<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>An Embedded 4th and 5th Order Runge-Kutta ODE Solver</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="An Embedded 4th and 5th Order Runge-Kutta ODE Solver"/>
<meta name="keywords" id="keywords" content=" Runge45 Ode Runge-kutta Runge Kutta solve differential equation an embedded 4th and 5th order runge-kutta ode solver syntax purpose operation sequence include xf fun t x f warning m ti tf xi e scalar fabs vector parallel mode example source code "/>
<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='_runge45_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="rombergmul.cpp.xml" target="_top">Prev</a>
</td><td><a href="runge45_1.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>library</option>
<option>Runge45</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>CppAD-></option>
<option>Install</option>
<option>Introduction</option>
<option>AD</option>
<option>ADFun</option>
<option>preprocessor</option>
<option>multi_thread</option>
<option>library</option>
<option>ipopt_solve</option>
<option>Example</option>
<option>speed</option>
<option>Appendix</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>library-></option>
<option>ErrorHandler</option>
<option>NearEqual</option>
<option>speed_test</option>
<option>SpeedTest</option>
<option>time_test</option>
<option>NumericType</option>
<option>CheckNumericType</option>
<option>SimpleVector</option>
<option>CheckSimpleVector</option>
<option>nan</option>
<option>pow_int</option>
<option>Poly</option>
<option>LuDetAndSolve</option>
<option>RombergOne</option>
<option>RombergMul</option>
<option>Runge45</option>
<option>Rosen34</option>
<option>OdeErrControl</option>
<option>OdeGear</option>
<option>OdeGearControl</option>
<option>CppAD_vector</option>
<option>thread_alloc</option>
<option>index_sort</option>
<option>BenderQuad</option>
<option>opt_val_hes</option>
<option>LuRatio</option>
</select>
</td>
<td>
<select onchange='choose_down0(this)'>
<option>Runge45-></option>
<option>runge45_1.cpp</option>
<option>runge45_2.cpp</option>
</select>
</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-></option>
<option>Syntax</option>
<option>Purpose</option>
<option>Operation Sequence</option>
<option>Include</option>
<option>xf</option>
<option>Fun</option>
<option>---..t</option>
<option>---..x</option>
<option>---..f</option>
<option>---..Warning</option>
<option>M</option>
<option>ti</option>
<option>tf</option>
<option>xi</option>
<option>e</option>
<option>Scalar</option>
<option>---..fabs</option>
<option>Vector</option>
<option>Parallel Mode</option>
<option>Example</option>
<option>Source Code</option>
</select>
</td>
</tr></table><br/>
<center><b><big><big>An Embedded 4th and 5th Order Runge-Kutta ODE Solver</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<br/>
<code><font color="blue"><span style='white-space: nowrap'># include <cppad/runge_45.hpp><br/>
</span></font></code>
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i><font color="blue"><span style='white-space: nowrap'> = Runge45(</span></font><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>M</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>ti</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>tf</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>xi</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i><font color="blue"><span style='white-space: nowrap'> = Runge45(</span></font><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>M</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>ti</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>tf</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>xi</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>e</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
This is an implementation of the
Cash-Karp embedded 4th and 5th order Runge-Kutta ODE solver
described in Section 16.2 of <a href="bib.xml#Numerical Recipes" target="_top"><span style='white-space: nowrap'>Numerical Recipes</span></a>
.
We use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
</mrow></math>
for the size of the vector
<code><i><font color="black"><span style='white-space: nowrap'>xi</span></font></i></code>
.
Let
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
</mrow></math>
denote the real numbers
and let
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mo stretchy="false">×</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">→</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
be a smooth function.
The return value
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
contains a 5th order
approximation for the value
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>tf</mi>
<mo stretchy="false">)</mo>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">:</mo>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>ti</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>tf</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">→</mo>
<msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi>
</mstyle></mrow>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
is defined by
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>
<mi mathvariant='italic'>ti</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>xi</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">'</mo>
<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>
If your set of ordinary differential equations
are stiff, an implicit method may be better
(perhaps <a href="rosen34.xml" target="_top"><span style='white-space: nowrap'>Rosen34</span></a>
.)
<br/>
<br/>
<b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b>
<br/>
The <a href="glossary.xml#Operation.Sequence" target="_top"><span style='white-space: nowrap'>operation sequence</span></a>
for
<code><i><font color="black"><span style='white-space: nowrap'>Runge</span></font></i></code>
does not depend on any of its
<code><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i></code>
input values provided that
the operation sequence for
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>.Ode(</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
does not on any of its
<code><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i></code>
inputs (see below).
<br/>
<br/>
<b><big><a name="Include" id="Include">Include</a></big></b>
<br/>
The file <code><font color="blue">cppad/runge_45.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="xf" id="xf">xf</a></big></b>
<br/>
The return value
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
has the prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'> </span></font><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and the size of
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
is equal to
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
(see description of <a href="runge45.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below).
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>tf</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>xf</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>O</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>h</mi>
<mn>6</mn>
</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>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>tf</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>ti</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>M</mi>
</mrow></math>
is the step size.
If
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
contains not a number <a href="nan.xml" target="_top"><span style='white-space: nowrap'>nan</span></a>
,
see the discussion for <a href="runge45.xml#Fun.f" target="_top"><span style='white-space: nowrap'>f</span></a>
.
<br/>
<br/>
<b><big><a name="Fun" id="Fun">Fun</a></big></b>
<br/>
The class
<code><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i></code>
and the object
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i></code>
satisfy the prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
The object
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i></code>
(and the class
<code><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i></code>
)
must have a member function named <code><font color="blue">Ode</font></code>
that supports the syntax
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>.Ode(</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></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>
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/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>t</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see description of <a href="runge45.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>
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/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and has size
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
(see description of <a href="runge45.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
below).
<br/>
<br/>
<b><a name="Fun.f" id="Fun.f">f</a></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
to
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>.Ode</span></font></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
On input and output,
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
is a vector of size
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
and the input values of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
do not matter.
On output,
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
is set equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
in the differential equation.
If any of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
have the value not a number <code><font color="blue">nan</font></code>
the routine <code><font color="blue">Runge45</font></code> returns with all the
elements of
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
and
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
equal to <code><font color="blue">nan</font></code>.
<br/>
<br/>
<b><a name="Fun.Warning" id="Fun.Warning">Warning</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>
must have a call by reference in its prototype; i.e.,
do not forget the <code><font color="blue">&</font></code> in the prototype for
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="M" id="M">M</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>M</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     size_t </span></font><i><font color="black"><span style='white-space: nowrap'>M</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the number of steps
to use when solving the differential equation.
This must be greater than or equal one.
The step size is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>h</mi>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>tf</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>ti</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>M</mi>
</mrow></math>
, thus
the larger
<code><i><font color="black"><span style='white-space: nowrap'>M</span></font></i></code>
, the more accurate the
return value
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
is as an approximation
for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>tf</mi>
<mo stretchy="false">)</mo>
</mrow></math>
.
<br/>
<br/>
<b><big><a name="ti" id="ti">ti</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>ti</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>ti</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see description of <a href="runge45.xml#Scalar" target="_top"><span style='white-space: nowrap'>Scalar</span></a>
below).
It specifies the initial time for
<code><i><font color="black"><span style='white-space: nowrap'>t</span></font></i></code>
in the
differential equation; i.e.,
the time corresponding to the value
<code><i><font color="black"><span style='white-space: nowrap'>xi</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="tf" id="tf">tf</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>tf</span></font></i></code>
has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>tf</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the final time for
<code><i><font color="black"><span style='white-space: nowrap'>t</span></font></i></code>
in the
differential equation; i.e.,
the time corresponding to the value
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
.
<br/>
<br/>
<b><big><a name="xi" id="xi">xi</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>xi</span></font></i></code>
has the prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>xi</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and the size of
<code><i><font color="black"><span style='white-space: nowrap'>xi</span></font></i></code>
is equal to
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
It specifies the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>ti</mi>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<br/>
<b><big><a name="e" id="e">e</a></big></b>
<br/>
The argument
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
is optional and has the prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'> &</span></font><i><font color="black"><span style='white-space: nowrap'>e</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
If
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
is present,
the size of
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
must be equal to
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
The input value of the elements of
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
does not matter.
On output
it contains an element by element
estimated bound for the absolute value of the error in
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
<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>
<mn>5</mn>
</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>
<mo stretchy="false">=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>tf</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>ti</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>M</mi>
</mrow></math>
is the step size.
If on output,
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
contains not a number <code><font color="blue">nan</font></code>,
see the discussion for <a href="runge45.xml#Fun.f" target="_top"><span style='white-space: nowrap'>f</span></a>
.
<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.
<br/>
<br/>
<b><a name="Scalar.fabs" id="Scalar.fabs">fabs</a></b>
<br/>
In addition, the following function 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>
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font><i><font color="black"><span style='white-space: nowrap'>a</span></font></i><font color="blue"><span style='white-space: nowrap'> = fabs(</span></font><i><font color="black"><span style='white-space: nowrap'>b</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
Note that this operation is only used for computing
<code><i><font color="black"><span style='white-space: nowrap'>e</span></font></i></code>
; hence
the operation sequence for
<code><i><font color="black"><span style='white-space: nowrap'>xf</span></font></i></code>
can still be independent of
the arguments to <code><font color="blue">Runge45</font></code> even if
<code><font color="blue"><span style='white-space: nowrap'><br/>
     fabs(</span></font><i><font color="black"><span style='white-space: nowrap'>b</span></font></i><font color="blue"><span style='white-space: nowrap'>) = std::max(-</span></font><i><font color="black"><span style='white-space: nowrap'>b</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>b</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
.
<br/>
<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
The type
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
must be a <a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
class with
<a href="simplevector.xml#Elements of Specified Type" target="_top"><span style='white-space: nowrap'>elements of type 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="Parallel Mode" id="Parallel Mode">Parallel Mode</a></big></b>
<br/>
For each set of types
<a href="runge45.xml#Scalar" target="_top"><span style='white-space: nowrap'>Scalar</span></a>
,
<a href="runge45.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
, and
<a href="runge45.xml#Fun" target="_top"><span style='white-space: nowrap'>Fun</span></a>
,
the first call to <code><font color="blue">Runge45</font></code>
must not be <a href="ta_in_parallel.xml" target="_top"><span style='white-space: nowrap'>parallel</span></a>
execution mode.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="runge45_1.cpp.xml" target="_top"><span style='white-space: nowrap'>runge45_1.cpp</span></a>
contains a simple example and test of <code><font color="blue">Runge45</font></code>.
It returns true if it succeeds and false otherwise.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>The file
<a href="runge45_2.cpp.xml" target="_top"><span style='white-space: nowrap'>runge45_2.cpp</span></a>
contains an example using <code><font color="blue">Runge45</font></code>
in the context of algorithmic differentiation.
It also 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/runge_45.hpp</font></code>.
<hr/>Input File: cppad/runge_45.hpp
</body>
</html>
