008080">a1vector</span> <span style="font-weight: bold"><span style="color: #000000">ode_info_out</span></span><span style="color: #990000">(</span><span style="color: #993399">2</span> <span style="color: #990000">*</span> n_<span style="color: #990000">);</span>
CppAD<span style="color: #990000">::</span><span style="color: #008080">checkpoint<double></span> <span style="font-weight: bold"><span style="color: #000000">ode_check</span></span><span style="color: #990000">(</span>
<span style="color: #FF0000">"ode"</span><span style="color: #990000">,</span> ode_algo<span style="color: #990000">,</span> ode_info_in<span style="color: #990000">,</span> ode_info_out
<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// set the independent variables for recording</span></span>
CppAD<span style="color: #990000">::</span><span style="font-weight: bold"><span style="color: #000000">Independent</span></span><span style="color: #990000">(</span> ax <span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// repack to get dependence on ax</span></span>
<span style="font-weight: bold"><span style="color: #000000">pack</span></span><span style="color: #990000">(</span>ode_info_in<span style="color: #990000">,</span> ax<span style="color: #990000">,</span> az0<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// Now run the checkpoint algorithm n_step times</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> k <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> k <span style="color: #990000"><</span> n_step<span style="color: #990000">;</span> k<span style="color: #990000">++)</span>
<span style="color: #FF0000">{</span> <span style="font-weight: bold"><span style="color: #000000">ode_check</span></span><span style="color: #990000">(</span>ode_info_in<span style="color: #990000">,</span> ode_info_out<span style="color: #990000">);</span>
ode_info_in <span style="color: #990000">=</span> ode_info_out<span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// Unpack the results (must use ax1 so do not overwrite ax)</span></span>
<span style="color: #008080">a1vector</span> <span style="font-weight: bold"><span style="color: #000000">ax1</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">az1</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #000000">unpack</span></span><span style="color: #990000">(</span>ode_info_out<span style="color: #990000">,</span> ax1<span style="color: #990000">,</span> az1<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// We could record a complicated funciton of x and z(T, x) in f,</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// but make this example simpler we record x -> z(T, x).</span></span>
CppAD<span style="color: #990000">::</span><span style="color: #008080">ADFun<double></span> <span style="font-weight: bold"><span style="color: #000000">f</span></span><span style="color: #990000">(</span>ax<span style="color: #990000">,</span> az1<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// check function values</span></span>
<span style="color: #008080">a0vector</span> <span style="font-weight: bold"><span style="color: #000000">x</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">z1</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> j <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> j <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> j<span style="color: #990000">++)</span>
x<span style="color: #990000">[</span>j<span style="color: #990000">]</span> <span style="color: #990000">=</span> <span style="color: #009900">double</span><span style="color: #990000">(</span>j <span style="color: #990000">+</span> <span style="color: #993399">1</span><span style="color: #990000">);</span>
z1 <span style="color: #990000">=</span> f<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">Forward</span></span><span style="color: #990000">(</span><span style="color: #993399">0</span><span style="color: #990000">,</span> x<span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// separate calculation of z(t, x)</span></span>
<span style="color: #008080">a0vector</span> <span style="font-weight: bold"><span style="color: #000000">check_z1</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
check_z1<span style="color: #990000">[</span><span style="color: #993399">0</span><span style="color: #990000">]</span> <span style="color: #990000">=</span> x<span style="color: #990000">[</span><span style="color: #993399">0</span><span style="color: #990000">]</span> <span style="color: #990000">*</span> T<span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">1</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
check_z1<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> x<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">*</span> T <span style="color: #990000">*</span> check_z1<span style="color: #990000">[</span>i<span style="color: #990000">-</span><span style="color: #993399">1</span><span style="color: #990000">]</span> <span style="color: #990000">/</span> <span style="color: #009900">double</span><span style="color: #990000">(</span>i<span style="color: #990000">+</span><span style="color: #993399">1</span><span style="color: #990000">);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// expected accuracy for each component of of z(t, x)</span></span>
<span style="color: #008080">a0vector</span> <span style="font-weight: bold"><span style="color: #000000">acc</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
<span style="color: #FF0000">{</span> <span style="font-weight: bold"><span style="color: #0000FF">if</span></span><span style="color: #990000">(</span> i <span style="color: #990000"><</span> <span style="color: #993399">4</span> <span style="color: #990000">)</span>
<span style="color: #FF0000">{</span> <span style="font-style: italic"><span style="color: #9A1900">// Runge-Kutta methos is exact for this case</span></span>
acc<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> <span style="color: #993399">10</span><span style="color: #990000">.</span> <span style="color: #990000">*</span> eps<span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
<span style="font-weight: bold"><span style="color: #0000FF">else</span></span>
<span style="color: #FF0000">{</span> acc<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> <span style="color: #993399">1.0</span><span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> k <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> k <span style="color: #990000"><</span> <span style="color: #993399">5</span><span style="color: #990000">;</span> k<span style="color: #990000">++)</span>
acc<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">*=</span> x<span style="color: #990000">[</span>k<span style="color: #990000">]</span> <span style="color: #990000">*</span> delta_t_<span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">// check z1(T, x)</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
ok <span style="color: #990000">&=</span> <span style="font-weight: bold"><span style="color: #000000">NearEqual</span></span><span style="color: #990000">(</span>z1<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">,</span> check_z1<span style="color: #990000">[</span>i<span style="color: #990000">],</span> acc<span style="color: #990000">[</span>i<span style="color: #990000">],</span> acc<span style="color: #990000">[</span>i<span style="color: #990000">]);</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// Now use f to compute a derivative. For this 'simple' example it is</span></span>
<span style="font-style: italic"><span style="color: #9A1900">// the derivative of z_{n-1} (T, x) respect to x of the</span></span>
<span style="color: #008080">a0vector</span> <span style="font-weight: bold"><span style="color: #000000">w</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">),</span> <span style="font-weight: bold"><span style="color: #000000">dw</span></span><span style="color: #990000">(</span>n_<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> i <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> i <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> i<span style="color: #990000">++)</span>
<span style="color: #FF0000">{</span> w<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> <span style="color: #993399">0.0</span><span style="color: #990000">;</span>
<span style="font-weight: bold"><span style="color: #0000FF">if</span></span><span style="color: #990000">(</span> i <span style="color: #990000">==</span> n_ <span style="color: #990000">-</span> <span style="color: #993399">1</span> <span style="color: #990000">)</span>
w<span style="color: #990000">[</span>i<span style="color: #990000">]</span> <span style="color: #990000">=</span> <span style="color: #993399">1.0</span><span style="color: #990000">;</span>
<span style="color: #FF0000">}</span>
dw <span style="color: #990000">=</span> f<span style="color: #990000">.</span><span style="font-weight: bold"><span style="color: #000000">Reverse</span></span><span style="color: #990000">(</span><span style="color: #993399">1</span><span style="color: #990000">,</span> w<span style="color: #990000">);</span>
<span style="font-weight: bold"><span style="color: #0000FF">for</span></span><span style="color: #990000">(</span><span style="color: #008080">size_t</span> j <span style="color: #990000">=</span> <span style="color: #993399">0</span><span style="color: #990000">;</span> j <span style="color: #990000"><</span> n_<span style="color: #990000">;</span> j<span style="color: #990000">++)</span>
<span style="color: #FF0000">{</span> <span style="color: #009900">double</span> check <span style="color: #990000">=</span> z1<span style="color: #990000">[</span>n_ <span style="color: #990000">-</span> <span style="color: #993399">1</span><span style="color: #990000">]</span> <span style="color: #990000">/</span> x<span style="color: #990000">[</span>j<span style="color: #990000">];</span>
ok <span style="color: #990000">&=</span> <span style="font-weight: bold"><span style="color: #000000">NearEqual</span></span><span style="color: #990000">(</span>dw<span style="color: #990000">[</span>j<span style="color: #990000">]</span> <span style="color: #990000">,</span> check<span style="color: #990000">,</span> <span style="color: #993399">100</span><span style="color: #990000">.*</span>eps<span style="color: #990000">,</span> <span style="color: #993399">100</span><span style="color: #990000">.*</span>eps<span style="color: #990000">);</span>
<span style="color: #FF0000">}</span>
<span style="font-style: italic"><span style="color: #9A1900">//</span></span>
<span style="font-weight: bold"><span style="color: #0000FF">return</span></span> ok<span style="color: #990000">;</span>
<span style="color: #FF0000">}</span></tt></pre>
</font></code>
<hr/>Input File: example/atomic/ode.cpp
</body>
</html>
