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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Concepts</title>
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<link rel="stylesheet" href="../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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<link rel="home" href="../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="up" href="../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="prev" href="odeint_in_detail/using_boost__range.html" title="Using boost::range">
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="boost_numeric_odeint.concepts"></a><a class="link" href="concepts.html" title="Concepts">Concepts</a>
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</h2></div></div></div>
|
||||
<div class="toc"><dl>
|
||||
<dt><span class="section"><a href="concepts/system.html">System</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/symplectic_system.html">Symplectic
|
||||
System</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/simple_symplectic_system.html">Simple
|
||||
Symplectic System</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/implicit_system.html">Implicit
|
||||
System</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/stepper.html">Stepper</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/error_stepper.html">Error Stepper</a></span></dt>
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||||
<dt><span class="section"><a href="concepts/controlled_stepper.html">Controlled
|
||||
Stepper</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/dense_output_stepper.html">Dense
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||||
Output Stepper</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/state_algebra_operations.html">State
|
||||
Algebra Operations</a></span></dt>
|
||||
<dt><span class="section"><a href="concepts/state_wrapper.html">State Wrapper</a></span></dt>
|
||||
</dl></div>
|
||||
<a name="odeint.concepts"></a>
|
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</div>
|
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
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and Mario Mulansky<p>
|
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Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
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</p>
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</div></td>
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</tr></table>
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<hr>
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<div class="spirit-nav">
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<a accesskey="p" href="odeint_in_detail/using_boost__range.html"><img src="../images/prev.png" alt="Prev"></a><a accesskey="u" href="../index.html"><img src="../images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../images/home.png" alt="Home"></a><a accesskey="n" href="concepts/system.html"><img src="../images/next.png" alt="Next"></a>
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@ -1,227 +0,0 @@
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<html>
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<head>
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||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Controlled Stepper</title>
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<link rel="stylesheet" href="../../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="up" href="../concepts.html" title="Concepts">
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<link rel="prev" href="error_stepper.html" title="Error Stepper">
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<link rel="next" href="dense_output_stepper.html" title="Dense Output Stepper">
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</head>
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<div class="spirit-nav">
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<a accesskey="p" href="error_stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="dense_output_stepper.html"><img src="../../images/next.png" alt="Next"></a>
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</div>
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||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.controlled_stepper"></a><a class="link" href="controlled_stepper.html" title="Controlled Stepper">Controlled
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||||
Stepper</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
This concept specifies the interface a controlled stepper has to fulfill
|
||||
to be used within <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
|
||||
functions</a>.
|
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</p>
|
||||
<h5>
|
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<a name="boost_numeric_odeint.concepts.controlled_stepper.h0"></a>
|
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<span><a name="boost_numeric_odeint.concepts.controlled_stepper.description"></a></span><a class="link" href="controlled_stepper.html#boost_numeric_odeint.concepts.controlled_stepper.description">Description</a>
|
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</h5>
|
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<p>
|
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A controlled stepper following this Controlled Stepper concept provides the
|
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possibility to perform one step of the solution <span class="emphasis"><em>x(t)</em></span>
|
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of an ODE with step-size <span class="emphasis"><em>dt</em></span> to obtain <span class="emphasis"><em>x(t+dt)</em></span>
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with a given step-size <span class="emphasis"><em>dt</em></span>. Depending on an error estimate
|
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of the solution the step might be rejected and a smaller step-size is suggested.
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</p>
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<h5>
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<a name="boost_numeric_odeint.concepts.controlled_stepper.h1"></a>
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<span><a name="boost_numeric_odeint.concepts.controlled_stepper.associated_types"></a></span><a class="link" href="controlled_stepper.html#boost_numeric_odeint.concepts.controlled_stepper.associated_types">Associated
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types</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" type="disc">
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<li class="listitem">
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<p><span class="bold"><strong>state_type</strong></span></p>
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<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">state_type</span></code></p>
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<p>The
|
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type characterizing the state of the ODE, hence <span class="emphasis"><em>x</em></span>.</p>
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</li>
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<li class="listitem">
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<p><span class="bold"><strong>deriv_type</strong></span></p>
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<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">deriv_type</span></code></p>
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<p>The
|
||||
type characterizing the derivative of the ODE, hence <span class="emphasis"><em>d x/dt</em></span>.</p>
|
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</li>
|
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<li class="listitem">
|
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<p><span class="bold"><strong>time_type</strong></span></p>
|
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<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span></code></p>
|
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<p>The
|
||||
type characterizing the dependent variable of the ODE, hence the time
|
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<span class="emphasis"><em>t</em></span>.</p>
|
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</li>
|
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<li class="listitem">
|
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<p><span class="bold"><strong>value_type</strong></span></p>
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<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">value_type</span></code></p>
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<p>The
|
||||
numerical data type which is used within the stepper, something like
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<code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>,
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<code class="computeroutput"><span class="identifier">complex</span><span class="special">&</span><span class="identifier">lt</span><span class="special">;</span> <span class="keyword">double</span> <span class="special">&</span><span class="identifier">gt</span><span class="special">;</span></code>.</p>
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</li>
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<li class="listitem">
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<p><span class="bold"><strong>stepper_category</strong></span></p>
|
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<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">stepper_category</span></code></p>
|
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<p>A
|
||||
tag type characterizing the category of the stepper. This type must be
|
||||
convertible to <code class="computeroutput"><span class="identifier">controlled_stepper_tag</span></code>.</p>
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</li>
|
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</ul></div>
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<h5>
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<a name="boost_numeric_odeint.concepts.controlled_stepper.h2"></a>
|
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<span><a name="boost_numeric_odeint.concepts.controlled_stepper.notation"></a></span><a class="link" href="controlled_stepper.html#boost_numeric_odeint.concepts.controlled_stepper.notation">Notation</a>
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</h5>
|
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">ControlledStepper</span></code></span></dt>
|
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<dd><p>
|
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A type that is a model of Controlled Stepper
|
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
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<dd><p>
|
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A type representing the state <span class="emphasis"><em>x</em></span> of the ODE
|
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</p></dd>
|
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<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
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<dd><p>
|
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A type representing the time <span class="emphasis"><em>t</em></span> of the ODE
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</p></dd>
|
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<dt><span class="term"><code class="computeroutput"><span class="identifier">stepper</span></code></span></dt>
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<dd><p>
|
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An object of type <code class="computeroutput"><span class="identifier">ControlledStepper</span></code>
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</p></dd>
|
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<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
|
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<dd><p>
|
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Object of type <code class="computeroutput"><span class="identifier">State</span></code>
|
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</p></dd>
|
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<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code>, <code class="computeroutput"><span class="identifier">dt</span></code></span></dt>
|
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<dd><p>
|
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Objects of type <code class="computeroutput"><span class="identifier">Time</span></code>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
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<dd><p>
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An object defining the ODE, should be a model of <a class="link" href="system.html" title="System">System</a>,
|
||||
<a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a>, <a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>.
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
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<h5>
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<a name="boost_numeric_odeint.concepts.controlled_stepper.h3"></a>
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<span><a name="boost_numeric_odeint.concepts.controlled_stepper.valid_expressions"></a></span><a class="link" href="controlled_stepper.html#boost_numeric_odeint.concepts.controlled_stepper.valid_expressions">Valid
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||||
Expressions</a>
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||||
</h5>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
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</p>
|
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</th>
|
||||
<th>
|
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<p>
|
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Expression
|
||||
</p>
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||||
</th>
|
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<th>
|
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<p>
|
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Type
|
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</p>
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</th>
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<th>
|
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<p>
|
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Semantics
|
||||
</p>
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||||
</th>
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</tr></thead>
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<tbody><tr>
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||||
<td>
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<p>
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Do step
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</p>
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</td>
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<td>
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<p>
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</p>
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<pre xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" class="table-programlisting"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">try_step</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">)</span></pre>
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<p>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">controlled_step_result</span></code>
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</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
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||||
Tries one step of step size <code class="computeroutput"><span class="identifier">dt</span></code>.
|
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If the step was successful, <code class="computeroutput"><span class="identifier">success</span></code>
|
||||
is returned, the resulting state is written to <code class="computeroutput"><span class="identifier">x</span></code>,
|
||||
the new time is stored in <code class="computeroutput"><span class="identifier">t</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">dt</span></code> now contains
|
||||
a new (possibly larger) step-size for the next step. If the error
|
||||
was too big, <code class="computeroutput"><span class="identifier">rejected</span></code>
|
||||
is returned and the results are neglected - <code class="computeroutput"><span class="identifier">x</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">t</span></code> are unchanged
|
||||
and <code class="computeroutput"><span class="identifier">dt</span></code> now contains
|
||||
a reduced step-size to be used for the next try.
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.controlled_stepper.h4"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.controlled_stepper.models"></a></span><a class="link" href="controlled_stepper.html#boost_numeric_odeint.concepts.controlled_stepper.models">Models</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">controlled_error_stepper</span><span class="special"><</span> <span class="identifier">runge_kutta_cash_karp54</span>
|
||||
<span class="special">></span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">controlled_error_stepper_fsal</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span>
|
||||
<span class="special">></span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">controlled_error_stepper</span><span class="special"><</span> <span class="identifier">runge_kutta_fehlberg78</span>
|
||||
<span class="special">></span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4_controller</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">bulirsch_stoer</span></code>
|
||||
</li>
|
||||
</ul></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="error_stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="dense_output_stepper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,311 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Dense Output Stepper</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
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<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="controlled_stepper.html" title="Controlled Stepper">
|
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<link rel="next" href="state_algebra_operations.html" title="State Algebra Operations">
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</head>
|
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
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<div class="spirit-nav">
|
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<a accesskey="p" href="controlled_stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="state_algebra_operations.html"><img src="../../images/next.png" alt="Next"></a>
|
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</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.dense_output_stepper"></a><a class="link" href="dense_output_stepper.html" title="Dense Output Stepper">Dense
|
||||
Output Stepper</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
This concept specifies the interface a dense output stepper has to fulfill
|
||||
to be used within <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
|
||||
functions</a>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.dense_output_stepper.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.dense_output_stepper.description"></a></span><a class="link" href="dense_output_stepper.html#boost_numeric_odeint.concepts.dense_output_stepper.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
A dense output stepper following this Dense Output Stepper concept provides
|
||||
the possibility to perform a single step of the solution <span class="emphasis"><em>x(t)</em></span>
|
||||
of an ODE to obtain <span class="emphasis"><em>x(t+dt)</em></span>. The step-size <code class="computeroutput"><span class="identifier">dt</span></code> might be adjusted automatically due
|
||||
to error control. Dense output steppers also can interpolate the solution
|
||||
to calculate the state <span class="emphasis"><em>x(t')</em></span> at any point <span class="emphasis"><em>t
|
||||
<= t' <= t+dt</em></span>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.dense_output_stepper.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.dense_output_stepper.associated_types"></a></span><a class="link" href="dense_output_stepper.html#boost_numeric_odeint.concepts.dense_output_stepper.associated_types">Associated
|
||||
types</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>state_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">state_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the state of the ODE, hence <span class="emphasis"><em>x</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>deriv_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">deriv_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the derivative of the ODE, hence <span class="emphasis"><em>d x/dt</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>time_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the dependent variable of the ODE, hence the time
|
||||
<span class="emphasis"><em>t</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>value_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">value_type</span></code></p>
|
||||
<p>The
|
||||
numerical data type which is used within the stepper, something like
|
||||
<code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">complex</span><span class="special">&</span><span class="identifier">lt</span><span class="special">;</span> <span class="keyword">double</span> <span class="special">&</span><span class="identifier">gt</span><span class="special">;</span></code>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>stepper_category</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">stepper_category</span></code></p>
|
||||
<p>A
|
||||
tag type characterizing the category of the stepper. This type must be
|
||||
convertible to <code class="computeroutput"><span class="identifier">dense_output_stepper_tag</span></code>.</p>
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.dense_output_stepper.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.dense_output_stepper.notation"></a></span><a class="link" href="dense_output_stepper.html#boost_numeric_odeint.concepts.dense_output_stepper.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Stepper</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of Dense Output Stepper
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the state <span class="emphasis"><em>x</em></span> of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">stepper</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object of type <code class="computeroutput"><span class="identifier">Stepper</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">x0</span></code>, <code class="computeroutput"><span class="identifier">x</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">t0</span></code>, <code class="computeroutput"><span class="identifier">dt0</span></code>, <code class="computeroutput"><span class="identifier">t</span></code></span></dt>
|
||||
<dd><p>
|
||||
Objects of type <code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object defining the ODE, should be a model of <a class="link" href="system.html" title="System">System</a>,
|
||||
<a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a>, <a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>.
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.dense_output_stepper.h3"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.dense_output_stepper.valid_expressions"></a></span><a class="link" href="dense_output_stepper.html#boost_numeric_odeint.concepts.dense_output_stepper.valid_expressions">Valid
|
||||
Expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Initialize integration
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">initialize</span><span class="special">(</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">dt0</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
void
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Initializes the stepper with initial values <code class="computeroutput"><span class="identifier">x0</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">t0</span></code> and <code class="computeroutput"><span class="identifier">dt0</span></code>.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Do step
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span>
|
||||
<span class="identifier">sys</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span>
|
||||
<span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span> <span class="special">,</span>
|
||||
<span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Performs one step using the ODE defined by <code class="computeroutput"><span class="identifier">sys</span></code>.
|
||||
The step-size might be changed internally due to error control.
|
||||
This function returns a pair containing <code class="computeroutput"><span class="identifier">t</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">t</span><span class="special">+</span><span class="identifier">dt</span></code> representing the interval
|
||||
for which interpolation can be performed.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Do interpolation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">calc_state</span><span class="special">(</span>
|
||||
<span class="identifier">t_inter</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Performs the interpolation to calculate /x(t<sub>inter</sub>/) where /t <=
|
||||
t<sub>inter</sub> <= t+dt/.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get current time
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">current_time</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">const</span> <span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span><span class="special">&</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns the current time <span class="emphasis"><em>t+dt</em></span> of the stepper,
|
||||
that is the end time of the last step and the starting time for
|
||||
the next call of <code class="computeroutput"><span class="identifier">do_step</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get current state
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">current_state</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">const</span> <span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">state_type</span><span class="special">&</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns the current state of the stepper, that is <span class="emphasis"><em>x(t+dt)</em></span>,
|
||||
the state at the time returned by <code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">current_time</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.dense_output_stepper.h4"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.dense_output_stepper.models"></a></span><a class="link" href="dense_output_stepper.html#boost_numeric_odeint.concepts.dense_output_stepper.models">Models</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">dense_output_controlled_explicit_fsal</span><span class="special"><</span> <span class="identifier">controlled_error_stepper_fsal</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span>
|
||||
<span class="special">></span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">bulirsch_stoer_dense_out</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4_dense_output</span></code>
|
||||
</li>
|
||||
</ul></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="controlled_stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="state_algebra_operations.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,361 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Error Stepper</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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||||
<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="stepper.html" title="Stepper">
|
||||
<link rel="next" href="controlled_stepper.html" title="Controlled Stepper">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="controlled_stepper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.error_stepper"></a><a class="link" href="error_stepper.html" title="Error Stepper">Error Stepper</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
This concepts specifies the interface an error stepper has to fulfill to
|
||||
be used within a ControlledErrorStepper. An error stepper must always fullfil
|
||||
the stepper concept. This can trivially implemented by
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">System</span> <span class="special">></span>
|
||||
<span class="identifier">error_stepper</span><span class="special">::</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">System</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="identifier">dt</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">state_type</span> <span class="identifier">xerr</span><span class="special">;</span>
|
||||
<span class="comment">// allocate xerr</span>
|
||||
<span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">xerr</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.error_stepper.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.error_stepper.description"></a></span><a class="link" href="error_stepper.html#boost_numeric_odeint.concepts.error_stepper.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
An error stepper following this Error Stepper concept is capable of doing
|
||||
one step of the solution <span class="emphasis"><em>x(t)</em></span> of an ODE with step-size
|
||||
<span class="emphasis"><em>dt</em></span> to obtain <span class="emphasis"><em>x(t+dt)</em></span> and also computing
|
||||
an error estimate <span class="emphasis"><em>x<sub>err</sub></em></span> of the result. Error Steppers
|
||||
can be Runge Kutta steppers, symplectic steppers as well as implicit steppers.
|
||||
Based on the stepper type, the ODE is defined as <a class="link" href="system.html" title="System">System</a>,
|
||||
<a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a>, <a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.error_stepper.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.error_stepper.refinement_of"></a></span><a class="link" href="error_stepper.html#boost_numeric_odeint.concepts.error_stepper.refinement_of">Refinement
|
||||
of</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
DefaultConstructable
|
||||
</li>
|
||||
<li class="listitem">
|
||||
CopyConstructable
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Stepper
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.error_stepper.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.error_stepper.associated_types"></a></span><a class="link" href="error_stepper.html#boost_numeric_odeint.concepts.error_stepper.associated_types">Associated
|
||||
types</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>state_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">state_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the state of the ODE, hence <span class="emphasis"><em>x</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>deriv_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">deriv_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the derivative of the ODE, hence <span class="emphasis"><em>d x/dt</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>time_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the dependent variable of the ODE, hence the time
|
||||
<span class="emphasis"><em>t</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>value_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">value_type</span></code></p>
|
||||
<p>The
|
||||
numerical data type which is used within the stepper, something like
|
||||
<code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">complex</span><span class="special">&</span><span class="identifier">lt</span><span class="special">;</span> <span class="keyword">double</span> <span class="special">&</span><span class="identifier">gt</span><span class="special">;</span></code>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>order_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">order_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the order of the ODE, typically <code class="computeroutput"><span class="keyword">unsigned</span>
|
||||
<span class="keyword">short</span></code>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>stepper_category</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">stepper_category</span></code></p>
|
||||
<p>A
|
||||
tag type characterizing the category of the stepper. This type must be
|
||||
convertible to <code class="computeroutput"><span class="identifier">error_stepper_tag</span></code>.</p>
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.error_stepper.h3"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.error_stepper.notation"></a></span><a class="link" href="error_stepper.html#boost_numeric_odeint.concepts.error_stepper.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">ErrorStepper</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of Error Stepper
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the state <span class="emphasis"><em>x</em></span> of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Error</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the error calculated by the stepper, usually same
|
||||
as <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the time <span class="emphasis"><em>t</em></span> of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">stepper</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object of type <code class="computeroutput"><span class="identifier">ErrorStepper</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">xerr</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">Error</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code>, <code class="computeroutput"><span class="identifier">dt</span></code></span></dt>
|
||||
<dd><p>
|
||||
Objects of type <code class="computeroutput"><span class="identifier">Time</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object defining the ODE, should be a model of either <a class="link" href="system.html" title="System">System</a>,
|
||||
<a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a>, <a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>.
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.error_stepper.h4"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.error_stepper.valid_expressions"></a></span><a class="link" href="error_stepper.html#boost_numeric_odeint.concepts.error_stepper.valid_expressions">Valid
|
||||
Expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get the stepper order
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">order</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">order_type</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns the order of the stepper for one step without error estimation.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get the stepper order
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">stepper_order</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">order_type</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns the order of the stepper for one error estimation step
|
||||
which is used for error calculation.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get the error order
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">errorr_order</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">order_type</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns the order of the error step which is used for error calculation.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Do step
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span>
|
||||
<span class="identifier">sys</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span>
|
||||
<span class="identifier">t</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Performs one step of step size <code class="computeroutput"><span class="identifier">dt</span></code>.
|
||||
The newly obtained state is written in-place to <code class="computeroutput"><span class="identifier">x</span></code>.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Do step with error estimation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span>
|
||||
<span class="identifier">sys</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span>
|
||||
<span class="identifier">t</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">xerr</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Performs one step of step size <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
with error estimation. The newly obtained state is written in-place
|
||||
to <code class="computeroutput"><span class="identifier">x</span></code> and the estimated
|
||||
error to <code class="computeroutput"><span class="identifier">xerr</span></code>.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.error_stepper.h5"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.error_stepper.models"></a></span><a class="link" href="error_stepper.html#boost_numeric_odeint.concepts.error_stepper.models">Models</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_cash_karp54</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_fehlberg78</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4</span></code>
|
||||
</li>
|
||||
</ul></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="controlled_stepper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,175 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Implicit System</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="simple_symplectic_system.html" title="Simple Symplectic System">
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<link rel="next" href="stepper.html" title="Stepper">
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</head>
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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||||
<div class="spirit-nav">
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<a accesskey="p" href="simple_symplectic_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="stepper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.implicit_system"></a><a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.implicit_system.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.implicit_system.description"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
This concept describes how to define a ODE that can be solved by an implicit
|
||||
routine. Implicit routines need not only the function <span class="emphasis"><em>f(x,t)</em></span>
|
||||
but also the Jacobian <span class="emphasis"><em>df/dx = A(x,t)</em></span>. <span class="emphasis"><em>A</em></span>
|
||||
is a matrix and implicit routines need to solve the linear problem <span class="emphasis"><em>Ax
|
||||
= b</em></span>. In odeint this is implemented with use of <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.UBlas</a>,
|
||||
therefore, the <span class="emphasis"><em>state_type</em></span> implicit routines is <span class="emphasis"><em>ublas::vector</em></span>
|
||||
and the matrix is defined as <span class="emphasis"><em>ublas::matrix</em></span>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.implicit_system.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.implicit_system.notation"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of <code class="computeroutput"><span class="identifier">Implicit_System</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the time of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object of type <code class="computeroutput"><span class="identifier">System</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type ublas::vector
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">dxdt</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type ublas::vector
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">jacobi</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type ublas::matrix
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">Time</span></code>
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.implicit_system.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.implicit_system.valid_expressions"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.valid_expressions">Valid
|
||||
Expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Calculate <span class="emphasis"><em>dx/dt := f(x,t)</em></span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span>
|
||||
<span class="identifier">dxdt</span> <span class="special">,</span>
|
||||
<span class="identifier">t</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span><span class="identifier">t</span><span class="special">)</span></code>,
|
||||
the result is stored into dxdt
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Calculate <span class="emphasis"><em>A := df/dx (x,t)</em></span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span>
|
||||
<span class="identifier">jacobi</span> <span class="special">,</span>
|
||||
<span class="identifier">t</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates the Jacobian of <span class="emphasis"><em>f</em></span> at <span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>t</em></span>,
|
||||
the result is stored into <code class="computeroutput"><span class="identifier">jacobi</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="simple_symplectic_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="stepper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,179 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Simple Symplectic System</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
||||
<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="symplectic_system.html" title="Symplectic System">
|
||||
<link rel="next" href="implicit_system.html" title="Implicit System">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="symplectic_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="implicit_system.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.simple_symplectic_system"></a><a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||
Symplectic System</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.simple_symplectic_system.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.simple_symplectic_system.description"></a></span><a class="link" href="simple_symplectic_system.html#boost_numeric_odeint.concepts.simple_symplectic_system.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
In most Hamiltonian systems the kinetic term is a quadratic term in the momentum
|
||||
<span class="emphasis"><em>H<sub>kin</sub> = p^2 / 2m</em></span> and in many cases it is possible to rescale
|
||||
coordinates and set <span class="emphasis"><em>m=1</em></span> which leads to a trivial equation
|
||||
of motion:
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>q'(t) = f(p) = p. </em></span>
|
||||
</p>
|
||||
<p>
|
||||
while for <span class="emphasis"><em>p'</em></span> we still have the general form
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>p'(t) = g(q) </em></span>
|
||||
</p>
|
||||
<p>
|
||||
As this case is very frequent we introduced a concept where only the nontrivial
|
||||
equation for <span class="emphasis"><em>p'</em></span> has to be provided to the symplectic
|
||||
stepper. We call this concept <span class="emphasis"><em>Simple_Symplectic_System</em></span>
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.simple_symplectic_system.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.simple_symplectic_system.notation"></a></span><a class="link" href="simple_symplectic_system.html#boost_numeric_odeint.concepts.simple_symplectic_system.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term">System</span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of Simple_Symplectic_System
|
||||
</p></dd>
|
||||
<dt><span class="term">Coor</span></dt>
|
||||
<dd><p>
|
||||
The type of the coordinate <span class="emphasis"><em>q</em></span>
|
||||
</p></dd>
|
||||
<dt><span class="term">MomentumDeriv</span></dt>
|
||||
<dd><p>
|
||||
The type of the derivative of momentum <span class="emphasis"><em>p'</em></span>
|
||||
</p></dd>
|
||||
<dt><span class="term">sys</span></dt>
|
||||
<dd><p>
|
||||
An object that models System
|
||||
</p></dd>
|
||||
<dt><span class="term">q</span></dt>
|
||||
<dd><p>
|
||||
Object of type Coor
|
||||
</p></dd>
|
||||
<dt><span class="term">dpdt</span></dt>
|
||||
<dd><p>
|
||||
Object of type MomentumDeriv
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.simple_symplectic_system.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.simple_symplectic_system.valid_expressions"></a></span><a class="link" href="simple_symplectic_system.html#boost_numeric_odeint.concepts.simple_symplectic_system.valid_expressions">Valid
|
||||
Expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Check for pair
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">is_pair</span><span class="special"><</span>
|
||||
<span class="identifier">System</span> <span class="special">>::</span><span class="identifier">type</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">false_</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Check if System is a pair, should be evaluated to false in this
|
||||
case.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Calculate <span class="emphasis"><em>dp/dt = g(q)</em></span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">sys</span><span class="special">(</span>
|
||||
<span class="identifier">q</span> <span class="special">,</span>
|
||||
<span class="identifier">dpdt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <span class="emphasis"><em>g(q)</em></span>, the result is stored into
|
||||
<code class="computeroutput"><span class="identifier">dpdt</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="symplectic_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="implicit_system.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,827 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>State Algebra Operations</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
||||
<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="dense_output_stepper.html" title="Dense Output Stepper">
|
||||
<link rel="next" href="state_wrapper.html" title="State Wrapper">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="dense_output_stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="state_wrapper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations"></a><a class="link" href="state_algebra_operations.html" title="State Algebra Operations">State
|
||||
Algebra Operations</a>
|
||||
</h3></div></div></div>
|
||||
<div class="toc"><dl>
|
||||
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations">Operations</a></span></dt>
|
||||
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra">Algebra</a></span></dt>
|
||||
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.pre_defined_implementations">Pre-Defined
|
||||
implementations</a></span></dt>
|
||||
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.example_expressions">Example
|
||||
expressions</a></span></dt>
|
||||
</dl></div>
|
||||
<div class="note"><table border="0" summary="Note">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../images/note.png"></td>
|
||||
<th align="left">Note</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
The following does not apply to implicit steppers like implicit_euler or
|
||||
Rosenbrock 4 as there the <code class="computeroutput"><span class="identifier">state_type</span></code>
|
||||
can not be changed from <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span></code>
|
||||
and no algebra/operations are used.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_algebra_operations.description"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
The <code class="computeroutput"><span class="identifier">State</span></code>, <code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code>
|
||||
together define a concept describing how the mathematical vector operations
|
||||
required for the stepper algorithms are performed. The typical vector operation
|
||||
done within steppers is
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em><span class="bold"><strong>y</strong></span> = Σ α<sub>i</sub> <span class="bold"><strong>x<sub>i</sub></strong></span></em></span>.
|
||||
</p>
|
||||
<p>
|
||||
The <code class="computeroutput"><span class="identifier">State</span></code> represents the
|
||||
state variable of an ODE, usually denoted with <span class="emphasis"><em>x</em></span>. Algorithmically,
|
||||
the state is often realized as a <code class="computeroutput"><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span></code> or <code class="computeroutput"><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">></span></code>,
|
||||
however, the genericity of odeint enables you to basically use anything as
|
||||
a state type. The algorithmic counterpart of such mathematical expressions
|
||||
is divided into two parts. First, the <code class="computeroutput"><span class="identifier">Algebra</span></code>
|
||||
is used to account for the vector character of the equation. In the case
|
||||
of a <code class="computeroutput"><span class="identifier">vector</span></code> as state type
|
||||
this means the <code class="computeroutput"><span class="identifier">Algebra</span></code> is
|
||||
responsible for iteration over all vector elements. Second, the <code class="computeroutput"><span class="identifier">Operations</span></code> are used to represent the actual
|
||||
operation applied to each of the vector elements. So the <code class="computeroutput"><span class="identifier">Algebra</span></code>
|
||||
iterates over all elements of the <code class="computeroutput"><span class="identifier">State</span></code>s
|
||||
and calls an operation taken from the <code class="computeroutput"><span class="identifier">Operations</span></code>
|
||||
for each element. This is where <code class="computeroutput"><span class="identifier">State</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code> have to work together to make
|
||||
odeint running. Please have a look at the <code class="computeroutput"><span class="identifier">range_algebra</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">default_operations</span></code> to
|
||||
see an example how this is implemented.
|
||||
</p>
|
||||
<p>
|
||||
In the following we describe how <code class="computeroutput"><span class="identifier">State</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code> are used together within the
|
||||
stepper implementations.
|
||||
</p>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.operations"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations" title="Operations">Operations</a>
|
||||
</h4></div></div></div>
|
||||
<h6>
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.notation"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations.notation">Notation</a>
|
||||
</h6>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Operations</span></code></span></dt>
|
||||
<dd><p>
|
||||
The operations type
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Value1</span></code>, ... ,
|
||||
<code class="computeroutput"><span class="identifier">ValueN</span></code></span></dt>
|
||||
<dd><p>
|
||||
Types representing the value or time type of stepper
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Scale</span></code></span></dt>
|
||||
<dd><p>
|
||||
Type of the scale operation
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">scale</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">Scale</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="literal">ScaleSum<span class="emphasis"><em>N</em></span></code></span></dt>
|
||||
<dd><p>
|
||||
Type that represents a general scale_sum operation, <code class="literal"><span class="emphasis"><em>N</em></span></code>
|
||||
should be replaced by a number from 1 to 14.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="literal">scale_sum/N/</code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="literal">ScaleSum<span class="emphasis"><em>N</em></span></code>,
|
||||
<code class="literal"><span class="emphasis"><em>N</em></span></code> should be replaced by a
|
||||
number from 1 to 14.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code></span></dt>
|
||||
<dd><p>
|
||||
Type of the scale sum swap operation
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">scale_sum_swap2</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">a1</span><span class="special">,</span>
|
||||
<span class="identifier">a2</span><span class="special">,</span>
|
||||
<span class="special">...</span></code></span></dt>
|
||||
<dd><p>
|
||||
Objects of type <code class="computeroutput"><span class="identifier">Value1</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">Value2</span></code>, ...
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">y</span><span class="special">,</span>
|
||||
<span class="identifier">x1</span><span class="special">,</span>
|
||||
<span class="identifier">x2</span><span class="special">,</span>
|
||||
<span class="special">...</span></code></span></dt>
|
||||
<dd><p>
|
||||
Objects of <code class="computeroutput"><span class="identifier">State</span></code>'s
|
||||
value type
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h6>
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions">Valid
|
||||
Expressions</a>
|
||||
</h6>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get scale operation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Operations</span><span class="special">::</span><span class="identifier">scale</span><span class="special"><</span>
|
||||
<span class="identifier">Value</span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Scale</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Get <code class="computeroutput"><span class="identifier">Scale</span></code> from
|
||||
<code class="computeroutput"><span class="identifier">Operations</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Scale</span></code> constructor
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Scale</span><span class="special"><</span>
|
||||
<span class="identifier">Value</span> <span class="special">>(</span>
|
||||
<span class="identifier">a</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Scale</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Constructs a <code class="computeroutput"><span class="identifier">Scale</span></code>
|
||||
object
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Scale</span></code> operation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">scale</span><span class="special">(</span>
|
||||
<span class="identifier">x</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <code class="computeroutput"><span class="identifier">x</span> <span class="special">*=</span> <span class="identifier">a</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get general <code class="computeroutput"><span class="identifier">scale_sum</span></code>
|
||||
operation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="literal">Operations::scale_sum/N/< Value1 , ... , ValueN ></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="literal">ScaleSum/N/</code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Get the <code class="literal">ScaleSum/N/</code> type from <code class="computeroutput"><span class="identifier">Operations</span></code>, <code class="literal"><span class="emphasis"><em>N</em></span></code>
|
||||
should be replaced by a number from 1 to 14.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">scale_sum</span></code> constructor
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="literal">ScaleSum/N/< Value1 , ... , ValueN >( a1 , ...
|
||||
, aN )</code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="literal">ScaleSum/N/</code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Constructs a <code class="computeroutput"><span class="identifier">scale_sum</span></code>
|
||||
object given <code class="literal"><span class="emphasis"><em>N</em></span></code> parameter
|
||||
values with <code class="literal"><span class="emphasis"><em>N</em></span></code> between
|
||||
1 and 14.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">scale_sum</span></code> operation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="literal">scale_sum/N/( y , x1 , ... , xN )</code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> <span class="identifier">a1</span><span class="special">*</span><span class="identifier">x1</span>
|
||||
<span class="special">+</span> <span class="identifier">a2</span><span class="special">*</span><span class="identifier">x2</span>
|
||||
<span class="special">+</span> <span class="special">...</span>
|
||||
<span class="special">+</span> <span class="identifier">aN</span><span class="special">*</span><span class="identifier">xN</span></code>.
|
||||
Note that this is an <code class="literal"><span class="emphasis"><em>N</em></span>+1</code>-ary
|
||||
function call.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get scale sum swap operation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Operations</span><span class="special">::</span><span class="identifier">scale_sum_swap2</span><span class="special"><</span>
|
||||
<span class="identifier">Value1</span> <span class="special">,</span>
|
||||
<span class="identifier">Value2</span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Get scale sum swap from operations
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
|
||||
constructor
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">ScaleSumSwap2</span><span class="special"><</span> <span class="identifier">Value1</span>
|
||||
<span class="special">,</span> <span class="identifier">Value2</span>
|
||||
<span class="special">>(</span> <span class="identifier">a1</span>
|
||||
<span class="special">,</span> <span class="identifier">a2</span>
|
||||
<span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Constructor
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
|
||||
operation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">scale_sum_swap2</span><span class="special">(</span> <span class="identifier">x1</span>
|
||||
<span class="special">,</span> <span class="identifier">x2</span>
|
||||
<span class="special">,</span> <span class="identifier">x3</span>
|
||||
<span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <code class="computeroutput"><span class="identifier">tmp</span> <span class="special">=</span> <span class="identifier">x1</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">x1</span> <span class="special">=</span>
|
||||
<span class="identifier">a1</span><span class="special">*</span><span class="identifier">x2</span> <span class="special">+</span>
|
||||
<span class="identifier">a2</span><span class="special">*</span><span class="identifier">x3</span></code> and <code class="computeroutput"><span class="identifier">x2</span>
|
||||
<span class="special">=</span> <span class="identifier">tmp</span></code>.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra" title="Algebra">Algebra</a>
|
||||
</h4></div></div></div>
|
||||
<h6>
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.notation"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra.notation">Notation</a>
|
||||
</h6>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
|
||||
<dd><p>
|
||||
The state type
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Algebra</span></code></span></dt>
|
||||
<dd><p>
|
||||
The algebra type
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="literal">Operation/N/</code></span></dt>
|
||||
<dd><p>
|
||||
An <code class="literal"><span class="emphasis"><em>N</em></span></code>-ary operation type,
|
||||
<code class="literal"><span class="emphasis"><em>N</em></span></code> should be a number from
|
||||
1 to 14.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">algebra</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">Algebra</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="literal">operation/N/</code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="literal">Operation/N</code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">y</span><span class="special">,</span>
|
||||
<span class="identifier">x1</span><span class="special">,</span>
|
||||
<span class="identifier">x2</span><span class="special">,</span>
|
||||
<span class="special">...</span></code></span></dt>
|
||||
<dd><p>
|
||||
Objects of type <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h6>
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions">Valid
|
||||
Expressions</a>
|
||||
</h6>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Vector Operation with arity 2
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">algebra</span><span class="special">.</span><span class="identifier">for_each2</span><span class="special">(</span>
|
||||
<span class="identifier">y</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span>
|
||||
<span class="identifier">operation2</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
void
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calls <code class="computeroutput"><span class="identifier">operation2</span><span class="special">(</span> <span class="identifier">y_i</span>
|
||||
<span class="special">,</span> <span class="identifier">x_i</span>
|
||||
<span class="special">)</span></code> for each element <code class="computeroutput"><span class="identifier">y_i</span></code> of <code class="computeroutput"><span class="identifier">y</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">x_i</span></code> of <code class="computeroutput"><span class="identifier">x</span></code>.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Vector Operation with arity 3
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">algebra</span><span class="special">.</span><span class="identifier">for_each3</span><span class="special">(</span>
|
||||
<span class="identifier">y</span> <span class="special">,</span>
|
||||
<span class="identifier">x1</span> <span class="special">,</span>
|
||||
<span class="identifier">x2</span> <span class="special">,</span>
|
||||
<span class="identifier">operation3</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
void
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calls <code class="computeroutput"><span class="identifier">operation3</span><span class="special">(</span> <span class="identifier">y_i</span>
|
||||
<span class="special">,</span> <span class="identifier">x1_i</span>
|
||||
<span class="special">,</span> <span class="identifier">x2_i</span>
|
||||
<span class="special">)</span></code> for each element <code class="computeroutput"><span class="identifier">y_i</span></code> of <code class="computeroutput"><span class="identifier">y</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">x1_i</span></code> of
|
||||
<code class="computeroutput"><span class="identifier">x1</span></code> and <code class="computeroutput"><span class="identifier">x2_i</span></code> of <code class="computeroutput"><span class="identifier">x2</span></code>.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Vector Operation with arity <code class="literal"><span class="emphasis"><em>N</em></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="literal">algebra.for_each/N/( y , x1 , ... , xN , operation/N/
|
||||
)</code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
void
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calls <code class="literal">operation/N/( y_i , x1_i , ... , xN_i )</code>
|
||||
for each element <code class="computeroutput"><span class="identifier">y_i</span></code>
|
||||
of <code class="computeroutput"><span class="identifier">y</span></code> and <code class="computeroutput"><span class="identifier">x1_i</span></code> of <code class="computeroutput"><span class="identifier">x1</span></code>
|
||||
and so on. <code class="literal"><span class="emphasis"><em>N</em></span></code> should be
|
||||
replaced by a number between 1 and 14.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.pre_defined_implementations"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.pre_defined_implementations" title="Pre-Defined implementations">Pre-Defined
|
||||
implementations</a>
|
||||
</h4></div></div></div>
|
||||
<p>
|
||||
As standard configuration odeint uses the <code class="computeroutput"><span class="identifier">range_algebra</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">default_operations</span></code>
|
||||
which suffices most situations. However, a few more possibilities exist
|
||||
either to gain better performance or to ensure interoperability with other
|
||||
libraries. In the following we list the existing <code class="computeroutput"><span class="identifier">Algebra</span></code>/<code class="computeroutput"><span class="identifier">Operations</span></code> configurations that can be
|
||||
used in the steppers.
|
||||
</p>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Algebra</span></code>
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">Operations</span></code>
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Remarks
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Anything supporting <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>,
|
||||
like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">list</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>,... based on a <code class="computeroutput"><span class="identifier">value_type</span></code> that supports operators
|
||||
+,* (typically <code class="computeroutput"><span class="keyword">double</span></code>)
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">range_algebra</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">default_operations</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Standard implementation, applicable for most typical situations.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code> based on a <code class="computeroutput"><span class="identifier">value_type</span></code> that supports operators
|
||||
+,*
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">array_algebra</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">default_operations</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Special implementation for boost::array with better performance
|
||||
than <code class="computeroutput"><span class="identifier">range_algebra</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Anything that defines operators + within itself and * with scalar
|
||||
(Mathematically spoken, anything that is a vector space).
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">default_operations</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
For the use of <a class="link" href="controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a>, the template <code class="computeroutput"><span class="identifier">vector_space_reduce</span></code>
|
||||
has to be instantiated.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span></code>, <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">host_vector</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">thrust_algebra</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">thrust_operations</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
For running odeint on CUDA devices by using <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code> or anything which allocates
|
||||
the elements in a C-like manner
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">mkl_operations</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Using the <a href="http://software.intel.com/en-us/articles/intel-mkl/" target="_top">Intel
|
||||
Math Kernel Library</a> in odeint for maximum performance.
|
||||
Currently, only the RK4 stepper is supported.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.state_algebra_operations.example_expressions"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.example_expressions" title="Example expressions">Example
|
||||
expressions</a>
|
||||
</h4></div></div></div>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Vector operation
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">algebra</span><span class="special">.</span><span class="identifier">for_each3</span><span class="special">(</span>
|
||||
<span class="identifier">y</span> <span class="special">,</span>
|
||||
<span class="identifier">x1</span> <span class="special">,</span>
|
||||
<span class="identifier">x2</span> <span class="special">,</span>
|
||||
<span class="identifier">Operations</span><span class="special">::</span><span class="identifier">scale_sum2</span><span class="special"><</span>
|
||||
<span class="identifier">Value1</span> <span class="special">,</span>
|
||||
<span class="identifier">Value2</span> <span class="special">>(</span>
|
||||
<span class="identifier">a1</span> <span class="special">,</span>
|
||||
<span class="identifier">a2</span> <span class="special">)</span>
|
||||
<span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
void
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <span class="emphasis"><em><span class="bold"><strong>y</strong></span> = a1
|
||||
<span class="bold"><strong>x1</strong></span> + a2 <span class="bold"><strong>x2</strong></span></em></span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="dense_output_stepper.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="state_wrapper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,278 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>State Wrapper</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
||||
<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="state_algebra_operations.html" title="State Algebra Operations">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="state_algebra_operations.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.state_wrapper"></a><a class="link" href="state_wrapper.html" title="State Wrapper">State Wrapper</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.state_wrapper.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_wrapper.description"></a></span><a class="link" href="state_wrapper.html#boost_numeric_odeint.concepts.state_wrapper.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
The <code class="computeroutput"><span class="identifier">State</span> <span class="identifier">Wrapper</span></code>
|
||||
concept describes the way odeint creates temporary state objects to store
|
||||
intermediate results within the stepper's <code class="computeroutput"><span class="identifier">do_step</span></code>
|
||||
methods.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.state_wrapper.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_wrapper.notation"></a></span><a class="link" href="state_wrapper.html#boost_numeric_odeint.concepts.state_wrapper.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is the <code class="computeroutput"><span class="identifier">state_type</span></code>
|
||||
of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">WrappedState</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of State Wrapper for the state type <code class="computeroutput"><span class="identifier">State</span></code>.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">w</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">WrappedState</span></code>
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.state_wrapper.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.state_wrapper.valid_expressions"></a></span><a class="link" href="state_wrapper.html#boost_numeric_odeint.concepts.state_wrapper.valid_expressions">Valid
|
||||
Expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get resizeability
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">is_resizeable</span><span class="special"><</span>
|
||||
<span class="identifier">State</span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">false_type</span></code> or <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">true_type</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">true_type</span></code> if the <code class="computeroutput"><span class="identifier">State</span></code> is resizeable, <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">false_type</span></code> otherwise.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Create <code class="computeroutput"><span class="identifier">WrappedState</span></code>
|
||||
type
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">state_wrapper</span><span class="special"><</span>
|
||||
<span class="identifier">State</span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">WrappedState</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Creates the type for a <code class="computeroutput"><span class="identifier">WrappedState</span></code>
|
||||
for the state type <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Constructor
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">WrappedState</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">WrappedState</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Constructs a state wrapper with an empty state
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Copy Constructor
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">WrappedState</span><span class="special">(</span>
|
||||
<span class="identifier">w</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">WrappedState</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Constructs a state wrapper with a state of the same size as the
|
||||
state in <code class="computeroutput"><span class="identifier">w</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get state
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">w</span><span class="special">.</span><span class="identifier">m_v</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns the <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
object of this state wrapper.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Check size
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">w</span><span class="special">.</span><span class="identifier">same_size</span><span class="special">(</span>
|
||||
<span class="identifier">x</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">bool</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns <code class="computeroutput"><span class="keyword">true</span></code> if <code class="computeroutput"><span class="identifier">x</span></code> and <code class="computeroutput"><span class="identifier">w</span><span class="special">.</span><span class="identifier">m_v</span></code>
|
||||
have the same size, <code class="computeroutput"><span class="keyword">false</span></code>
|
||||
otherwise. <span class="bold"><strong>Only required if <code class="computeroutput"><span class="identifier">State</span></code> is resizeable.</strong></span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Resize
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">w</span><span class="special">.</span><span class="identifier">resize</span><span class="special">(</span>
|
||||
<span class="identifier">x</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">bool</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
If <code class="computeroutput"><span class="identifier">w</span><span class="special">.</span><span class="identifier">m_v</span></code> and <code class="computeroutput"><span class="identifier">x</span></code>
|
||||
have different sizes, <code class="computeroutput"><span class="identifier">w</span><span class="special">.</span><span class="identifier">m_v</span></code>
|
||||
is resized to the size of <code class="computeroutput"><span class="identifier">x</span></code>
|
||||
and the method returns <code class="computeroutput"><span class="keyword">true</span></code>.
|
||||
Otherwise the method returns <code class="computeroutput"><span class="keyword">false</span></code>.
|
||||
<span class="bold"><strong>Only required if <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
is resizeable.</strong></span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="state_algebra_operations.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,271 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Stepper</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
||||
<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="implicit_system.html" title="Implicit System">
|
||||
<link rel="next" href="error_stepper.html" title="Error Stepper">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="implicit_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="error_stepper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.stepper"></a><a class="link" href="stepper.html" title="Stepper">Stepper</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
This concepts specifies the interface a simple stepper has to fulfill to
|
||||
be used within the <a class="link" href="../odeint_in_detail/integrate_functions.html" title="Integrate functions">integrate
|
||||
functions</a>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.stepper.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.stepper.description"></a></span><a class="link" href="stepper.html#boost_numeric_odeint.concepts.stepper.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
The basic stepper concept. A basic stepper following this Stepper concept
|
||||
is able to perform a single step of the solution <span class="emphasis"><em>x(t)</em></span>
|
||||
of an ODE to obtain <span class="emphasis"><em>x(t+dt)</em></span> using a given step size
|
||||
<span class="emphasis"><em>dt</em></span>. Basic steppers can be Runge Kutta steppers, symplectic
|
||||
steppers as well as implicit steppers. Depending on the actual stepper, the
|
||||
ODE is defined as <a class="link" href="system.html" title="System">System</a>,
|
||||
<a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a>, <a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>. Note that all error steppers are also basic steppers.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.stepper.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.stepper.refinement_of"></a></span><a class="link" href="stepper.html#boost_numeric_odeint.concepts.stepper.refinement_of">Refinement
|
||||
of</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
DefaultConstructable
|
||||
</li>
|
||||
<li class="listitem">
|
||||
CopyConstructable
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.stepper.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.stepper.associated_types"></a></span><a class="link" href="stepper.html#boost_numeric_odeint.concepts.stepper.associated_types">Associated
|
||||
types</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>state_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">state_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the state of the ODE, hence <span class="emphasis"><em>x</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>deriv_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">deriv_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the derivative of the ODE, hence <span class="emphasis"><em>d x/dt</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>time_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">time_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the dependent variable of the ODE, hence the time
|
||||
<span class="emphasis"><em>t</em></span>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>value_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">value_type</span></code></p>
|
||||
<p>The
|
||||
numerical data type which is used within the stepper, something like
|
||||
<code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">complex</span><span class="special">&</span><span class="identifier">lt</span><span class="special">;</span> <span class="keyword">double</span> <span class="special">&</span><span class="identifier">gt</span><span class="special">;</span></code>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>order_type</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">order_type</span></code></p>
|
||||
<p>The
|
||||
type characterizing the order of the ODE, typically <code class="computeroutput"><span class="keyword">unsigned</span>
|
||||
<span class="keyword">short</span></code>.</p>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<p><span class="bold"><strong>stepper_category</strong></span></p>
|
||||
<p><code class="computeroutput"><span class="identifier">Stepper</span><span class="special">::</span><span class="identifier">stepper_category</span></code></p>
|
||||
<p>A
|
||||
tag type characterizing the category of the stepper. This type must be
|
||||
convertible to <code class="computeroutput"><span class="identifier">stepper_tag</span></code>.</p>
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.stepper.h3"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.stepper.notation"></a></span><a class="link" href="stepper.html#boost_numeric_odeint.concepts.stepper.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Stepper</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of Stepper
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the state <span class="emphasis"><em>x</em></span> of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the time <span class="emphasis"><em>t</em></span> of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">stepper</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object of type <code class="computeroutput"><span class="identifier">Stepper</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code>, <code class="computeroutput"><span class="identifier">dt</span></code></span></dt>
|
||||
<dd><p>
|
||||
Objects of type <code class="computeroutput"><span class="identifier">Time</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object defining the ODE. Depending on the Stepper this might be
|
||||
a model of <a class="link" href="system.html" title="System">System</a>,
|
||||
<a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a>, <a class="link" href="simple_symplectic_system.html" title="Simple Symplectic System">Simple
|
||||
Symplectic System</a> or <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.stepper.h4"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.stepper.valid_expressions"></a></span><a class="link" href="stepper.html#boost_numeric_odeint.concepts.stepper.valid_expressions">Valid Expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Get the order
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">order</span><span class="special">()</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">order_type</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns the order of the stepper.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Do step
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">stepper</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span>
|
||||
<span class="identifier">sys</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span>
|
||||
<span class="identifier">t</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Performs one step of step size <code class="computeroutput"><span class="identifier">dt</span></code>.
|
||||
The newly obtained state is written in place in <code class="computeroutput"><span class="identifier">x</span></code>.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.stepper.h5"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.stepper.models"></a></span><a class="link" href="stepper.html#boost_numeric_odeint.concepts.stepper.models">Models</a>
|
||||
</h5>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta4</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">euler</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_cash_karp54</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_fehlberg78</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">modified_midpoint</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4</span></code>
|
||||
</li>
|
||||
</ul></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="implicit_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="error_stepper.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,233 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Symplectic System</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
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<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="system.html" title="System">
|
||||
<link rel="next" href="simple_symplectic_system.html" title="Simple Symplectic System">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="simple_symplectic_system.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.symplectic_system"></a><a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.symplectic_system.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.symplectic_system.description"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
This concept describes how to define a symplectic system written with generalized
|
||||
coordinate <code class="computeroutput"><span class="identifier">q</span></code> and generalized
|
||||
momentum <code class="computeroutput"><span class="identifier">p</span></code>:
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>q'(t) = f(p) </em></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>p'(t) = g(q) </em></span>
|
||||
</p>
|
||||
<p>
|
||||
Such a situation is typically found for Hamiltonian systems with a separable
|
||||
Hamiltonian:
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>H(p,q) = H<sub>kin</sub>(p) + V(q) </em></span>
|
||||
</p>
|
||||
<p>
|
||||
which gives the equations of motion:
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>q'(t) = dH<sub>kin</sub> / dp = f(p) </em></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>p'(t) = dV / dq = g(q) </em></span>
|
||||
</p>
|
||||
<p>
|
||||
The algorithmic implementation of this situation is described by a pair of
|
||||
callable objects for <span class="emphasis"><em>f</em></span> and <span class="emphasis"><em>g</em></span> with
|
||||
a specific parameter signature. Such a system should be implemented as a
|
||||
std::pair of functions or a functors. Symplectic systems are used in symplectic
|
||||
steppers like <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.symplectic_system.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.symplectic_system.notation"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of Symplectic_System
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Coor</span></code></span></dt>
|
||||
<dd><p>
|
||||
The type of the coordinate <span class="emphasis"><em>q</em></span>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Momentum</span></code></span></dt>
|
||||
<dd><p>
|
||||
The type of the momentum <span class="emphasis"><em>p</em></span>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">CoorDeriv</span></code></span></dt>
|
||||
<dd><p>
|
||||
The type of the derivative of coordinate <span class="emphasis"><em>q'</em></span>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">MomentumDeriv</span></code></span></dt>
|
||||
<dd><p>
|
||||
The type of the derivative of momentum <span class="emphasis"><em>p'</em></span>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object of the type <code class="computeroutput"><span class="identifier">System</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">q</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type Coor
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">p</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type Momentum
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">dqdt</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type CoorDeriv
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">dpdt</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type MomentumDeriv
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.symplectic_system.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.symplectic_system.valid_expressions"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.valid_expressions">Valid
|
||||
expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Check for pair
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">is_pair</span><span class="special"><</span>
|
||||
<span class="identifier">System</span> <span class="special">>::</span><span class="identifier">type</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">true_</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Check if System is a pair
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Calculate <span class="emphasis"><em>dq/dt = f(p)</em></span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span>
|
||||
<span class="identifier">p</span> <span class="special">,</span>
|
||||
<span class="identifier">dqdt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <span class="emphasis"><em>f(p)</em></span>, the result is stored into
|
||||
<code class="computeroutput"><span class="identifier">dqdt</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Calculate <span class="emphasis"><em>dp/dt = g(q)</em></span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span>
|
||||
<span class="identifier">q</span> <span class="special">,</span>
|
||||
<span class="identifier">dpdt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates <span class="emphasis"><em>g(q)</em></span>, the result is stored into
|
||||
<code class="computeroutput"><span class="identifier">dpdt</span></code>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="simple_symplectic_system.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,150 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>System</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
||||
<link rel="up" href="../concepts.html" title="Concepts">
|
||||
<link rel="prev" href="../concepts.html" title="Concepts">
|
||||
<link rel="next" href="symplectic_system.html" title="Symplectic System">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="../concepts.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="symplectic_system.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.concepts.system"></a><a class="link" href="system.html" title="System">System</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.system.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.system.description"></a></span><a class="link" href="system.html#boost_numeric_odeint.concepts.system.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
The System concept models the algorithmic implementation of the rhs. of the
|
||||
ODE <span class="emphasis"><em>x' = f(x,t)</em></span>. The only requirement for this concept
|
||||
is that it should be callable with a specific parameter syntax (see below).
|
||||
A System is typically implemented as a function or a functor. Systems fulfilling
|
||||
this concept are required by all Runge-Kutta steppers as well as the Bulirsch-Stoer
|
||||
steppers. However, symplectic and implicit steppers work with other system
|
||||
concepts, see <a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
|
||||
System</a> and <a class="link" href="implicit_system.html" title="Implicit System">Implicit
|
||||
System</a>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.system.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.system.notation"></a></span><a class="link" href="system.html#boost_numeric_odeint.concepts.system.notation">Notation</a>
|
||||
</h5>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type that is a model of System
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the state <span class="emphasis"><em>x</em></span> of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Deriv</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the derivative <span class="emphasis"><em>x'</em></span> of the ODE
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
|
||||
<dd><p>
|
||||
A type representing the time
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
|
||||
<dd><p>
|
||||
An object of type <code class="computeroutput"><span class="identifier">System</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">State</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">dxdt</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">Deriv</span></code>
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code></span></dt>
|
||||
<dd><p>
|
||||
Object of type <code class="computeroutput"><span class="identifier">Time</span></code>
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.concepts.system.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.concepts.system.valid_expressions"></a></span><a class="link" href="system.html#boost_numeric_odeint.concepts.system.valid_expressions">Valid expressions</a>
|
||||
</h5>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Name
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Type
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Semantics
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Calculate <span class="emphasis"><em>dx/dt := f(x,t)</em></span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">sys</span><span class="special">(</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span>
|
||||
<span class="identifier">dxdt</span> <span class="special">,</span>
|
||||
<span class="identifier">t</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="keyword">void</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Calculates f(x,t), the result is stored into dxdt
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
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<a accesskey="p" href="../concepts.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="symplectic_system.html"><img src="../../images/next.png" alt="Next"></a>
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<a name="boost_numeric_odeint.getting_started"></a><a class="link" href="getting_started.html" title="Getting started">Getting started</a>
|
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</h2></div></div></div>
|
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<div class="toc"><dl>
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<dt><span class="section"><a href="getting_started/overview.html">Overview</a></span></dt>
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<dt><span class="section"><a href="getting_started/usage__compilation__headers.html">Usage,
|
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Compilation, Headers</a></span></dt>
|
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<dt><span class="section"><a href="getting_started/short_example.html">Short
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Example</a></span></dt>
|
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</dl></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
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<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
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<a accesskey="p" href="usage__compilation__headers.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../getting_started.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="../tutorial.html"><img src="../../images/next.png" alt="Next"></a>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
|
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<a name="boost_numeric_odeint.getting_started.short_example"></a><a class="link" href="short_example.html" title="Short Example">Short
|
||||
Example</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
Imaging, you want to numerically integrate a harmonic oscillator with friction.
|
||||
The equations of motion are given by <span class="emphasis"><em>x'' = -x + γ x'</em></span>.
|
||||
Odeint only deals with first order ODEs that have no higher derivatives than
|
||||
x' involved. However, any higher order ODE can be transformed to a system
|
||||
of first order ODEs by introducing the new variables <span class="emphasis"><em>q=x</em></span>
|
||||
and <span class="emphasis"><em>p=x'</em></span> such that <span class="emphasis"><em>w=(q,p)</em></span>. To
|
||||
apply numerical integration one first has to design the right hand side of
|
||||
the equation <span class="emphasis"><em>w' = f(w) = (p,-q+γ p)</em></span>:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">/* The type of container used to hold the state vector */</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">gam</span> <span class="special">=</span> <span class="number">0.15</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">/* The rhs of x' = f(x) */</span>
|
||||
<span class="keyword">void</span> <span class="identifier">harmonic_oscillator</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">gam</span><span class="special">*</span><span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Here we chose <code class="computeroutput"><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span></code>
|
||||
as the state type, but others are also possible, for example <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span><span class="number">2</span><span class="special">></span></code>. odeint is designed in such a way that
|
||||
you can easily use your own state types. Next, the ODE is defined which is
|
||||
in this case a simple function calculating <span class="emphasis"><em>f(x)'</em></span>. The
|
||||
parameter signature of this function is crucial: the integration methods
|
||||
will always call them in the form <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span>
|
||||
<span class="identifier">dxdt</span><span class="special">,</span>
|
||||
<span class="identifier">t</span><span class="special">)</span></code>
|
||||
(there are exceptions for some special routines). So, even if there is no
|
||||
explicit time dependence, one has to define <code class="computeroutput"><span class="identifier">t</span></code>
|
||||
as a function parameter.
|
||||
</p>
|
||||
<p>
|
||||
Now, we have to define the initial state from which the integration should
|
||||
start:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">(</span><span class="number">2</span><span class="special">);</span>
|
||||
<span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="comment">// start at x=1.0, p=0.0</span>
|
||||
<span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
For the integration itself we'll use the <code class="computeroutput">integrate</code>
|
||||
function, which is a convenient way to get quick results. It is based on
|
||||
the error-controlled <code class="computeroutput">runge_kutta_rk5_ck</code>
|
||||
stepper (5th order) and uses adaptive step-size.
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.1</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The integrate function expects as parameters the rhs of the ode as defined
|
||||
above, the initial state <code class="computeroutput"><span class="identifier">x</span></code>,
|
||||
the start-and end-time of the integration as well as the initial time step=size.
|
||||
Note, that <code class="computeroutput">integrate</code>
|
||||
uses an adaptive step-size during the integration steps so the time points
|
||||
will not be equally spaced. The integration returns the number of steps that
|
||||
were applied and updates x which is set to the approximate solution of the
|
||||
ODE at the end of integration.
|
||||
</p>
|
||||
<p>
|
||||
It is, of course, also possible to represent the ode system as a class. The
|
||||
rhs must then be implemented as a functor having defined the ()-operator:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">/* The rhs of x' = f(x) defined as a class */</span>
|
||||
<span class="keyword">class</span> <span class="identifier">harm_osc</span> <span class="special">{</span>
|
||||
|
||||
<span class="keyword">double</span> <span class="identifier">m_gam</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
<span class="identifier">harm_osc</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">gam</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_gam</span><span class="special">(</span><span class="identifier">gam</span><span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()</span> <span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">m_gam</span><span class="special">*</span><span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
which can be used via
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">harm_osc</span> <span class="identifier">ho</span><span class="special">(</span><span class="number">0.15</span><span class="special">);</span>
|
||||
<span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">ho</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.1</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
You surely have already noticed that during the integration a lot of steps
|
||||
had to be done. You might wonder if you could access them do observe the
|
||||
solution during the iteration. Yes, this is possible, of course. All you
|
||||
have to do is to provide a reasonable observer. An example is
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">push_back_state_and_time</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>&</span> <span class="identifier">m_states</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">>&</span> <span class="identifier">m_times</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">push_back_state_and_time</span><span class="special">(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="special">&</span><span class="identifier">states</span> <span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="special">&</span><span class="identifier">times</span> <span class="special">)</span>
|
||||
<span class="special">:</span> <span class="identifier">m_states</span><span class="special">(</span> <span class="identifier">states</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_times</span><span class="special">(</span> <span class="identifier">times</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">m_states</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span>
|
||||
<span class="identifier">m_times</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
which stores the intermediate steps in a container. Note, the argument structure
|
||||
of the ()-operator: odeint calls the observer exactly in this way, providing
|
||||
the current state and time. Now, you only have to pass this container to
|
||||
the integration function:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">vector</span><span class="special"><</span><span class="identifier">state_type</span><span class="special">></span> <span class="identifier">x_vec</span><span class="special">;</span>
|
||||
<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">times</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="number">0.1</span> <span class="special">,</span>
|
||||
<span class="identifier">push_back_state_and_time</span><span class="special">(</span> <span class="identifier">x_vec</span> <span class="special">,</span> <span class="identifier">times</span> <span class="special">)</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">/* output */</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span><span class="special">;</span> <span class="identifier">i</span><span class="special"><=</span><span class="identifier">steps</span><span class="special">;</span> <span class="identifier">i</span><span class="special">++</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">times</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="char">'\t'</span> <span class="special"><<</span> <span class="identifier">x_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">][</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="char">'\t'</span> <span class="special"><<</span> <span class="identifier">x_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">][</span><span class="number">1</span><span class="special">]</span> <span class="special"><<</span> <span class="char">'\n'</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
That is all. Of course, you can use functional libraries like <a href="http://www.boost.org/doc/libs/release/doc/html/lambda.html" target="_top">Boost.Lambda</a>
|
||||
or <a href="http://www.boost.org/doc/libs/1_46_1/libs/spirit/phoenix/doc/html/index.html" target="_top">Boost.Phoenix</a>
|
||||
to ease the creation of observer functions.
|
||||
</p>
|
||||
<p>
|
||||
The full cpp file for this example can be found here: <a href="../../../../examples/harmonic_oscillator.cpp" target="_top">../../examples/harmonic_oscillator.cpp</a>
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
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<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Usage, Compilation, Headers</title>
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<div class="titlepage"><div><div><h3 class="title">
|
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<a name="boost_numeric_odeint.getting_started.usage__compilation__headers"></a><a class="link" href="usage__compilation__headers.html" title="Usage, Compilation, Headers">Usage,
|
||||
Compilation, Headers</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
odeint is completely header-only, no linking against pre-compiled code is
|
||||
required. It can be include by
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">numeric</span><span class="special">/</span><span class="identifier">odeint</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<p>
|
||||
which includes all headers of the library. All functions and classes from
|
||||
odeint live in the namespace
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
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<hr>
|
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<div class="spirit-nav">
|
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|
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<html>
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<head>
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||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>odeint in detail</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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</div>
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<div class="section">
|
||||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail"></a><a class="link" href="odeint_in_detail.html" title="odeint in detail">odeint in detail</a>
|
||||
</h2></div></div></div>
|
||||
<div class="toc"><dl>
|
||||
<dt><span class="section"><a href="odeint_in_detail/steppers.html">Steppers</a></span></dt>
|
||||
<dt><span class="section"><a href="odeint_in_detail/generation_functions.html">Generation
|
||||
functions</a></span></dt>
|
||||
<dt><span class="section"><a href="odeint_in_detail/integrate_functions.html">Integrate
|
||||
functions</a></span></dt>
|
||||
<dt><span class="section"><a href="odeint_in_detail/iterators.html">Iterators</a></span></dt>
|
||||
<dt><span class="section"><a href="odeint_in_detail/state_types__algebras_and_operations.html">State
|
||||
types, algebras and operations</a></span></dt>
|
||||
<dt><span class="section"><a href="odeint_in_detail/using_boost__ref.html">Using
|
||||
boost::ref</a></span></dt>
|
||||
<dt><span class="section"><a href="odeint_in_detail/using_boost__range.html">Using
|
||||
boost::range</a></span></dt>
|
||||
</dl></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
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|
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|
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@ -1,301 +0,0 @@
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||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
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<title>Generation functions</title>
|
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<link rel="stylesheet" href="../../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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<link rel="prev" href="steppers.html" title="Steppers">
|
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<link rel="next" href="integrate_functions.html" title="Integrate functions">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<div class="spirit-nav">
|
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<a accesskey="p" href="steppers.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="integrate_functions.html"><img src="../../images/next.png" alt="Next"></a>
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</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.generation_functions"></a><a class="link" href="generation_functions.html" title="Generation functions">Generation
|
||||
functions</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
In the <a class="link" href="../tutorial.html" title="Tutorial">Tutorial</a> we have
|
||||
learned how we can use the generation functions <code class="computeroutput"><span class="identifier">make_controlled</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">make_dense_output</span></code> to
|
||||
create controlled and dense output stepper from a simple stepper or an error
|
||||
stepper. The syntax of these two functions is very simple:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">stepper1</span> <span class="special">=</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">);</span>
|
||||
<span class="keyword">auto</span> <span class="identifier">stepper2</span> <span class="special">=</span> <span class="identifier">make_dense_output</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The first two parameters are the absolute and the relative error tolerances
|
||||
and the third parameter is the stepper. In C++03 you can infer the type from
|
||||
the <code class="computeroutput"><span class="identifier">result_of</span></code> mechanism:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">result_of</span><span class="special">::</span><span class="identifier">make_controlled</span><span class="special"><</span> <span class="identifier">stepper_type</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">stepper3</span> <span class="special">=</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">);</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">result_of</span><span class="special">::</span><span class="identifier">make_dense_output</span><span class="special"><</span> <span class="identifier">stepper_type</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">stepper4</span> <span class="special">=</span> <span class="identifier">make_dense_output</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
To use your own steppers with the <code class="computeroutput"><span class="identifier">make_controlled</span></code>
|
||||
or <code class="computeroutput"><span class="identifier">make_dense_output</span></code> you
|
||||
need to specialize two class templates. Suppose your steppers are called
|
||||
<code class="computeroutput"><span class="identifier">custom_stepper</span></code>, <code class="computeroutput"><span class="identifier">custom_controller</span></code> and <code class="computeroutput"><span class="identifier">custom_dense_output</span></code>.
|
||||
Then, the first class you need to specialize is <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">get_controller</span></code>,
|
||||
a meta function returning the type of the controller:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">numeric</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">odeint</span> <span class="special">{</span>
|
||||
|
||||
<span class="keyword">template</span><span class="special"><></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">get_controller</span><span class="special"><</span> <span class="identifier">custom_stepper</span> <span class="special">></span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">custom_controller</span> <span class="identifier">type</span><span class="special">;</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="special">}</span> <span class="special">}</span> <span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The second one is a factory class <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">controller_factory</span></code>
|
||||
which constructs the controller from the tolerances and the stepper. In our
|
||||
dummy implementation this class is
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">numeric</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">odeint</span> <span class="special">{</span>
|
||||
|
||||
<span class="keyword">template</span><span class="special"><></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">controller_factory</span><span class="special"><</span> <span class="identifier">custom_stepper</span> <span class="special">,</span> <span class="identifier">custom_controller</span> <span class="special">></span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">custom_controller</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">double</span> <span class="identifier">abs_tol</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">rel_tol</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">custom_stepper</span> <span class="special">&</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">return</span> <span class="identifier">custom_controller</span><span class="special">();</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="special">}</span> <span class="special">}</span> <span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
This is all to use the <code class="computeroutput"><span class="identifier">make_controlled</span></code>
|
||||
mechanism. Now you can use your controller via
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">stepper5</span> <span class="special">=</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">custom_stepper</span><span class="special">()</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
For the dense_output_stepper everything works similar. Here you have to specialize
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">get_dense_output</span></code> and <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">::</span><span class="identifier">dense_output_factory</span></code>.
|
||||
These two classes have the same syntax as their relatives <code class="computeroutput"><span class="identifier">get_controller</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">controller_factory</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
Of course, all controllers and dense-output steppers in odeint can be used
|
||||
with these mechanisms. In the table below you will find, which steppers is
|
||||
constructed from <code class="computeroutput"><span class="identifier">make_controlled</span></code>
|
||||
or <code class="computeroutput"><span class="identifier">make_dense_output</span></code> if applied
|
||||
on a stepper from odeint:
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.generation_functions.generation_functions_make_controlled__abs_error___rel_error___stepper__"></a><p class="title"><b>Table 1.8. Generation functions make_controlled( abs_error , rel_error , stepper
|
||||
)</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Generation functions make_controlled( abs_error , rel_error , stepper
|
||||
)">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Stepper
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Result of make_controlled
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Remarks
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_cash_karp54</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">runge_kutta_cash_karp54</span>
|
||||
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special"><...></span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="emphasis"><em>a<sub>x</sub>=1</em></span>, <span class="emphasis"><em>a<sub>dxdt</sub>=1</em></span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_fehlberg78</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">runge_kutta_fehlberg78</span>
|
||||
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special"><...></span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="emphasis"><em>a<sub>x</sub>=1</em></span>, <span class="emphasis"><em>a<sub>dxdt</sub>=1</em></span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span>
|
||||
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special"><...></span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="emphasis"><em>a <sub>x</sub>=1</em></span>, <span class="emphasis"><em>a<sub>dxdt</sub>=1</em></span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4_controlled</span><span class="special"><</span> <span class="identifier">rosenbrock4</span>
|
||||
<span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
-
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.generation_functions.generation_functions_make_dense_output__abs_error___rel_error___stepper__"></a><p class="title"><b>Table 1.9. Generation functions make_dense_output( abs_error , rel_error , stepper
|
||||
)</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Generation functions make_dense_output( abs_error , rel_error , stepper
|
||||
)">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Stepper
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Result of make_dense_output
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Remarks
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">dense_output_runge_kutta</span><span class="special"><</span> <span class="identifier">controlled_runge_kutta</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span>
|
||||
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special"><...></span> <span class="special">></span>
|
||||
<span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="emphasis"><em>a <sub>x</sub>=1</em></span>, <span class="emphasis"><em>a<sub>dxdt</sub>=1</em></span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4</span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">rosenbrock4_dense_output</span><span class="special"><</span> <span class="identifier">rosenbrock4_controller</span><span class="special"><</span> <span class="identifier">rosenbrock4</span>
|
||||
<span class="special">></span> <span class="special">></span></code>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
-
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break">
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="steppers.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="integrate_functions.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,340 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Integrate functions</title>
|
||||
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
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<link rel="up" href="../odeint_in_detail.html" title="odeint in detail">
|
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<link rel="prev" href="generation_functions.html" title="Generation functions">
|
||||
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|
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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||||
<div class="spirit-nav">
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||||
<a accesskey="p" href="generation_functions.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="iterators.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions"></a><a class="link" href="integrate_functions.html" title="Integrate functions">Integrate
|
||||
functions</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
Integrate functions perform the time evolution of a given ODE from some starting
|
||||
time <span class="emphasis"><em>t<sub>0</sub></em></span> to a given end time <span class="emphasis"><em>t<sub>1</sub></em></span>
|
||||
and starting at state <span class="emphasis"><em>x<sub>0</sub></em></span> by subsequent calls of a given
|
||||
stepper's <code class="computeroutput"><span class="identifier">do_step</span></code> function.
|
||||
Additionally, the user can provide an __observer to analyze the state during
|
||||
time evolution. There are five different integrate functions which have different
|
||||
strategies on when to call the observer function during integration. All
|
||||
of the integrate functions except <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code>
|
||||
can be called with any stepper following one of the stepper concepts: <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> , <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
|
||||
Stepper</a> , <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a> , <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
|
||||
Output Stepper</a>. Depending on the abilities of the stepper, the integrate
|
||||
functions make use of step-size control or dense output.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h0"></a>
|
||||
<span><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls">Equidistant
|
||||
observer calls</a>
|
||||
</h5>
|
||||
<p>
|
||||
If observer calls at equidistant time intervals <span class="emphasis"><em>dt</em></span> are
|
||||
needed, the <code class="computeroutput"><span class="identifier">integrate_const</span></code>
|
||||
or <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> function
|
||||
should be used. We start with explaining <code class="computeroutput"><span class="identifier">integrate_const</span></code>:
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">t1</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">t1</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">observer</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
These integrate the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
|
||||
with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>.
|
||||
Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and
|
||||
<code class="computeroutput"><span class="identifier">x0</span></code> and ends at some <span class="emphasis"><em>t'
|
||||
= t<sub>0</sub> + n dt</em></span> with <span class="emphasis"><em>n</em></span> such that <span class="emphasis"><em>t<sub>1</sub> -
|
||||
dt < t' <= t<sub>1</sub></em></span>. <code class="computeroutput"><span class="identifier">x0</span></code>
|
||||
is changed to the approximative solution <span class="emphasis"><em>x(t')</em></span> at the
|
||||
end of integration. If provided, the <code class="computeroutput"><span class="identifier">observer</span></code>
|
||||
is invoked at times <span class="emphasis"><em>t<sub>0</sub></em></span>, <span class="emphasis"><em>t<sub>0</sub> + dt</em></span>,
|
||||
<span class="emphasis"><em>t<sub>0</sub> + 2dt</em></span>, ... ,<span class="emphasis"><em>t'</em></span>. <code class="computeroutput"><span class="identifier">integrate_const</span></code> returns the number of steps
|
||||
performed during the integration. Note that if you are using a simple <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
|
||||
Stepper</a> and want to make exactly <code class="computeroutput"><span class="identifier">n</span></code>
|
||||
steps you should prefer the <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code>
|
||||
function below.
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error Stepper</a>
|
||||
then <code class="computeroutput"><span class="identifier">dt</span></code> is also the step
|
||||
size used for integration and the observer is called just after every
|
||||
step.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
is the initial step size. The actual step size will change due to error
|
||||
control during time evolution. However, if an observer is provided the
|
||||
step size will be adjusted such that the algorithm always calculates
|
||||
<span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n dt</em></span> and calls
|
||||
the observer at that point. Note that the use of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a> is reasonable here only if <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
is considerably larger than typical step sizes used by the stepper.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense Output
|
||||
Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
is the initial step size. The actual step size will be adjusted during
|
||||
integration due to error control. If an observer is provided dense output
|
||||
is used to calculate <span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n
|
||||
dt</em></span>.
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h1"></a>
|
||||
<span><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps">Integrate
|
||||
a given number of steps</a>
|
||||
</h5>
|
||||
<p>
|
||||
This function is very similar to <code class="computeroutput"><span class="identifier">integrate_const</span></code>
|
||||
above. The only difference is that it does not take the end time as parameter,
|
||||
but rather the number of steps. The integration is then performed until the
|
||||
time <code class="computeroutput"><span class="identifier">t0</span><span class="special">+</span><span class="identifier">n</span><span class="special">*</span><span class="identifier">dt</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">n</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">n</span> <span class="special">,</span> <span class="identifier">observer</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
|
||||
with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>
|
||||
starting at <span class="emphasis"><em>x<sub>0</sub></em></span> and <span class="emphasis"><em>t<sub>0</sub></em></span>. If provided,
|
||||
<code class="computeroutput"><span class="identifier">observer</span></code> is called after
|
||||
every step and at the beginning with <code class="computeroutput"><span class="identifier">t0</span></code>,
|
||||
similar as above. The approximate result for <span class="emphasis"><em>x( t<sub>0</sub> + n dt )</em></span>
|
||||
is stored in <code class="computeroutput"><span class="identifier">x0</span></code>. This function
|
||||
returns the end time <code class="computeroutput"><span class="identifier">t0</span> <span class="special">+</span> <span class="identifier">n</span><span class="special">*</span><span class="identifier">dt</span></code>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h2"></a>
|
||||
<span><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step">Observer
|
||||
calls at each step</a>
|
||||
</h5>
|
||||
<p>
|
||||
If the observer should be called at each time step then the <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> function should be used.
|
||||
Note that in the case of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a> or <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
|
||||
Output Stepper</a> this leads to non-equidistant observer calls as the
|
||||
step size changes.
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">t1</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">t1</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">observer</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
|
||||
with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>.
|
||||
Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and
|
||||
<code class="computeroutput"><span class="identifier">x0</span></code> and ends at <span class="emphasis"><em>t<sub>1</sub></em></span>.
|
||||
<code class="computeroutput"><span class="identifier">x0</span></code> is changed to the approximative
|
||||
solution <span class="emphasis"><em>x(t<sub>1</sub>)</em></span> at the end of integration. If provided,
|
||||
the <code class="computeroutput"><span class="identifier">observer</span></code> is called after
|
||||
each step (and before the first step at <code class="computeroutput"><span class="identifier">t0</span></code>).
|
||||
<code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> returns
|
||||
the number of steps performed during the integration.
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error Stepper</a>
|
||||
then <code class="computeroutput"><span class="identifier">dt</span></code> is the step size
|
||||
used for integration and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code>
|
||||
behaves like <code class="computeroutput"><span class="identifier">integrate_const</span></code>
|
||||
except that for the last step the step size is reduced to ensure we end
|
||||
exactly at <code class="computeroutput"><span class="identifier">t1</span></code>. If provided,
|
||||
the observer is called at each step.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
is the initial step size. The actual step size is changed according to
|
||||
error control of the stepper. For the last step, the step size will be
|
||||
reduced to ensure we end exactly at <code class="computeroutput"><span class="identifier">t1</span></code>.
|
||||
If provided, the observer is called after each time step (and before
|
||||
the first step at <code class="computeroutput"><span class="identifier">t0</span></code>).
|
||||
</li>
|
||||
<li class="listitem">
|
||||
If stepper is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
|
||||
Output Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
is the initial step size and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code>
|
||||
behaves just like for <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a> above. No dense output is used.
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h3"></a>
|
||||
<span><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points">Observer
|
||||
calls at given time points</a>
|
||||
</h5>
|
||||
<p>
|
||||
If the observer should be called at some user given time points the <code class="computeroutput"><span class="identifier">integrate_times</span></code> function should be used.
|
||||
The times for observer calls are provided as a sequence of time values. The
|
||||
sequence is either defined via two iterators pointing to begin and end of
|
||||
the sequence or in terms of a <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
object.
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">times_start</span> <span class="special">,</span>
|
||||
<span class="identifier">times_end</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">observer</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span>
|
||||
<span class="identifier">stepper</span> <span class="special">,</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">time_range</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">observer</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
|
||||
with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>.
|
||||
Integration starts at <code class="computeroutput"><span class="special">*</span><span class="identifier">times_start</span></code>
|
||||
and ends exactly at <code class="computeroutput"><span class="special">*(</span><span class="identifier">times_end</span><span class="special">-</span><span class="number">1</span><span class="special">)</span></code>.
|
||||
<code class="computeroutput"><span class="identifier">x0</span></code> contains the approximate
|
||||
solution at the end point of integration. This function requires an observer
|
||||
which is invoked at the subsequent times <code class="computeroutput"><span class="special">*</span><span class="identifier">times_start</span><span class="special">++</span></code>
|
||||
until <code class="computeroutput"><span class="identifier">times_start</span> <span class="special">==</span>
|
||||
<span class="identifier">times_end</span></code>. If called with a <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
<code class="computeroutput"><span class="identifier">time_range</span></code> the function behaves
|
||||
the same with <code class="computeroutput"><span class="identifier">times_start</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">time_range</span>
|
||||
<span class="special">)</span></code> and <code class="computeroutput"><span class="identifier">times_end</span>
|
||||
<span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">end</span><span class="special">(</span>
|
||||
<span class="identifier">time_range</span> <span class="special">)</span></code>.
|
||||
<code class="computeroutput"><span class="identifier">integrate_times</span></code> returns the
|
||||
number of steps performed during the integration.
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error Stepper</a>
|
||||
<code class="computeroutput"><span class="identifier">dt</span></code> is the step size used
|
||||
for integration. However, whenever a time point from the sequence is
|
||||
approached the step size <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
will be reduced to obtain the state <span class="emphasis"><em>x(t)</em></span> exactly
|
||||
at the time point.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
|
||||
Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
is the initial step size. The actual step size is adjusted during integration
|
||||
according to error control. However, if a time point from the sequence
|
||||
is approached the step size is reduced to obtain the state <span class="emphasis"><em>x(t)</em></span>
|
||||
exactly at the time point.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense Output
|
||||
Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
|
||||
is the initial step size. The actual step size is adjusted during integration
|
||||
according to error control. Dense output is used to obtain the states
|
||||
<span class="emphasis"><em>x(t)</em></span> at the time points from the sequence.
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h4"></a>
|
||||
<span><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function">Convenience
|
||||
integrate function</a>
|
||||
</h5>
|
||||
<p>
|
||||
Additionally to the sophisticated integrate function above odeint also provides
|
||||
a simple <code class="computeroutput"><span class="identifier">integrate</span></code> routine
|
||||
which uses a dense output stepper based on <code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
|
||||
with standard error bounds <span class="emphasis"><em>10<sup>-6</sup></em></span> for the steps.
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">t1</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
<code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span>
|
||||
<span class="identifier">system</span> <span class="special">,</span>
|
||||
<span class="identifier">x0</span> <span class="special">,</span>
|
||||
<span class="identifier">t0</span> <span class="special">,</span>
|
||||
<span class="identifier">t1</span> <span class="special">,</span>
|
||||
<span class="identifier">dt</span> <span class="special">,</span>
|
||||
<span class="identifier">observer</span> <span class="special">)</span></code>
|
||||
</p>
|
||||
<p>
|
||||
This function behaves exactly like <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code>
|
||||
above but no stepper has to be provided. It also returns the number of steps
|
||||
performed during the integration.
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="generation_functions.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="iterators.html"><img src="../../images/next.png" alt="Next"></a>
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||||
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||||
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|
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|
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||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Using boost::range</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="up" href="../odeint_in_detail.html" title="odeint in detail">
|
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<link rel="prev" href="using_boost__ref.html" title="Using boost::ref">
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<link rel="next" href="../concepts.html" title="Concepts">
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</head>
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<div class="spirit-nav">
|
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<a accesskey="p" href="using_boost__ref.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="../concepts.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.using_boost__range"></a><a class="link" href="using_boost__range.html" title="Using boost::range">Using
|
||||
boost::range</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
Most steppers in odeint also accept the state give as a range. A range is
|
||||
sequence of values modeled by a range concept. See <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
for an overview over existing concepts and examples of ranges. This means
|
||||
that the <code class="computeroutput"><span class="identifier">state_type</span></code> of the
|
||||
stepper must not necessarily be used to call the <code class="computeroutput"><span class="identifier">do_step</span></code>
|
||||
method.
|
||||
</p>
|
||||
<p>
|
||||
One use-case for <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
in odeint has been shown in <a class="link" href="../tutorial/chaotic_systems_and_lyapunov_exponents.html" title="Chaotic systems and Lyapunov exponents">Chaotic
|
||||
System</a> where the state consists of two parts: one for the original
|
||||
system and one for the perturbations. The ranges are used to initialize (solve)
|
||||
only the system part where the perturbation part is not touched, that is
|
||||
a range consisting only of the system part is used. After that the complete
|
||||
state including the perturbations is solved.
|
||||
</p>
|
||||
<p>
|
||||
Another use case is a system consisting of coupled units where you want to
|
||||
initialize each unit separately with the ODE of the uncoupled unit. An example
|
||||
is a chain of coupled van-der-Pol-oscillators which are initialized uniformly
|
||||
from the uncoupled van-der-Pol-oscillator. Then you can use <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
to solve only one individual oscillator in the chain.
|
||||
</p>
|
||||
<p>
|
||||
In short, you can <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
to use one state within two system functions which expect states with different
|
||||
sizes.
|
||||
</p>
|
||||
<p>
|
||||
An example was given in the <a class="link" href="../tutorial/chaotic_systems_and_lyapunov_exponents.html" title="Chaotic systems and Lyapunov exponents">Chaotic
|
||||
System</a> tutorial. Using Boost.Range usually means that your system
|
||||
function needs to adapt to the iterators of Boost.Range. That is, your function
|
||||
is called with a range and you need to get the iterators from that range.
|
||||
This can easily be done. You have to implement your system as a class or
|
||||
a struct and you have to templatize the <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>. Then you can use the <code class="computeroutput"><span class="identifier">range_iterator</span></code>-meta
|
||||
function and <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span></code> and <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">end</span></code> to
|
||||
obtain the iterators of your range:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">sys</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Deriv</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x_</span> <span class="special">,</span> <span class="identifier">Deriv</span> <span class="special">&</span><span class="identifier">dxdt_</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x_</span> <span class="special">);</span>
|
||||
<span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="identifier">Deriv</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">dxdt</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt_</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">// fill dxdt</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
If your range is a random access-range you can also apply the bracket operator
|
||||
to the iterator to access the elements in the range:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">sys</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Deriv</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x_</span> <span class="special">,</span> <span class="identifier">Deriv</span> <span class="special">&</span><span class="identifier">dxdt_</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x_</span> <span class="special">);</span>
|
||||
<span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="identifier">Deriv</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">dxdt</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt_</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">f1</span><span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">);</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">f2</span><span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The following two tables show which steppers and which algebras are compatible
|
||||
with <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.using_boost__range.steppers_supporting_boost_range"></a><p class="title"><b>Table 1.10. Steppers supporting Boost.Range</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Steppers supporting Boost.Range">
|
||||
<colgroup><col></colgroup>
|
||||
<thead><tr><th>
|
||||
<p>
|
||||
Stepper
|
||||
</p>
|
||||
</th></tr></thead>
|
||||
<tbody>
|
||||
<tr><td>
|
||||
<p>
|
||||
adams_bashforth_moulton
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
bulirsch_stoer_dense_out
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
bulirsch_stoer
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
controlled_runge_kutta
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
dense_output_runge_kutta
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
euler
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
explicit_error_generic_rk
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
explicit_generic_rk
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
rosenbrock4_controller
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
rosenbrock4_dense_output
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
rosenbrock4
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
runge_kutta4_classic
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
runge_kutta4
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
runge_kutta_cash_karp54_classic
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
runge_kutta_cash_karp54
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
runge_kutta_dopri5
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
runge_kutta_fehlberg78
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
symplectic_euler
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
symplectic_rkn_sb3a_mclachlan
|
||||
</p>
|
||||
</td></tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.using_boost__range.algebras_supporting_boost_range"></a><p class="title"><b>Table 1.11. Algebras supporting Boost.Range</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Algebras supporting Boost.Range">
|
||||
<colgroup><col></colgroup>
|
||||
<thead><tr><th>
|
||||
<p>
|
||||
algebra
|
||||
</p>
|
||||
</th></tr></thead>
|
||||
<tbody>
|
||||
<tr><td>
|
||||
<p>
|
||||
range_algebra
|
||||
</p>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<p>
|
||||
thrust_algebra
|
||||
</p>
|
||||
</td></tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break">
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="using_boost__ref.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="../concepts.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
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|
||||
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|
||||
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|
||||
@ -1,79 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Using boost::ref</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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|
||||
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|
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|
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|
||||
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|
||||
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|
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||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.odeint_in_detail.using_boost__ref"></a><a class="link" href="using_boost__ref.html" title="Using boost::ref">Using
|
||||
boost::ref</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
In odeint all system functions and observers are passed by value. For example,
|
||||
if you call a <code class="computeroutput"><span class="identifier">do_step</span></code> method
|
||||
of a particular stepper or the integration functions, your system and your
|
||||
stepper will be passed by value:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">rk4</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span> <span class="comment">// pass sys by value</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
This behavior is suitable for most systems, especially if your system does
|
||||
not contain any data or only a few parameters. However, in some cases you
|
||||
might contain some large amount of data with you system function and passing
|
||||
them by value is not desired since the data would be copied.
|
||||
</p>
|
||||
<p>
|
||||
In such cases you can easily use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span></code> (and
|
||||
its relative <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">cref</span></code>) which passes its argument by reference
|
||||
(or constant reference). odeint will unpack the arguments and no copying
|
||||
at all of your system object will take place:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">rk4</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">sys</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span> <span class="comment">// pass sys as references</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The same mechanism can be used for the observers in the integrate functions.
|
||||
</p>
|
||||
<div class="tip"><table border="0" summary="Tip">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../images/tip.png"></td>
|
||||
<th align="left">Tip</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
If you are using C++11 you can also use <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">ref</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cref</span></code>
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="state_types__algebras_and_operations.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="using_boost__range.html"><img src="../../images/next.png" alt="Next"></a>
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@ -1,51 +0,0 @@
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||||
<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
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<title>Tutorial</title>
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<link rel="stylesheet" href="../boostbook.css" type="text/css">
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<link rel="prev" href="getting_started/short_example.html" title="Short Example">
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||
<a name="boost_numeric_odeint.tutorial"></a><a class="link" href="tutorial.html" title="Tutorial">Tutorial</a>
|
||||
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|
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<div class="toc"><dl>
|
||||
<dt><span class="section"><a href="tutorial/harmonic_oscillator.html">Harmonic
|
||||
oscillator</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/solar_system.html">Solar system</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/chaotic_systems_and_lyapunov_exponents.html">Chaotic
|
||||
systems and Lyapunov exponents</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/stiff_systems.html">Stiff systems</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/special_topics.html">Special
|
||||
topics</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/using_cuda_and_thrust.html">Using
|
||||
Cuda and Thrust</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/using_opencl_and_vexcl.html">Using
|
||||
OpenCL and VexCL</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/all_examples.html">All examples</a></span></dt>
|
||||
<dt><span class="section"><a href="tutorial/references.html">References</a></span></dt>
|
||||
</dl></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
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<div class="spirit-nav">
|
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<a accesskey="p" href="getting_started/short_example.html"><img src="../images/prev.png" alt="Prev"></a><a accesskey="u" href="../index.html"><img src="../images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../images/home.png" alt="Home"></a><a accesskey="n" href="tutorial/harmonic_oscillator.html"><img src="../images/next.png" alt="Next"></a>
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</body>
|
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</html>
|
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@ -1,376 +0,0 @@
|
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<html>
|
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<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
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<title>All examples</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.all_examples"></a><a class="link" href="all_examples.html" title="All examples">All examples</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
The following table gives an overview over all examples.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="boost_numeric_odeint.tutorial.all_examples.examples_overview"></a><p class="title"><b>Table 1.5. Examples Overview</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Examples Overview">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
File
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Brief Description
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/harmonic_oscillator.cpp" target="_top">harmonic_oscillator.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The harmonic oscillator examples gives a brief introduction to
|
||||
odeint and shows the usage of the classical Runge-Kutta-solvers.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/simple1d.cpp" target="_top">simple1d.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Integrating a simple, one-dimensional ODE showing the usage of
|
||||
integrate- and generate-functions.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/elliptic_functions.cpp" target="_top">elliptic_functions.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Example calculating the elliptic functions using Bulirsch-Stoer
|
||||
and Runge-Kutta-Dopri5 Steppers with dense output.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/solar_system.cpp" target="_top">solar_system.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The solar system example shows the usage of the symplectic solvers.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/chaotic_system.cpp" target="_top">chaotic_system.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The chaotic system examples integrates the Lorenz system and calculates
|
||||
the Lyapunov exponents.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/stiff_system.cpp" target="_top">stiff_system.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The stiff system example shows the usage of the stiff solvers using
|
||||
the Jacobian of the system function.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/van_der_pol_stiff.cpp" target="_top">van_der_pol_stiff.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
This stiff system example again shows the usage of the stiff solvers
|
||||
by integrating the van der Pol osscillator.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/stuart_landau.cpp" target="_top">stuart_landau.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The Stuart-Landau example shows how odeint can be used with complex
|
||||
state types.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/fpu.cpp" target="_top">fpu.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The Fermi-Pasta-Ulam (FPU) example shows how odeint can be used
|
||||
to integrate lattice systems.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/phase_oscillator_ensemble.cpp" target="_top">phase_oscillator_ensemble.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The phase oscillator ensemble example shows how globally coupled
|
||||
oscillators can be analyzed and how statistical measures can be
|
||||
computed during integration.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/stepper_details.cpp" target="_top">stepper_details.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Trivial example showing the usability of the several stepper classes.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/harmonic_oscillator_units.cpp" target="_top">harmonic_oscillator_units.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
This examples shows how <a href="http://www.boost.org/doc/libs/release/libs/units/index.html" target="_top">Boost.Units</a>
|
||||
can be used with odeint.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/two_dimensional_phase_lattice.cpp" target="_top">two_dimensional_phase_lattice.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The 2D phase oscillator example shows how a two-dimensional lattice
|
||||
works with odeint and how matrix types can be used as state types
|
||||
in odeint.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/resizing_lattice.cpp" target="_top">resizing_lattice.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Shows the strength of odeint's memory management by simulating
|
||||
a Hamiltonian system on an expanding lattice.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/list_lattice.cpp" target="_top">list_lattice.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Example of a phase lattice integration using <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">list</span></code>
|
||||
as state type.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/lorenz_point.cpp" target="_top">lorenz_point.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Alternative way of integrating lorenz by using a self defined point3d
|
||||
data type as state type.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/lorenz_gmpxx.cpp" target="_top">lorenz_gmpxx.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
This examples integrates the Lorenz system by means of an arbitrary
|
||||
precision type.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/my_vector.cpp" target="_top">my_vector.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Simple example showing how to get odeint to work with a self-defined
|
||||
vector type.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/thrust/phase_oscillator_ensemble.cu" target="_top">phase_oscillator_ensemble.cu</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The Thrust phase oscillator ensemble example shows how globally
|
||||
coupled oscillators can be analyzed with Thrust and CUDA, employing
|
||||
the power of modern graphic devices.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/thrust/phase_oscillator_chain.cu" target="_top">phase_oscillator_chain.cu</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The Thrust phase oscillator chain example shows how chains of nearest
|
||||
neighbor coupled oscillators can be integrated with Thrust and
|
||||
odeint.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/thrust/lorenz_parameters.cu" target="_top">lorenz_parameters.cu</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The Lorenz parameters examples show how ensembles of ordinary differential
|
||||
equations can be solved by means of Thrust to study the dependence
|
||||
of an ODE on some parameters.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/mtl/gauss_packet.cpp" target="_top">gauss_packet.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The MTL-Gauss-packet example shows how the MTL can be easily used
|
||||
with odeint.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/stochastic_euler.cpp" target="_top">stochastic_euler.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Implementation of a custom stepper - the stochastic euler - for
|
||||
solving stochastic differential equations.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
<a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/generation_functions.cpp" target="_top">generation_functions.cpp</a>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Shows skeletal code on how to implemente own factory functions.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break">
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="using_opencl_and_vexcl.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="references.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,278 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Chaotic systems and Lyapunov exponents</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="up" href="../tutorial.html" title="Tutorial">
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||||
<link rel="prev" href="solar_system.html" title="Solar system">
|
||||
<link rel="next" href="stiff_systems.html" title="Stiff systems">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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||||
<div class="spirit-nav">
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||||
<a accesskey="p" href="solar_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="stiff_systems.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.chaotic_systems_and_lyapunov_exponents"></a><a class="link" href="chaotic_systems_and_lyapunov_exponents.html" title="Chaotic systems and Lyapunov exponents">Chaotic
|
||||
systems and Lyapunov exponents</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
odeint can easily be used to investigate the properties of chaotic deterministic
|
||||
systems. In mathematical terms chaotic refers to an exponential growth of
|
||||
perturbations <span class="emphasis"><em>δ x</em></span>. In order to observe this exponential
|
||||
growth one usually solves the equations for the tangential dynamics which
|
||||
is again an ordinary differential equation. These equations are linear but
|
||||
time dependent and can be obtained via
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>d δ x / dt = J(x) δ x</em></span>
|
||||
</p>
|
||||
<p>
|
||||
where <span class="emphasis"><em>J</em></span> is the Jacobian of the system under consideration.
|
||||
<span class="emphasis"><em>δ x</em></span> can also be interpreted as a perturbation of the original
|
||||
system. In principle <span class="emphasis"><em>n</em></span> of these perturbations exit,
|
||||
they form a hypercube and evolve in the time. The Lyapunov exponents are
|
||||
then defined as logarithmic growth rates of the perturbations. If one Lyapunov
|
||||
exponent is larger then zero nearby trajectories diverge exponentially hence
|
||||
they are chaotic. If the largest Lyapunov exponent is zero one is usually
|
||||
faced with periodic motion. In the case of a largest Lyapunov exponent smaller
|
||||
then zero the converges the a fixed point. More information's about Lyapunov
|
||||
exponents and nonlinear dynamical systems can be found in many textbooks,
|
||||
see for example: E. Ott "Chaos is Dynamical Systems", Cambridge.
|
||||
</p>
|
||||
<p>
|
||||
To calculate the Lyapunov exponents numerically one usually solves the equations
|
||||
of motion for <span class="emphasis"><em>n</em></span> perturbations and orthonormalizes them
|
||||
every <span class="emphasis"><em>k</em></span> steps. The Lyapunov exponent is the average
|
||||
the logarithm of the stretching factor of each perturbation.
|
||||
</p>
|
||||
<p>
|
||||
To demonstrate how one can use odeint to determine the Lyapunov exponents
|
||||
we choose the Lorenz system. It is one of the most studied dynamical systems
|
||||
in the nonlinear dynamics community. For the standard parameters it possesses
|
||||
a strange attractor with non-integer dimension. The Lyapunov exponents take
|
||||
values of approximately 0.9, 0 and -12.
|
||||
</p>
|
||||
<p>
|
||||
The implementation of the Lorenz system is
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">10.0</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">R</span> <span class="special">=</span> <span class="number">28.0</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">8.0</span> <span class="special">/</span> <span class="number">3.0</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="number">3</span> <span class="special">></span> <span class="identifier">lorenz_state_type</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="identifier">lorenz</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">lorenz_state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">lorenz_state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">);</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">];</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
We need also to integrate the set of the perturbations. This is done in parallel
|
||||
to the original system, hence within one system function. Of course, we want
|
||||
to use the above definition of the Lorenz system, hence the definition of
|
||||
the system function including the Lorenz system itself and the perturbation
|
||||
could look like:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">num_of_lyap</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">N</span> <span class="special">=</span> <span class="identifier">n</span> <span class="special">+</span> <span class="identifier">n</span><span class="special">*</span><span class="identifier">num_of_lyap</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">num_of_lyap</span> <span class="special">></span> <span class="identifier">lyap_type</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="identifier">lorenz_with_lyap</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">lorenz</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">l</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">l</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">l</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">pert</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">3</span> <span class="special">+</span> <span class="identifier">l</span> <span class="special">*</span> <span class="number">3</span><span class="special">;</span>
|
||||
<span class="keyword">double</span> <span class="special">*</span><span class="identifier">dpert</span> <span class="special">=</span> <span class="identifier">dxdt</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">3</span> <span class="special">+</span> <span class="identifier">l</span> <span class="special">*</span> <span class="number">3</span><span class="special">;</span>
|
||||
<span class="identifier">dpert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">+</span> <span class="number">10.0</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="identifier">dpert</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="special">(</span> <span class="identifier">R</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">2</span><span class="special">];</span>
|
||||
<span class="identifier">dpert</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">b</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">2</span><span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The perturbations are stored linearly in the <code class="computeroutput"><span class="identifier">state_type</span></code>
|
||||
behind the state of the Lorenz system. The problem that lorenz() and lorenz_with_lyap() have different
|
||||
state types. A simple trick to over come this problem is put the Lorenz system
|
||||
inside a functor with templatized arguments:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">StateIn</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">StateOut</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Value</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">StateIn</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">StateOut</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">Value</span> <span class="identifier">t</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">);</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">];</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="identifier">lorenz_with_lyap</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">lorenz</span><span class="special">()(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="special">...</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
This works fine and <code class="computeroutput"><span class="identifier">lorenz_with_lyap</span></code>
|
||||
can be used for example via
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">;</span>
|
||||
<span class="comment">// initialize x</span>
|
||||
<span class="identifier">explicit_rk4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk4</span><span class="special">;</span>
|
||||
<span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz_with_lyap</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span> <span class="number">1000</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
This code snippet performs 1000 steps with constant step size 0.01.
|
||||
</p>
|
||||
<p>
|
||||
A real world use case for the calculation of the Lyapunov exponents of Lorenz
|
||||
system would always include some transient steps, just to ensure that the
|
||||
current state lies on the attractor, hence it would look like
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">;</span>
|
||||
<span class="comment">// initialize x</span>
|
||||
<span class="identifier">explicit_rk4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk4</span><span class="special">;</span>
|
||||
<span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span> <span class="number">1000</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
The problem is now, that <code class="computeroutput"><span class="identifier">x</span></code>
|
||||
is the full state containing also the perturbations and <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code>
|
||||
does not know that it should only use 3 elements. In detail, odeint and its
|
||||
steppers determine the length of the system under consideration by determining
|
||||
the length of the state. In the classical solvers, e.g. from Numerical Recipes,
|
||||
the problem was solved by pointer to the state and an appropriate length,
|
||||
something similar to
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">lorenz</span><span class="special">(</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span><span class="special">,</span> <span class="keyword">void</span><span class="special">*</span> <span class="identifier">params</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="special">...</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="keyword">int</span> <span class="identifier">system_length</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
|
||||
<span class="identifier">rk4</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">system_length</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">lorenz</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
But odeint supports a similar and much more sophisticated concept: <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>.
|
||||
To make the steppers and the system ready to work with <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
the system has to by changed:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Deriv</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x_</span> <span class="special">,</span> <span class="identifier">Deriv</span> <span class="special">&</span><span class="identifier">dxdt_</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x_</span> <span class="special">);</span>
|
||||
<span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="identifier">Deriv</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">dxdt</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt_</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">);</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">];</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
This is in principle all. Now, we only have to call <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code>
|
||||
with a range including only the first 3 components of <span class="emphasis"><em>x</em></span>:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// perform 10000 transient steps</span>
|
||||
<span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="identifier">n</span> <span class="special">)</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="number">10000</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Having integrated a sufficient number of transients steps we are now able
|
||||
to calculate the Lyapunov exponents:
|
||||
</p>
|
||||
<div class="orderedlist"><ol class="orderedlist" type="1">
|
||||
<li class="listitem">
|
||||
We initialize the perturbations. They are stored linearly behind the
|
||||
state of the Lorenz system. The perturbations are initialized such that
|
||||
<span class="emphasis"><em>p <sub>​ij</sub> = δ <sub>​ij</sub></em></span>, where <span class="emphasis"><em>p <sub>​ij</sub></em></span> is the
|
||||
<span class="emphasis"><em>j</em></span>-component of the <span class="emphasis"><em>i</em></span>.-th perturbation
|
||||
and <span class="emphasis"><em>δ <sub>​ij</sub></em></span> is the Kronecker symbol.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Integrate 100 steps of the full system with perturbations
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Orthonormalize the perturbation using Gram-Schmidt orthonormalization
|
||||
algorithm.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Repeat step 2 and 3. Every 10000 steps write the current Lyapunov exponent.
|
||||
</li>
|
||||
</ol></div>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">fill</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()+</span><span class="identifier">n</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">n</span><span class="special">+</span><span class="identifier">n</span><span class="special">*</span><span class="identifier">i</span><span class="special">+</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
||||
<span class="identifier">fill</span><span class="special">(</span> <span class="identifier">lyap</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">lyap</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
||||
|
||||
<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||
<span class="identifier">size_t</span> <span class="identifier">count</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
|
||||
<span class="keyword">while</span><span class="special">(</span> <span class="keyword">true</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
|
||||
<span class="identifier">t</span> <span class="special">=</span> <span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz_with_lyap</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="number">100</span> <span class="special">);</span>
|
||||
<span class="identifier">gram_schmidt</span><span class="special"><</span> <span class="identifier">num_of_lyap</span> <span class="special">>(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">lyap</span> <span class="special">,</span> <span class="identifier">n</span> <span class="special">);</span>
|
||||
<span class="special">++</span><span class="identifier">count</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">if</span><span class="special">(</span> <span class="special">!(</span><span class="identifier">count</span> <span class="special">%</span> <span class="number">100000</span><span class="special">)</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">t</span><span class="special">;</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">lyap</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="identifier">t</span> <span class="special">;</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The full code can be found here: <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/chaotic_system.cpp" target="_top">chaotic_system.cpp</a>
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="solar_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="stiff_systems.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
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|
||||
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||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>References</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="up" href="../tutorial.html" title="Tutorial">
|
||||
<link rel="prev" href="all_examples.html" title="All examples">
|
||||
<link rel="next" href="../odeint_in_detail.html" title="odeint in detail">
|
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</head>
|
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="all_examples.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="../odeint_in_detail.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.references"></a><a class="link" href="references.html" title="References">References</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
<span class="bold"><strong>General information about numerical integration of
|
||||
ordinary differential equations:</strong></span>
|
||||
</p>
|
||||
<p>
|
||||
[1] Press William H et al., Numerical Recipes 3rd Edition: The Art of Scientific
|
||||
Computing, 3rd ed. (Cambridge University Press, 2007).
|
||||
</p>
|
||||
<p>
|
||||
[2] Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner, Solving Ordinary
|
||||
Differential Equations I: Nonstiff Problems, 2nd ed. (Springer, Berlin, 2009).
|
||||
</p>
|
||||
<p>
|
||||
[3] Ernst Hairer and Gerhard Wanner, Solving Ordinary Differential Equations
|
||||
II: Stiff and Differential-Algebraic Problems, 2nd ed. (Springer, Berlin,
|
||||
2010).
|
||||
</p>
|
||||
<p>
|
||||
<span class="bold"><strong>Symplectic integration of numerical integration:</strong></span>
|
||||
</p>
|
||||
<p>
|
||||
[4] Ernst Hairer, Gerhard Wanner, and Christian Lubich, Geometric Numerical
|
||||
Integration: Structure-Preserving Algorithms for Ordinary Differential Equations,
|
||||
2nd ed. (Springer-Verlag Gmbh, 2006).
|
||||
</p>
|
||||
<p>
|
||||
[5] Leimkuhler Benedict and Reich Sebastian, Simulating Hamiltonian Dynamics
|
||||
(Cambridge University Press, 2005).
|
||||
</p>
|
||||
<p>
|
||||
<span class="bold"><strong>Special symplectic methods:</strong></span>
|
||||
</p>
|
||||
<p>
|
||||
[6] Haruo Yoshida, “Construction of higher order symplectic integrators,”
|
||||
Physics Letters A 150, no. 5 (November 12, 1990): 262-268.
|
||||
</p>
|
||||
<p>
|
||||
[7] Robert I. McLachlan, “On the numerical integration of ordinary differential
|
||||
equations by symmetric composition methods,” SIAM J. Sci. Comput. 16, no.
|
||||
1 (1995): 151-168.
|
||||
</p>
|
||||
<p>
|
||||
<span class="bold"><strong>Special systems:</strong></span>
|
||||
</p>
|
||||
<p>
|
||||
[8] <a href="http://www.scholarpedia.org/article/Fermi-Pasta-Ulam_nonlinear_lattice_oscillations" target="_top">Fermi-Pasta-Ulam
|
||||
nonlinear lattice oscillations</a>
|
||||
</p>
|
||||
<p>
|
||||
[9] Arkady Pikovsky, Michael Rosemblum, and Jürgen Kurths, Synchronization:
|
||||
A Universal Concept in Nonlinear Sciences. (Cambridge University Press, 2001).
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="all_examples.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="../odeint_in_detail.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,300 +0,0 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Solar system</title>
|
||||
<link rel="stylesheet" href="../../boostbook.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
|
||||
<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
|
||||
<link rel="up" href="../tutorial.html" title="Tutorial">
|
||||
<link rel="prev" href="harmonic_oscillator.html" title="Harmonic oscillator">
|
||||
<link rel="next" href="chaotic_systems_and_lyapunov_exponents.html" title="Chaotic systems and Lyapunov exponents">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="harmonic_oscillator.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="chaotic_systems_and_lyapunov_exponents.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.solar_system"></a><a class="link" href="solar_system.html" title="Solar system">Solar system</a>
|
||||
</h3></div></div></div>
|
||||
<div class="toc"><dl>
|
||||
<dt><span class="section"><a href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.gravitation_and_energy_conservation">Gravitation
|
||||
and energy conservation</a></span></dt>
|
||||
<dt><span class="section"><a href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.define_the_system_function">Define
|
||||
the system function</a></span></dt>
|
||||
</dl></div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.solar_system.gravitation_and_energy_conservation"></a><a class="link" href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.gravitation_and_energy_conservation" title="Gravitation and energy conservation">Gravitation
|
||||
and energy conservation</a>
|
||||
</h4></div></div></div>
|
||||
<p>
|
||||
The next example in this tutorial is a simulation of the outer solar system,
|
||||
consisting of the sun, Jupiter, Saturn, Uranus, Neptune and Pluto.
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../solar_system.jpg" alt="solar_system"></span>
|
||||
</p>
|
||||
<p>
|
||||
Each planet and of course the sun will be represented by mass points. The
|
||||
interaction force between each object is the gravitational force which
|
||||
can be written as
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>F<sub>​ij</sub> = -γ m<sub>​i</sub> m<sub>​j</sub> ( q<sub>​i</sub> - q<sub>​j</sub> ) / | q<sub>​i</sub> - q<sub>​j</sub> | <sup>3</sup></em></span>
|
||||
</p>
|
||||
<p>
|
||||
where <span class="emphasis"><em>γ</em></span> is the gravitational constant, <span class="emphasis"><em>m<sub>​i</sub></em></span>
|
||||
and <span class="emphasis"><em>m<sub>​j</sub></em></span> are the masses and <span class="emphasis"><em>q<sub>​i</sub></em></span>
|
||||
and <span class="emphasis"><em>q<sub>​j</sub></em></span> are the locations of the two objects. The equations
|
||||
of motion are then
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>dq<sub>​i</sub> / dt = p<sub>​i</sub></em></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>dp<sub>​i</sub> / dt = 1 / m<sub>​i</sub> Σ<sub>​ji</sub> F<sub>​ij</sub></em></span>
|
||||
</p>
|
||||
<p>
|
||||
where <span class="emphasis"><em>p<sub>​i</sub></em></span> is the momenta of object <span class="emphasis"><em>i</em></span>.
|
||||
The equations of motion can also be derived from the Hamiltonian
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>H = Σ<sub>​i</sub> p<sub>​i</sub><sup>2</sup> / ( 2 m<sub>​i</sub> ) + Σ<sub>​j</sub> V( q<sub>​i</sub> , q<sub>​j</sub> )</em></span>
|
||||
</p>
|
||||
<p>
|
||||
with the interaction potential <span class="emphasis"><em>V(q<sub>​i</sub>,q<sub>​j</sub>)</em></span>. The Hamiltonian
|
||||
equations give the equations of motion
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>dq<sub>​i</sub> / dt = dH / dp<sub>​i</sub></em></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>dp<sub>​i</sub> / dt = -dH / dq<sub>​i</sub></em></span>
|
||||
</p>
|
||||
<p>
|
||||
In time independent Hamiltonian system the energy and the phase space volume
|
||||
are conserved and special integration methods have to be applied in order
|
||||
to ensure these conservation laws. The odeint library provides classes
|
||||
for separable Hamiltonian systems, which can be written in the form <span class="emphasis"><em>H
|
||||
= Σ
|
||||
p<sub>​i</sub><sup>2</sup> / (2m<sub>​i</sub>) + H<sub>​q</sub>(q)</em></span>, where <span class="emphasis"><em>H<sub>​q</sub>(q)</em></span> only
|
||||
depends on the coordinates. Although this functional form might look a
|
||||
bit arbitrary, it covers nearly all classical mechanical systems with inertia
|
||||
and without dissipation, or where the equations of motion can be written
|
||||
in the form <span class="emphasis"><em>dq<sub>​i</sub> / dt = p<sub>​i</sub></em></span> / m<sub>​i</sub> , <span class="emphasis"><em>dp<sub>​i</sub> / dt =
|
||||
f( q<sub>​i</sub> )</em></span>.
|
||||
</p>
|
||||
<div class="note"><table border="0" summary="Note">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../images/note.png"></td>
|
||||
<th align="left">Note</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
A short physical note: While the two-body-problem is known to be integrable,
|
||||
that means it can be solved with purely analytic techniques, already
|
||||
the three-body-problem is not solveable. This was found in the end of
|
||||
the 19th century by H. Poincare which lead to the whole new subject of
|
||||
<a href="http://en.wikipedia.org/wiki/Chaos_theory" target="_top">Chaos Theory</a>.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.solar_system.define_the_system_function"></a><a class="link" href="solar_system.html#boost_numeric_odeint.tutorial.solar_system.define_the_system_function" title="Define the system function">Define
|
||||
the system function</a>
|
||||
</h4></div></div></div>
|
||||
<p>
|
||||
To implement this system we define a 3D point type which will represent
|
||||
the space as well as the velocity. Therefore, we use the operators from
|
||||
<a href="http://www.boost.org/doc/libs/release/doc/html/operators.html" target="_top">Boost.Operators</a>:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">/*the point type */</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Dim</span> <span class="special">></span>
|
||||
<span class="keyword">class</span> <span class="identifier">point</span> <span class="special">:</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">additive1</span><span class="special"><</span> <span class="identifier">point</span><span class="special"><</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">Dim</span> <span class="special">></span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">additive2</span><span class="special"><</span> <span class="identifier">point</span><span class="special"><</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">Dim</span> <span class="special">></span> <span class="special">,</span> <span class="identifier">T</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiplicative2</span><span class="special"><</span> <span class="identifier">point</span><span class="special"><</span> <span class="identifier">T</span> <span class="special">,</span> <span class="identifier">Dim</span> <span class="special">></span> <span class="special">,</span> <span class="identifier">T</span>
|
||||
<span class="special">></span> <span class="special">></span> <span class="special">></span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
|
||||
<span class="keyword">const</span> <span class="keyword">static</span> <span class="identifier">size_t</span> <span class="identifier">dim</span> <span class="special">=</span> <span class="identifier">Dim</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">point</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">dim</span> <span class="special">></span> <span class="identifier">point_type</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// ...</span>
|
||||
<span class="comment">// constructors</span>
|
||||
|
||||
<span class="comment">// ...</span>
|
||||
<span class="comment">// operators</span>
|
||||
|
||||
<span class="keyword">private</span><span class="special">:</span>
|
||||
|
||||
<span class="identifier">T</span> <span class="identifier">m_val</span><span class="special">[</span><span class="identifier">dim</span><span class="special">];</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="comment">//...</span>
|
||||
<span class="comment">// more operators</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The next step is to define a container type storing the values of <span class="emphasis"><em>q</em></span>
|
||||
and <span class="emphasis"><em>p</em></span> and to define system functions. As container
|
||||
type we use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// we simulate 5 planets and the sun</span>
|
||||
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">typedef</span> <span class="identifier">point</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="number">3</span> <span class="special">></span> <span class="identifier">point_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="identifier">point_type</span> <span class="special">,</span> <span class="identifier">n</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">n</span> <span class="special">></span> <span class="identifier">mass_type</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The <code class="computeroutput"><span class="identifier">container_type</span></code> is different
|
||||
from the state type of the ODE. The state type of the ode is simply a
|
||||
<code class="computeroutput"><span class="identifier">pair</span><span class="special"><</span>
|
||||
<span class="identifier">container_type</span> <span class="special">,</span>
|
||||
<span class="identifier">container_type</span> <span class="special">></span></code>
|
||||
since it needs the information about the coordinates and the momenta.
|
||||
</p>
|
||||
<p>
|
||||
Next we define the system's equations. As we will use a stepper that accounts
|
||||
for the Hamiltonian (energy-preserving) character of the system, we have
|
||||
to define the rhs different from the usual case where it is just a single
|
||||
function. The stepper will make use of the separable character, which means
|
||||
the system will be defined by two objects representing <span class="emphasis"><em>f(p) =
|
||||
-dH/dq</em></span> and <span class="emphasis"><em>g(q) = dH/dp</em></span>:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">gravitational_constant</span> <span class="special">=</span> <span class="number">2.95912208286e-4</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">solar_system_coor</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&</span><span class="identifier">m_masses</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">solar_system_coor</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&</span><span class="identifier">masses</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_masses</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">dqdt</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
||||
<span class="identifier">dqdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="identifier">m_masses</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">solar_system_momentum</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&</span><span class="identifier">m_masses</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">solar_system_momentum</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">mass_type</span> <span class="special">&</span><span class="identifier">masses</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_masses</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">dpdt</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">j</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">j</span><span class="special"><</span><span class="identifier">i</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">j</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">point_type</span> <span class="identifier">diff</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">j</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
||||
<span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="identifier">abs</span><span class="special">(</span> <span class="identifier">diff</span> <span class="special">);</span>
|
||||
<span class="identifier">diff</span> <span class="special">*=</span> <span class="special">(</span> <span class="identifier">gravitational_constant</span> <span class="special">*</span> <span class="identifier">m_masses</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">m_masses</span><span class="special">[</span><span class="identifier">j</span><span class="special">]</span> <span class="special">/</span> <span class="identifier">d</span> <span class="special">/</span> <span class="identifier">d</span> <span class="special">/</span> <span class="identifier">d</span> <span class="special">);</span>
|
||||
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">diff</span><span class="special">;</span>
|
||||
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">j</span><span class="special">]</span> <span class="special">-=</span> <span class="identifier">diff</span><span class="special">;</span>
|
||||
|
||||
<span class="special">}</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
In general a three body-system is chaotic, hence we can not expect that
|
||||
arbitrary initial conditions of the system will lead to a dynamic which
|
||||
is comparable with the solar system. That is we have to define proper initial
|
||||
conditions, which are taken from the book of Hairer, Wannier, Lubich.
|
||||
</p>
|
||||
<p>
|
||||
As mentioned above, we need to use some special integrators in order to
|
||||
conserve phase space volume. There is a well known family of such integrators,
|
||||
the so-called Runge-Kutta-Nystroem solvers, which we apply here in terms
|
||||
of a <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>
|
||||
stepper:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">symplectic_rkn_sb3a_mclachlan</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">100.0</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">integrate_const</span><span class="special">(</span>
|
||||
<span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">,</span>
|
||||
<span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">solar_system_coor</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">solar_system_momentum</span><span class="special">(</span> <span class="identifier">masses</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">p</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="number">0.0</span> <span class="special">,</span> <span class="number">200000.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">cout</span> <span class="special">)</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
These integration routine was used to produce the above sketch of the solar
|
||||
system. Note, that there are two particularities in this example. First,
|
||||
the state of the symplectic stepper is not <code class="computeroutput"><span class="identifier">container_type</span></code>
|
||||
but a pair of <code class="computeroutput"><span class="identifier">container_type</span></code>.
|
||||
Hence, we must pass such a pair to the integrate function. Since, we want
|
||||
to pass them as references we can simply pack them into <a href="http://www.boost.org/doc/libs/release/doc/html/ref.html" target="_top">Boost.Ref</a>.
|
||||
The second point is the observer, which is called with a state type, hence
|
||||
a pair of <code class="computeroutput"><span class="identifier">container_type</span></code>.
|
||||
The reference wrapper is also passed, but this is not a problem at all:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="identifier">m_out</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&</span><span class="identifier">out</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_out</span><span class="special">(</span> <span class="identifier">out</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
||||
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">;</span>
|
||||
<span class="identifier">m_out</span> <span class="special"><<</span> <span class="identifier">t</span><span class="special">;</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">m_out</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
||||
<span class="identifier">m_out</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The full example can be found here: <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/solar_system.cpp" target="_top">solar_system.cpp</a>
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
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|
||||
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||||
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<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.stiff_systems"></a><a class="link" href="stiff_systems.html" title="Stiff systems">Stiff systems</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
An important class of ordinary differential equations are so called stiff
|
||||
system which are characterized by two or more time scales of different order.
|
||||
Examples of such systems are found in chemical systems where reaction rates
|
||||
of individual sub-reaction might differ over large ranges, for example:
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>d S<sub>​1</sub> / dt = - 101 S<sub>​2</sub> - 100 S<sub>​1</sub></em></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>d S<sub>​2</sub> / dt = S<sub>​1</sub></em></span>
|
||||
</p>
|
||||
<p>
|
||||
To solve stiff systems efficiently using numerics the Jacobian
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>J = d f<sub>​i</sub> / d x<sub>​j</sub></em></span>
|
||||
</p>
|
||||
<p>
|
||||
is needed. Here is the definition of the above example
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">stiff_system</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">vector_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span> <span class="number">0</span> <span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="number">101.0</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span> <span class="number">0</span> <span class="special">]</span> <span class="special">-</span> <span class="number">100.0</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span> <span class="number">1</span> <span class="special">];</span>
|
||||
<span class="identifier">dxdt</span><span class="special">[</span> <span class="number">1</span> <span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">[</span> <span class="number">0</span> <span class="special">];</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">stiff_system_jacobi</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">vector_type</span> <span class="special">&</span> <span class="comment">/* x */</span> <span class="special">,</span> <span class="identifier">matrix_type</span> <span class="special">&</span><span class="identifier">J</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="special">&</span> <span class="comment">/* t */</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">&</span><span class="identifier">dfdt</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">J</span><span class="special">(</span> <span class="number">0</span> <span class="special">,</span> <span class="number">0</span> <span class="special">)</span> <span class="special">=</span> <span class="special">-</span><span class="number">101.0</span><span class="special">;</span>
|
||||
<span class="identifier">J</span><span class="special">(</span> <span class="number">0</span> <span class="special">,</span> <span class="number">1</span> <span class="special">)</span> <span class="special">=</span> <span class="special">-</span><span class="number">100.0</span><span class="special">;</span>
|
||||
<span class="identifier">J</span><span class="special">(</span> <span class="number">1</span> <span class="special">,</span> <span class="number">0</span> <span class="special">)</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
|
||||
<span class="identifier">J</span><span class="special">(</span> <span class="number">1</span> <span class="special">,</span> <span class="number">1</span> <span class="special">)</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||
<span class="identifier">dfdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||
<span class="identifier">dfdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The state type has to be a <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span></code>
|
||||
and the matrix type must by a <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span></code>
|
||||
since the stiff integrator only accepts these types. However, you might want
|
||||
use non-stiff intgrators on this system, too - we will do so later for demonstration.
|
||||
Therefore we want to use the same function also with other state_types, realized
|
||||
by templatizing the <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">stiff_system</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="special">...</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">stiff_system_jacobi</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Matrix</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">Matrix</span> <span class="special">&</span><span class="identifier">J</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="special">&</span><span class="identifier">t</span> <span class="special">,</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">dfdt</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="special">...</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Now you can use <code class="computeroutput"><span class="identifier">stiff_system</span></code>
|
||||
in combination with <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code> or <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>.
|
||||
In the example the explicit time derivative of <span class="emphasis"><em>f(x,t)</em></span>
|
||||
is introduced separately in the Jacobian. If <span class="emphasis"><em>df / dt = 0</em></span>
|
||||
simply fill <code class="computeroutput"><span class="identifier">dfdt</span></code> with zeros.
|
||||
</p>
|
||||
<p>
|
||||
A well know solver for stiff systems is the so called Rosenbrock method.
|
||||
It has a step size control and dense output facilities and can be used like
|
||||
all the other stepper:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">2</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">size_t</span> <span class="identifier">num_of_steps</span> <span class="special">=</span> <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_dense_output</span><span class="special"><</span> <span class="identifier">rosenbrock4</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="special">>(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">stiff_system</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">stiff_system_jacobi</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">50.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg2</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg1</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
During the integration 71 steps have been done. Comparing to a classical
|
||||
Runge-Kutta solver this is a very good result. For example the Dormand-Prince
|
||||
5 method with step size control and dense output yields 1531 steps.
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x2</span><span class="special">(</span> <span class="number">2</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">size_t</span> <span class="identifier">num_of_steps2</span> <span class="special">=</span> <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_dense_output</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">vector_type</span> <span class="special">></span> <span class="special">>(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">stiff_system</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x2</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">50.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg2</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg1</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Note, that we have used <a href="http://www.boost.org/doc/libs/1_46_1/libs/spirit/phoenix/doc/html/index.html" target="_top">Boost.Phoenix</a>,
|
||||
a great functional programming library to create and compose the observer.
|
||||
</p>
|
||||
<p>
|
||||
The full example can be found here: <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/stiff_system.cpp" target="_top">stiff_system.cpp</a>
|
||||
</p>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="chaotic_systems_and_lyapunov_exponents.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="special_topics.html"><img src="../../images/next.png" alt="Next"></a>
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||||
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</html>
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@ -1,691 +0,0 @@
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||||
<html>
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<head>
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||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Using Cuda and Thrust</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.75.2">
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<a accesskey="p" href="special_topics.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="using_opencl_and_vexcl.html"><img src="../../images/next.png" alt="Next"></a>
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</div>
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||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.using_cuda_and_thrust"></a><a class="link" href="using_cuda_and_thrust.html" title="Using Cuda and Thrust">Using
|
||||
Cuda and Thrust</a>
|
||||
</h3></div></div></div>
|
||||
<div class="toc"><dl>
|
||||
<dt><span class="section"><a href="using_cuda_and_thrust.html#boost_numeric_odeint.tutorial.using_cuda_and_thrust.phase_oscillator_ensemble">Phase
|
||||
oscillator ensemble</a></span></dt>
|
||||
<dt><span class="section"><a href="using_cuda_and_thrust.html#boost_numeric_odeint.tutorial.using_cuda_and_thrust.large_oscillator_chains">Large
|
||||
oscillator chains</a></span></dt>
|
||||
<dt><span class="section"><a href="using_cuda_and_thrust.html#boost_numeric_odeint.tutorial.using_cuda_and_thrust.parameter_studies">Parameter
|
||||
studies</a></span></dt>
|
||||
</dl></div>
|
||||
<p>
|
||||
Modern graphic cards (graphic processing units - GPUs) can be used to speed
|
||||
up the performance of time consuming algorithms by means of massive parallelization.
|
||||
They are designed to execute many operations in parallel. odeint can utilize
|
||||
the power of GPUs by means of CUDA and <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>,
|
||||
which is a STL-like interface for the native CUDA API.
|
||||
</p>
|
||||
<div class="important"><table border="0" summary="Important">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Important]" src="../../images/important.png"></td>
|
||||
<th align="left">Important</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
Thrust also supports parallelization using OpenMP. You can switch between
|
||||
CUDA-parallelization and OpenMP-parallelization by a simple compiler switch.
|
||||
Hence, this also provides an easy way to get basic OpenMP parallelization
|
||||
into odeint.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<p>
|
||||
To use odeint with CUDA a few points have to be taken into account. First
|
||||
of all, the problem has to be well chosen. It makes absolutely no sense to
|
||||
try to parallelize the code for a three dimensional system, it is simply
|
||||
to small and not worth the effort. One single function call (kernel execution)
|
||||
on the GPU is slow but you can do the operation on a huge set of data with
|
||||
only one call. We have experienced that the vector size over which is parallelized
|
||||
should be of the order of <span class="emphasis"><em>10<sup>6</sup></em></span> to make full use of the
|
||||
GPU. Secondly, you have to use <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>'s
|
||||
algorithms and functors when implementing the rhs the ODE. This might be
|
||||
tricky since it involves some kind of functional programming knowledge.
|
||||
</p>
|
||||
<p>
|
||||
Typical applications for CUDA and odeint are large systems, like lattices
|
||||
or discretizations of PDE, and parameter studies. We introduce now three
|
||||
examples which show how the power of GPUs can be used in combination with
|
||||
odeint.
|
||||
</p>
|
||||
<div class="important"><table border="0" summary="Important">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Important]" src="../../images/important.png"></td>
|
||||
<th align="left">Important</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
The full power of CUDA is only available for really large systems where
|
||||
the number of coupled ordinary differential equations is of order <span class="emphasis"><em>N=10<sup>6</sup></em></span>
|
||||
or larger. For smaller systems the CPU is usually much faster. You can
|
||||
also integrate an ensemble of different uncoupled ODEs in parallel as shown
|
||||
in the last example.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.using_cuda_and_thrust.phase_oscillator_ensemble"></a><a class="link" href="using_cuda_and_thrust.html#boost_numeric_odeint.tutorial.using_cuda_and_thrust.phase_oscillator_ensemble" title="Phase oscillator ensemble">Phase
|
||||
oscillator ensemble</a>
|
||||
</h4></div></div></div>
|
||||
<p>
|
||||
The first example is the phase oscillator ensemble from the previous section:
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>dφ<sub>​k</sub> / dt = ω<sub>​k</sub> + ε / N Σ<sub>​j</sub> sin( φ<sub>​j</sub> - φ<sub>​k</sub> ).</em></span>
|
||||
</p>
|
||||
<p>
|
||||
It has a phase transition at <span class="emphasis"><em>ε = 2</em></span> in the limit of infinite
|
||||
numbers of oscillators <span class="emphasis"><em>N</em></span>. In the case of finite <span class="emphasis"><em>N</em></span>
|
||||
this transition is smeared out but still clearly visible.
|
||||
</p>
|
||||
<p>
|
||||
<a href="http://code.google.com/p/thrust/" target="_top">Thrust</a> and Cuda are
|
||||
perfectly suited for such kinds of problems where one needs a large number
|
||||
of particles (oscillators). We start by defining the state type which is
|
||||
a <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span></code>. The content of this vector
|
||||
lives on the GPU. If you are not familiar with this we recommend reading
|
||||
the <span class="emphasis"><em>Getting started</em></span> section on the <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>
|
||||
website.
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">//change this to float if your device does not support double computation</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">double</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">//change this to host_vector< ... > of you want to run on CPU</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||
<span class="comment">// typedef thrust::host_vector< value_type > state_type;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Thrust follows a functional programming approach. If you want to perform
|
||||
a calculation on the GPU you usually have to call a global function like
|
||||
<code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">for_each</span></code>, <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">reduce</span></code>,
|
||||
... with an appropriate local functor which performs the basic operation.
|
||||
An example is
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">add_two</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">></span>
|
||||
<span class="identifier">__host__</span> <span class="identifier">__device__</span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="identifier">T</span> <span class="special">&</span><span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">t</span> <span class="special">+=</span> <span class="identifier">T</span><span class="special">(</span> <span class="number">2</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="comment">// ...</span>
|
||||
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">for_each</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">add_two</span><span class="special">()</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
This code generically adds two to every element in the container <code class="computeroutput"><span class="identifier">x</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
For the purpose of integrating the phase oscillator ensemble we need
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
|
||||
<li class="listitem">
|
||||
to calculate the system function, hence the r.h.s. of the ODE.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
this involves computing the mean field of the oscillator example, i.e.
|
||||
the values of <span class="emphasis"><em>R</em></span> and <span class="emphasis"><em>θ</em></span>
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
The mean field is calculated in a class <code class="computeroutput"><span class="identifier">mean_field_calculator</span></code>
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">mean_field_calculator</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">struct</span> <span class="identifier">sin_functor</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">unary_function</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">></span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">__host__</span> <span class="identifier">__device__</span>
|
||||
<span class="identifier">value_type</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="identifier">value_type</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">return</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">cos_functor</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">unary_function</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">></span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">__host__</span> <span class="identifier">__device__</span>
|
||||
<span class="identifier">value_type</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="identifier">value_type</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">return</span> <span class="identifier">cos</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">static</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">get_mean</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">sin_sum</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">reduce</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">sin_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">sin_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">cos_sum</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">reduce</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">cos_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">cos_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">cos_sum</span> <span class="special">/=</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">);</span>
|
||||
<span class="identifier">sin_sum</span> <span class="special">/=</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">value_type</span> <span class="identifier">K</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(</span> <span class="identifier">cos_sum</span> <span class="special">*</span> <span class="identifier">cos_sum</span> <span class="special">+</span> <span class="identifier">sin_sum</span> <span class="special">*</span> <span class="identifier">sin_sum</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">Theta</span> <span class="special">=</span> <span class="identifier">atan2</span><span class="special">(</span> <span class="identifier">sin_sum</span> <span class="special">,</span> <span class="identifier">cos_sum</span> <span class="special">);</span>
|
||||
|
||||
<span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">K</span> <span class="special">,</span> <span class="identifier">Theta</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Inside this class two member structures <code class="computeroutput"><span class="identifier">sin_functor</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">cos_functor</span></code> are defined.
|
||||
They compute the sine and the cosine of a value and they are used within
|
||||
a transform iterator to calculate the sum of <span class="emphasis"><em>sin(φ<sub>​k</sub>)</em></span>
|
||||
and <span class="emphasis"><em>cos(φ<sub>​k</sub>)</em></span>. The classifiers <code class="computeroutput"><span class="identifier">__host__</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">__device__</span></code> are CUDA
|
||||
specific and define a function or operator which can be executed on the
|
||||
GPU as well as on the CPU. The line
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">value_type</span> <span class="identifier">sin_sum</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">reduce</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">sin_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_transform_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">sin_functor</span><span class="special">()</span> <span class="special">)</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
performs the calculation of this sine-sum on the GPU (or on the CPU, depending
|
||||
on your thrust configuration).
|
||||
</p>
|
||||
<p>
|
||||
The system function is defined via
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">class</span> <span class="identifier">phase_oscillator_ensemble</span>
|
||||
<span class="special">{</span>
|
||||
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">sys_functor</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">m_K</span> <span class="special">,</span> <span class="identifier">m_Theta</span> <span class="special">,</span> <span class="identifier">m_epsilon</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">sys_functor</span><span class="special">(</span> <span class="identifier">value_type</span> <span class="identifier">K</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">Theta</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">epsilon</span> <span class="special">)</span>
|
||||
<span class="special">:</span> <span class="identifier">m_K</span><span class="special">(</span> <span class="identifier">K</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_Theta</span><span class="special">(</span> <span class="identifier">Theta</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_epsilon</span><span class="special">(</span> <span class="identifier">epsilon</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
||||
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">Tuple</span> <span class="special">></span>
|
||||
<span class="identifier">__host__</span> <span class="identifier">__device__</span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="identifier">Tuple</span> <span class="identifier">t</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">2</span><span class="special">>(</span><span class="identifier">t</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">1</span><span class="special">>(</span><span class="identifier">t</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">m_epsilon</span> <span class="special">*</span> <span class="identifier">m_K</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">m_Theta</span> <span class="special">-</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">0</span><span class="special">>(</span><span class="identifier">t</span><span class="special">)</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="comment">// ...</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()</span> <span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">dt</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">mean_field</span> <span class="special">=</span> <span class="identifier">mean_field_calculator</span><span class="special">::</span><span class="identifier">get_mean</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">for_each</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_zip_iterator</span><span class="special">(</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">dxdt</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">)</span> <span class="special">),</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_zip_iterator</span><span class="special">(</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">dxdt</span><span class="special">.</span><span class="identifier">end</span><span class="special">())</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">sys_functor</span><span class="special">(</span> <span class="identifier">mean_field</span><span class="special">.</span><span class="identifier">first</span> <span class="special">,</span> <span class="identifier">mean_field</span><span class="special">.</span><span class="identifier">second</span> <span class="special">,</span> <span class="identifier">m_epsilon</span> <span class="special">)</span>
|
||||
<span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="comment">// ...</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
This class is used within the <code class="computeroutput"><span class="identifier">do_step</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">integrate</span></code> method. It
|
||||
defines a member structure <code class="computeroutput"><span class="identifier">sys_functor</span></code>
|
||||
for the r.h.s. of each individual oscillator and the <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code> for the use in the steppers and integrators
|
||||
of odeint. The functor computes first the mean field of <span class="emphasis"><em>φ<sub>​k</sub></em></span>
|
||||
and secondly calculates the whole r.h.s. of the ODE using this mean field.
|
||||
Note, how nicely <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">tuple</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">zip_iterator</span></code> play together.
|
||||
</p>
|
||||
<p>
|
||||
Now, we are ready to put everything together. All we have to do for making
|
||||
odeint ready for using the GPU is to parametrize the stepper with the appropriate
|
||||
thrust algebra/operations:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Of course, you can also use a controlled or dense output stepper, e.g.
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Then, it is straightforward to integrate the phase ensemble by creating
|
||||
an instance of the rhs class and using an integration function:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">phase_oscillator_ensemble</span> <span class="identifier">ensemble</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">steps1</span> <span class="special">=</span> <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">ensemble</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">t_transients</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
We have to use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span></code> here in order to pass the rhs class
|
||||
as reference and not by value. This ensures that the natural frequencies
|
||||
of each oscillator are not copied when calling <code class="computeroutput"><span class="identifier">integrate_const</span></code>.
|
||||
In the full example the performance and results of the Runge-Kutta-4 and
|
||||
the Dopri5 solver are compared.
|
||||
</p>
|
||||
<p>
|
||||
The full example can be found at <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/thrust/phase_oscillator_ensemble.cu" target="_top">phase_oscillator_example.cu</a>.
|
||||
</p>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.using_cuda_and_thrust.large_oscillator_chains"></a><a class="link" href="using_cuda_and_thrust.html#boost_numeric_odeint.tutorial.using_cuda_and_thrust.large_oscillator_chains" title="Large oscillator chains">Large
|
||||
oscillator chains</a>
|
||||
</h4></div></div></div>
|
||||
<p>
|
||||
The next example is a large, one-dimensional chain of nearest-neighbor
|
||||
coupled phase oscillators with the following equations of motion:
|
||||
</p>
|
||||
<p>
|
||||
<span class="emphasis"><em>d φ<sub>​k</sub> / dt = ω<sub>​k</sub> + sin( φ<sub>​k+1</sub> - φ<sub>​k</sub> ) + sin( φ<sub>​k</sub> - φ<sub>​k-1</sub>)</em></span>
|
||||
</p>
|
||||
<p>
|
||||
In principle we can use all the techniques from the previous phase oscillator
|
||||
ensemble example, but we have to take special care about the coupling of
|
||||
the oscillators. To efficiently implement the coupling you can use a very
|
||||
elegant way employing Thrust's permutation iterator. A permutation iterator
|
||||
behaves like a normal iterator on a vector but it does not iterate along
|
||||
the usual order of the elements. It rather iterates along some permutation
|
||||
of the elements defined by some index map. To realize the nearest neighbor
|
||||
coupling we create one permutation iterator which travels one step behind
|
||||
a usual iterator and another permutation iterator which travels one step
|
||||
in front. The full system class is:
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">//change this to host_vector< ... > if you want to run on CPU</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span><span class="special"><</span> <span class="identifier">size_t</span> <span class="special">></span> <span class="identifier">index_vector_type</span><span class="special">;</span>
|
||||
<span class="comment">//typedef thrust::host_vector< value_type > state_type;</span>
|
||||
<span class="comment">//typedef thrust::host_vector< size_t > index_vector_type;</span>
|
||||
|
||||
<span class="keyword">class</span> <span class="identifier">phase_oscillators</span>
|
||||
<span class="special">{</span>
|
||||
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">sys_functor</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">Tuple</span> <span class="special">></span>
|
||||
<span class="identifier">__host__</span> <span class="identifier">__device__</span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="identifier">Tuple</span> <span class="identifier">t</span> <span class="special">)</span> <span class="comment">// this functor works on tuples of values</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">// first, unpack the tuple into value, neighbors and omega</span>
|
||||
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">phi</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">0</span><span class="special">>(</span><span class="identifier">t</span><span class="special">);</span>
|
||||
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">phi_left</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">1</span><span class="special">>(</span><span class="identifier">t</span><span class="special">);</span> <span class="comment">// left neighbor</span>
|
||||
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">phi_right</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">2</span><span class="special">>(</span><span class="identifier">t</span><span class="special">);</span> <span class="comment">// right neighbor</span>
|
||||
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">omega</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">3</span><span class="special">>(</span><span class="identifier">t</span><span class="special">);</span>
|
||||
<span class="comment">// the dynamical equation</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">4</span><span class="special">>(</span><span class="identifier">t</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">omega</span> <span class="special">+</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">phi_right</span> <span class="special">-</span> <span class="identifier">phi</span> <span class="special">)</span> <span class="special">+</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">phi</span> <span class="special">-</span> <span class="identifier">phi_left</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="identifier">phase_oscillators</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">omega</span> <span class="special">)</span>
|
||||
<span class="special">:</span> <span class="identifier">m_omega</span><span class="special">(</span> <span class="identifier">omega</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_N</span><span class="special">(</span> <span class="identifier">omega</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_prev</span><span class="special">(</span> <span class="identifier">omega</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_next</span><span class="special">(</span> <span class="identifier">omega</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">// build indices pointing to left and right neighbours</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">counting_iterator</span><span class="special"><</span><span class="identifier">size_t</span><span class="special">></span> <span class="identifier">c</span><span class="special">(</span> <span class="number">0</span> <span class="special">);</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span> <span class="identifier">c</span> <span class="special">,</span> <span class="identifier">c</span><span class="special">+</span><span class="identifier">m_N</span><span class="special">-</span><span class="number">1</span> <span class="special">,</span> <span class="identifier">m_prev</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()+</span><span class="number">1</span> <span class="special">);</span>
|
||||
<span class="identifier">m_prev</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="comment">// m_prev = { 0 , 0 , 1 , 2 , 3 , ... , N-1 }</span>
|
||||
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span> <span class="identifier">c</span><span class="special">+</span><span class="number">1</span> <span class="special">,</span> <span class="identifier">c</span><span class="special">+</span><span class="identifier">m_N</span> <span class="special">,</span> <span class="identifier">m_next</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">);</span>
|
||||
<span class="identifier">m_next</span><span class="special">[</span><span class="identifier">m_N</span><span class="special">-</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">m_N</span><span class="special">-</span><span class="number">1</span><span class="special">;</span> <span class="comment">// m_next = { 1 , 2 , 3 , ... , N-1 , N-1 }</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()</span> <span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">dt</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">for_each</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_zip_iterator</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span>
|
||||
<span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_permutation_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_prev</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_permutation_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_next</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span>
|
||||
<span class="identifier">dxdt</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span>
|
||||
<span class="special">)</span> <span class="special">),</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_zip_iterator</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span>
|
||||
<span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_permutation_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_prev</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_permutation_iterator</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_next</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span>
|
||||
<span class="identifier">dxdt</span><span class="special">.</span><span class="identifier">end</span><span class="special">())</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">sys_functor</span><span class="special">()</span>
|
||||
<span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="keyword">private</span><span class="special">:</span>
|
||||
|
||||
<span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">m_omega</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">m_N</span><span class="special">;</span>
|
||||
<span class="identifier">index_vector_type</span> <span class="identifier">m_prev</span><span class="special">;</span>
|
||||
<span class="identifier">index_vector_type</span> <span class="identifier">m_next</span><span class="special">;</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Note, how easy you can obtain the value for the left and right neighboring
|
||||
oscillator in the system functor using the permutation iterators. But,
|
||||
the call of the <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">for_each</span></code>
|
||||
function looks relatively complicated. Every term of the r.h.s. of the
|
||||
ODE is resembled by one iterator packed in exactly the same way as it is
|
||||
unpacked in the functor above.
|
||||
</p>
|
||||
<p>
|
||||
Now we put everything together. We create random initial conditions and
|
||||
decreasing frequencies such that we should get synchronization. We copy
|
||||
the frequencies and the initial conditions onto the device and finally
|
||||
initialize and perform the integration. As result we simply write out the
|
||||
current state, hence the phase of each oscillator.
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// create initial conditions and omegas on host:</span>
|
||||
<span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">x_host</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||
<span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">omega_host</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">N</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">x_host</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0</span> <span class="special">*</span> <span class="identifier">pi</span> <span class="special">*</span> <span class="identifier">drand48</span><span class="special">();</span>
|
||||
<span class="identifier">omega_host</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="special">(</span> <span class="identifier">N</span> <span class="special">-</span> <span class="identifier">i</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">epsilon</span><span class="special">;</span> <span class="comment">// decreasing frequencies</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="comment">// copy to device</span>
|
||||
<span class="identifier">state_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">x_host</span><span class="special">;</span>
|
||||
<span class="identifier">state_type</span> <span class="identifier">omega</span> <span class="special">=</span> <span class="identifier">omega_host</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// create stepper</span>
|
||||
<span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// create phase oscillator system function</span>
|
||||
<span class="identifier">phase_oscillators</span> <span class="identifier">sys</span><span class="special">(</span> <span class="identifier">omega</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">// integrate</span>
|
||||
<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">stepper</span> <span class="special">,</span> <span class="identifier">sys</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">>(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">,</span> <span class="string">"\n"</span> <span class="special">)</span> <span class="special">);</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The full example can be found at <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/thrust/phase_oscillator_chain.cu" target="_top">phase_oscillator_chain.cu</a>.
|
||||
</p>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h4 class="title">
|
||||
<a name="boost_numeric_odeint.tutorial.using_cuda_and_thrust.parameter_studies"></a><a class="link" href="using_cuda_and_thrust.html#boost_numeric_odeint.tutorial.using_cuda_and_thrust.parameter_studies" title="Parameter studies">Parameter
|
||||
studies</a>
|
||||
</h4></div></div></div>
|
||||
<p>
|
||||
Another important use case for <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>
|
||||
and Cuda are parameter studies of relatively small systems. Consider for
|
||||
example the three-dimensional Lorenz system from the chaotic systems example
|
||||
in the previous section which has three parameters. If you want to study
|
||||
the behavior of this system for different parameters you usually have to
|
||||
integrate the system for many parameter values. Using thrust and odeint
|
||||
you can do this integration in parallel, hence you integrate a whole ensemble
|
||||
of Lorenz systems where each individual realization has a different parameter
|
||||
value.
|
||||
</p>
|
||||
<p>
|
||||
In the following we will show how you can use <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>
|
||||
to integrate the above mentioned ensemble of Lorenz systems. We will vary
|
||||
only the parameter <span class="emphasis"><em>β</em></span> but it is straightforward to vary
|
||||
other parameters or even two or all three parameters. Furthermore, we will
|
||||
use the largest Lyapunov exponent to quantify the behavior of the system
|
||||
(chaoticity).
|
||||
</p>
|
||||
<p>
|
||||
We start by defining the range of the parameters we want to study. Of course,
|
||||
the state_type is again a <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span><span class="special"><</span> <span class="identifier">value_type</span>
|
||||
<span class="special">></span></code>.
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">beta_host</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">beta_min</span> <span class="special">=</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">beta_max</span> <span class="special">=</span> <span class="number">56.0</span><span class="special">;</span>
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">N</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
||||
<span class="identifier">beta_host</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">beta_min</span> <span class="special">+</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">)</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">beta_max</span> <span class="special">-</span> <span class="identifier">beta_min</span> <span class="special">)</span> <span class="special">/</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span> <span class="special">);</span>
|
||||
|
||||
<span class="identifier">state_type</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="identifier">beta_host</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The next thing we have to implement is the Lorenz system without perturbations.
|
||||
Later, a system with perturbations is also implemented in order to calculate
|
||||
the Lyapunov exponent. We will use an ansatz where each device function
|
||||
calculates one particular realization of the Lorenz ensemble
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz_system</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">struct</span> <span class="identifier">lorenz_functor</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">></span>
|
||||
<span class="identifier">__host__</span> <span class="identifier">__device__</span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="identifier">T</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">// unpack the parameter we want to vary and the Lorenz variables</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">R</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">3</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">0</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">1</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">z</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">2</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">4</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">)</span> <span class="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">y</span> <span class="special">-</span> <span class="identifier">x</span> <span class="special">);</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">5</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">)</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span> <span class="special">-</span> <span class="identifier">y</span> <span class="special">-</span> <span class="identifier">x</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">;</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">6</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">)</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">z</span> <span class="special">+</span> <span class="identifier">x</span> <span class="special">*</span> <span class="identifier">y</span> <span class="special">;</span>
|
||||
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="identifier">lorenz_system</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">N</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">beta</span> <span class="special">)</span>
|
||||
<span class="special">:</span> <span class="identifier">m_N</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_beta</span><span class="special">(</span> <span class="identifier">beta</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
||||
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Deriv</span> <span class="special">></span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">Deriv</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">for_each</span><span class="special">(</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_zip_iterator</span><span class="special">(</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">+</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">+</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">m_beta</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">+</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">+</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">m_N</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_zip_iterator</span><span class="special">(</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">+</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">+</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">+</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">m_beta</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">+</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">+</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">m_N</span> <span class="special">,</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">+</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">m_N</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
|
||||
<span class="identifier">lorenz_functor</span><span class="special">()</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="identifier">size_t</span> <span class="identifier">m_N</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">m_beta</span><span class="special">;</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
As <code class="computeroutput"><span class="identifier">state_type</span></code> a <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span></code> or a <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
of a <code class="computeroutput"><span class="identifier">device_vector</span></code> is used.
|
||||
The length of the state is <span class="emphasis"><em>3N</em></span> where <span class="emphasis"><em>N</em></span>
|
||||
is the number of systems. The system is encoded into this vector such that
|
||||
all <span class="emphasis"><em>x</em></span> components come first, then every <span class="emphasis"><em>y</em></span>
|
||||
components and finally every <span class="emphasis"><em>z</em></span> components. Implementing
|
||||
the device function is then a simple task, you only have to decompose the
|
||||
tuple originating from the zip iterators.
|
||||
</p>
|
||||
<p>
|
||||
Besides the system without perturbations we furthermore need to calculate
|
||||
the system including linearized equations governing the time evolution
|
||||
of small perturbations. Using the method from above this is straightforward,
|
||||
with a small difficulty that Thrust's tuples have a maximal arity of 10.
|
||||
But this is only a small problem since we can create a zip iterator packed
|
||||
with zip iterators. So the top level zip iterator contains one zip iterator
|
||||
for the state, one normal iterator for the parameter, and one zip iterator
|
||||
for the derivative. Accessing the elements of this tuple in the system
|
||||
function is then straightforward, you unpack the tuple with <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><>()</span></code>.
|
||||
We will not show the code here, it is to large. It can be found <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/thrust/lorenz_parameters.cu" target="_top">here</a> and
|
||||
is easy to understand.
|
||||
</p>
|
||||
<p>
|
||||
Furthermore, we need an observer which determines the norm of the perturbations,
|
||||
normalizes them and averages the logarithm of the norm. The device functor
|
||||
which is used within this observer is defined
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lyap_functor</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">></span>
|
||||
<span class="identifier">__host__</span> <span class="identifier">__device__</span>
|
||||
<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="identifier">T</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">value_type</span> <span class="special">&</span><span class="identifier">dx</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">0</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="special">&</span><span class="identifier">dy</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">1</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="special">&</span><span class="identifier">dz</span> <span class="special">=</span> <span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">2</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="identifier">value_type</span> <span class="identifier">norm</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(</span> <span class="identifier">dx</span> <span class="special">*</span> <span class="identifier">dx</span> <span class="special">+</span> <span class="identifier">dy</span> <span class="special">*</span> <span class="identifier">dy</span> <span class="special">+</span> <span class="identifier">dz</span> <span class="special">*</span> <span class="identifier">dz</span> <span class="special">);</span>
|
||||
<span class="identifier">dx</span> <span class="special">/=</span> <span class="identifier">norm</span><span class="special">;</span>
|
||||
<span class="identifier">dy</span> <span class="special">/=</span> <span class="identifier">norm</span><span class="special">;</span>
|
||||
<span class="identifier">dz</span> <span class="special">/=</span> <span class="identifier">norm</span><span class="special">;</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span> <span class="number">3</span> <span class="special">>(</span> <span class="identifier">t</span> <span class="special">)</span> <span class="special">+=</span> <span class="identifier">log</span><span class="special">(</span> <span class="identifier">norm</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
Note, that this functor manipulates the state, i.e. the perturbations.
|
||||
</p>
|
||||
<p>
|
||||
Now we complete the whole code to calculate the Lyapunov exponents. First,
|
||||
we have to define a state vector. This vector contains <span class="emphasis"><em>6N</em></span>
|
||||
entries, the state <span class="emphasis"><em>x,y,z</em></span> and its perturbations <span class="emphasis"><em>dx,dy,dz</em></span>.
|
||||
We initialize them such that <span class="emphasis"><em>x=y=z=10</em></span>, <span class="emphasis"><em>dx=1</em></span>,
|
||||
and <span class="emphasis"><em>dy=dz=0</em></span>. We define a stepper type, a controlled
|
||||
Runge-Kutta Dormand-Prince 5 stepper. We start with some integration to
|
||||
overcome the transient behavior. For this, we do not involve the perturbation
|
||||
and run the algorithm only on the state <span class="emphasis"><em>x,y,z</em></span> without
|
||||
any observer. Note, how <a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
|
||||
is used for partial integration of the state vector without perturbations
|
||||
(the first half of the whole state). After the transient, the full system
|
||||
with perturbations is integrated and the Lyapunov exponents are calculated
|
||||
and written to <code class="computeroutput"><span class="identifier">stdout</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">6</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">// initialize x,y,z</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">fill</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">// initial dx</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">fill</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">4</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">// initialize dy,dz</span>
|
||||
<span class="identifier">thrust</span><span class="special">::</span><span class="identifier">fill</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">4</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
||||
|
||||
|
||||
<span class="comment">// create error stepper, can be used with make_controlled or make_dense_output</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="identifier">thrust_algebra</span> <span class="special">,</span> <span class="identifier">thrust_operations</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
||||
|
||||
|
||||
<span class="identifier">lorenz_system</span> <span class="identifier">lorenz</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">,</span> <span class="identifier">beta</span> <span class="special">);</span>
|
||||
<span class="identifier">lorenz_perturbation_system</span> <span class="identifier">lorenz_perturbation</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">,</span> <span class="identifier">beta</span> <span class="special">);</span>
|
||||
<span class="identifier">lyap_observer</span> <span class="identifier">obs</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">,</span> <span class="number">1</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">// calculate transients</span>
|
||||
<span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">lorenz</span> <span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">N</span> <span class="special">)</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
||||
|
||||
<span class="comment">// calculate the Lyapunov exponents -- the main loop</span>
|
||||
<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
||||
<span class="keyword">while</span><span class="special">(</span> <span class="identifier">t</span> <span class="special"><</span> <span class="number">10000.0</span> <span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">lorenz_perturbation</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">+</span> <span class="number">1.0</span> <span class="special">,</span> <span class="number">0.1</span> <span class="special">);</span>
|
||||
<span class="identifier">t</span> <span class="special">+=</span> <span class="number">1.0</span><span class="special">;</span>
|
||||
<span class="identifier">obs</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="identifier">vector</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">></span> <span class="identifier">lyap</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
||||
<span class="identifier">obs</span><span class="special">.</span><span class="identifier">fill_lyap</span><span class="special">(</span> <span class="identifier">lyap</span> <span class="special">);</span>
|
||||
|
||||
<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">N</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">beta_host</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">lyap</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
The full example can be found at <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/thrust/lorenz_parameters.cu" target="_top">lorenz_parameters.cu</a>.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
|
||||
and Mario Mulansky<p>
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||
</p>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="special_topics.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="using_opencl_and_vexcl.html"><img src="../../images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,618 +0,0 @@
|
||||
/*=============================================================================
|
||||
Copyright (c) 2004 Joel de Guzman
|
||||
http://spirit.sourceforge.net/
|
||||
|
||||
Distributed under the Boost Software License, Version 1.0. (See accompany-
|
||||
ing file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
|
||||
=============================================================================*/
|
||||
|
||||
/*=============================================================================
|
||||
Body defaults
|
||||
=============================================================================*/
|
||||
|
||||
body
|
||||
{
|
||||
margin: 1em;
|
||||
font-family: sans-serif;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Paragraphs
|
||||
=============================================================================*/
|
||||
|
||||
p
|
||||
{
|
||||
text-align: left;
|
||||
font-size: 10pt;
|
||||
line-height: 1.15;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Program listings
|
||||
=============================================================================*/
|
||||
|
||||
/* Code on paragraphs */
|
||||
p tt.computeroutput
|
||||
{
|
||||
font-size: 9pt;
|
||||
}
|
||||
|
||||
pre.synopsis
|
||||
{
|
||||
font-size: 9pt;
|
||||
margin: 1pc 4% 0pc 4%;
|
||||
padding: 0.5pc 0.5pc 0.5pc 0.5pc;
|
||||
}
|
||||
|
||||
.programlisting,
|
||||
.screen
|
||||
{
|
||||
font-size: 9pt;
|
||||
display: block;
|
||||
margin: 1pc 4% 0pc 4%;
|
||||
padding: 0.5pc 0.5pc 0.5pc 0.5pc;
|
||||
}
|
||||
|
||||
/* Program listings in tables don't get borders */
|
||||
td .programlisting,
|
||||
td .screen
|
||||
{
|
||||
margin: 0pc 0pc 0pc 0pc;
|
||||
padding: 0pc 0pc 0pc 0pc;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Headings
|
||||
=============================================================================*/
|
||||
|
||||
h1, h2, h3, h4, h5, h6
|
||||
{
|
||||
text-align: left;
|
||||
margin: 1em 0em 0.5em 0em;
|
||||
font-weight: bold;
|
||||
}
|
||||
|
||||
h1 { font-size: 140%; }
|
||||
h2 {
|
||||
/* changed by Mario Mulansky to show logo instead of title */
|
||||
height: 127px;
|
||||
background: white url(logo.jpg) left no-repeat;
|
||||
text-indent: -999px;
|
||||
/* end change */
|
||||
font-weight: bold; font-size: 140%;
|
||||
}
|
||||
h3 { font-weight: bold; font-size: 130%; }
|
||||
h4 { font-weight: bold; font-size: 120%; }
|
||||
h5 { font-weight: normal; font-style: italic; font-size: 110%; }
|
||||
h6 { font-weight: normal; font-style: italic; font-size: 100%; }
|
||||
|
||||
/* Top page titles */
|
||||
title,
|
||||
h1.title,
|
||||
h2.title
|
||||
h3.title,
|
||||
h4.title,
|
||||
h5.title,
|
||||
h6.title,
|
||||
.refentrytitle
|
||||
{
|
||||
font-weight: bold;
|
||||
margin-bottom: 1pc;
|
||||
}
|
||||
|
||||
h1.title { font-size: 140% }
|
||||
h2.title { font-size: 140% }
|
||||
h3.title { font-size: 130% }
|
||||
h4.title { font-size: 120% }
|
||||
h5.title { font-size: 110% }
|
||||
h6.title { font-size: 100% }
|
||||
|
||||
.section h1
|
||||
{
|
||||
margin: 0em 0em 0.5em 0em;
|
||||
font-size: 140%;
|
||||
}
|
||||
|
||||
.section h2 { font-size: 140% }
|
||||
.section h3 { font-size: 130% }
|
||||
.section h4 { font-size: 120% }
|
||||
.section h5 { font-size: 110% }
|
||||
.section h6 { font-size: 100% }
|
||||
|
||||
/* Code on titles */
|
||||
h1 tt.computeroutput { font-size: 140% }
|
||||
h2 tt.computeroutput { font-size: 140% }
|
||||
h3 tt.computeroutput { font-size: 130% }
|
||||
h4 tt.computeroutput { font-size: 130% }
|
||||
h5 tt.computeroutput { font-size: 130% }
|
||||
h6 tt.computeroutput { font-size: 130% }
|
||||
|
||||
|
||||
/*=============================================================================
|
||||
Author
|
||||
=============================================================================*/
|
||||
|
||||
h3.author
|
||||
{
|
||||
font-size: 100%
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Lists
|
||||
=============================================================================*/
|
||||
|
||||
li
|
||||
{
|
||||
font-size: 10pt;
|
||||
line-height: 1.3;
|
||||
}
|
||||
|
||||
/* Unordered lists */
|
||||
ul
|
||||
{
|
||||
text-align: left;
|
||||
}
|
||||
|
||||
/* Ordered lists */
|
||||
ol
|
||||
{
|
||||
text-align: left;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Links
|
||||
=============================================================================*/
|
||||
|
||||
a
|
||||
{
|
||||
text-decoration: none; /* no underline */
|
||||
}
|
||||
|
||||
a:hover
|
||||
{
|
||||
text-decoration: underline;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Spirit style navigation
|
||||
=============================================================================*/
|
||||
|
||||
.spirit-nav
|
||||
{
|
||||
text-align: right;
|
||||
}
|
||||
|
||||
.spirit-nav a
|
||||
{
|
||||
color: white;
|
||||
padding-left: 0.5em;
|
||||
}
|
||||
|
||||
.spirit-nav img
|
||||
{
|
||||
border-width: 0px;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Copyright footer
|
||||
=============================================================================*/
|
||||
.copyright-footer
|
||||
{
|
||||
text-align: right;
|
||||
font-size: 70%;
|
||||
}
|
||||
|
||||
.copyright-footer p
|
||||
{
|
||||
text-align: right;
|
||||
font-size: 80%;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Table of contents
|
||||
=============================================================================*/
|
||||
|
||||
.toc
|
||||
{
|
||||
margin: 1pc 4% 0pc 4%;
|
||||
padding: 0.1pc 1pc 0.1pc 1pc;
|
||||
font-size: 80%;
|
||||
line-height: 1.15;
|
||||
}
|
||||
|
||||
.boost-toc
|
||||
{
|
||||
float: right;
|
||||
padding: 0.5pc;
|
||||
}
|
||||
|
||||
/* Code on toc */
|
||||
.toc .computeroutput { font-size: 120% }
|
||||
|
||||
/* No margin on nested menus */
|
||||
|
||||
.toc dl dl { margin: 0; }
|
||||
|
||||
/*=============================================================================
|
||||
Tables
|
||||
=============================================================================*/
|
||||
|
||||
.table-title,
|
||||
div.table p.title
|
||||
{
|
||||
margin-left: 4%;
|
||||
padding-right: 0.5em;
|
||||
padding-left: 0.5em;
|
||||
}
|
||||
|
||||
.informaltable table,
|
||||
.table table
|
||||
{
|
||||
width: 92%;
|
||||
margin-left: 4%;
|
||||
margin-right: 4%;
|
||||
}
|
||||
|
||||
div.informaltable table,
|
||||
div.table table
|
||||
{
|
||||
padding: 4px;
|
||||
}
|
||||
|
||||
/* Table Cells */
|
||||
div.informaltable table tr td,
|
||||
div.table table tr td
|
||||
{
|
||||
padding: 0.5em;
|
||||
text-align: left;
|
||||
font-size: 9pt;
|
||||
}
|
||||
|
||||
div.informaltable table tr th,
|
||||
div.table table tr th
|
||||
{
|
||||
padding: 0.5em 0.5em 0.5em 0.5em;
|
||||
border: 1pt solid white;
|
||||
font-size: 80%;
|
||||
}
|
||||
|
||||
table.simplelist
|
||||
{
|
||||
width: auto !important;
|
||||
margin: 0em !important;
|
||||
padding: 0em !important;
|
||||
border: none !important;
|
||||
}
|
||||
table.simplelist td
|
||||
{
|
||||
margin: 0em !important;
|
||||
padding: 0em !important;
|
||||
text-align: left !important;
|
||||
font-size: 9pt !important;
|
||||
border: none !important;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Blurbs
|
||||
=============================================================================*/
|
||||
|
||||
div.note,
|
||||
div.tip,
|
||||
div.important,
|
||||
div.caution,
|
||||
div.warning,
|
||||
p.blurb
|
||||
{
|
||||
font-size: 9pt; /* A little bit smaller than the main text */
|
||||
line-height: 1.2;
|
||||
display: block;
|
||||
margin: 1pc 4% 0pc 4%;
|
||||
padding: 0.5pc 0.5pc 0.5pc 0.5pc;
|
||||
}
|
||||
|
||||
p.blurb img
|
||||
{
|
||||
padding: 1pt;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Variable Lists
|
||||
=============================================================================*/
|
||||
|
||||
div.variablelist
|
||||
{
|
||||
margin: 1em 0;
|
||||
}
|
||||
|
||||
/* Make the terms in definition lists bold */
|
||||
div.variablelist dl dt,
|
||||
span.term
|
||||
{
|
||||
font-weight: bold;
|
||||
font-size: 10pt;
|
||||
}
|
||||
|
||||
div.variablelist table tbody tr td
|
||||
{
|
||||
text-align: left;
|
||||
vertical-align: top;
|
||||
padding: 0em 2em 0em 0em;
|
||||
font-size: 10pt;
|
||||
margin: 0em 0em 0.5em 0em;
|
||||
line-height: 1;
|
||||
}
|
||||
|
||||
div.variablelist dl dt
|
||||
{
|
||||
margin-bottom: 0.2em;
|
||||
}
|
||||
|
||||
div.variablelist dl dd
|
||||
{
|
||||
margin: 0em 0em 0.5em 2em;
|
||||
font-size: 10pt;
|
||||
}
|
||||
|
||||
div.variablelist table tbody tr td p,
|
||||
div.variablelist dl dd p
|
||||
{
|
||||
margin: 0em 0em 0.5em 0em;
|
||||
line-height: 1;
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Misc
|
||||
=============================================================================*/
|
||||
|
||||
/* Title of books and articles in bibliographies */
|
||||
span.title
|
||||
{
|
||||
font-style: italic;
|
||||
}
|
||||
|
||||
span.underline
|
||||
{
|
||||
text-decoration: underline;
|
||||
}
|
||||
|
||||
span.strikethrough
|
||||
{
|
||||
text-decoration: line-through;
|
||||
}
|
||||
|
||||
/* Copyright, Legal Notice */
|
||||
div div.legalnotice p
|
||||
{
|
||||
text-align: left
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Colors
|
||||
=============================================================================*/
|
||||
|
||||
@media screen
|
||||
{
|
||||
body {
|
||||
background-color: #FFFFFF;
|
||||
color: #000000;
|
||||
}
|
||||
|
||||
/* Syntax Highlighting */
|
||||
.keyword { color: #0000AA; }
|
||||
.identifier { color: #000000; }
|
||||
.special { color: #707070; }
|
||||
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|
||||
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|
||||
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|
||||
.string { color: teal; }
|
||||
.number { color: teal; }
|
||||
.white_bkd { background-color: #FFFFFF; }
|
||||
.dk_grey_bkd { background-color: #999999; }
|
||||
|
||||
/* Links */
|
||||
a, a .keyword, a .identifier, a .special, a .preprocessor
|
||||
a .char, a .comment, a .string, a .number
|
||||
{
|
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color: #005a9c;
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}
|
||||
|
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a:visited, a:visited .keyword, a:visited .identifier,
|
||||
a:visited .special, a:visited .preprocessor a:visited .char,
|
||||
a:visited .comment, a:visited .string, a:visited .number
|
||||
{
|
||||
color: #9c5a9c;
|
||||
}
|
||||
|
||||
h1 a, h2 a, h3 a, h4 a, h5 a, h6 a,
|
||||
h1 a:hover, h2 a:hover, h3 a:hover, h4 a:hover, h5 a:hover, h6 a:hover,
|
||||
h1 a:visited, h2 a:visited, h3 a:visited, h4 a:visited, h5 a:visited, h6 a:visited
|
||||
{
|
||||
text-decoration: none; /* no underline */
|
||||
color: #000000;
|
||||
}
|
||||
|
||||
/* Copyright, Legal Notice */
|
||||
.copyright
|
||||
{
|
||||
color: #666666;
|
||||
font-size: small;
|
||||
}
|
||||
|
||||
div div.legalnotice p
|
||||
{
|
||||
color: #666666;
|
||||
}
|
||||
|
||||
/* Program listing */
|
||||
pre.synopsis
|
||||
{
|
||||
border: 1px solid #DCDCDC;
|
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|
||||
|
||||
.programlisting,
|
||||
.screen
|
||||
{
|
||||
border: 1px solid #DCDCDC;
|
||||
}
|
||||
|
||||
td .programlisting,
|
||||
td .screen
|
||||
{
|
||||
border: 0px solid #DCDCDC;
|
||||
}
|
||||
|
||||
/* Blurbs */
|
||||
div.note,
|
||||
div.tip,
|
||||
div.important,
|
||||
div.caution,
|
||||
div.warning,
|
||||
p.blurb
|
||||
{
|
||||
border: 1px solid #DCDCDC;
|
||||
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|
||||
|
||||
/* Table of contents */
|
||||
.toc
|
||||
{
|
||||
border: 1px solid #DCDCDC;
|
||||
}
|
||||
|
||||
/* Tables */
|
||||
div.informaltable table tr td,
|
||||
div.table table tr td
|
||||
{
|
||||
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|
||||
}
|
||||
|
||||
div.informaltable table tr th,
|
||||
div.table table tr th
|
||||
{
|
||||
background-color: #F0F0F0;
|
||||
border: 1px solid #DCDCDC;
|
||||
}
|
||||
|
||||
.copyright-footer
|
||||
{
|
||||
color: #8F8F8F;
|
||||
}
|
||||
|
||||
/* Misc */
|
||||
span.highlight
|
||||
{
|
||||
color: #00A000;
|
||||
}
|
||||
}
|
||||
|
||||
@media print
|
||||
{
|
||||
/* Links */
|
||||
a
|
||||
{
|
||||
color: black;
|
||||
}
|
||||
|
||||
a:visited
|
||||
{
|
||||
color: black;
|
||||
}
|
||||
|
||||
.spirit-nav
|
||||
{
|
||||
display: none;
|
||||
}
|
||||
|
||||
/* Program listing */
|
||||
pre.synopsis
|
||||
{
|
||||
border: 1px solid gray;
|
||||
}
|
||||
|
||||
.programlisting,
|
||||
.screen
|
||||
{
|
||||
border: 1px solid gray;
|
||||
}
|
||||
|
||||
td .programlisting,
|
||||
td .screen
|
||||
{
|
||||
border: 0px solid #DCDCDC;
|
||||
}
|
||||
|
||||
/* Table of contents */
|
||||
.toc
|
||||
{
|
||||
border: 1px solid gray;
|
||||
}
|
||||
|
||||
.informaltable table,
|
||||
.table table
|
||||
{
|
||||
border: 1px solid gray;
|
||||
border-collapse: collapse;
|
||||
}
|
||||
|
||||
/* Tables */
|
||||
div.informaltable table tr td,
|
||||
div.table table tr td
|
||||
{
|
||||
border: 1px solid gray;
|
||||
}
|
||||
|
||||
div.informaltable table tr th,
|
||||
div.table table tr th
|
||||
{
|
||||
border: 1px solid gray;
|
||||
}
|
||||
|
||||
table.simplelist tr td
|
||||
{
|
||||
border: none !important;
|
||||
}
|
||||
|
||||
/* Misc */
|
||||
span.highlight
|
||||
{
|
||||
font-weight: bold;
|
||||
}
|
||||
}
|
||||
|
||||
/*=============================================================================
|
||||
Images
|
||||
=============================================================================*/
|
||||
|
||||
span.inlinemediaobject img
|
||||
{
|
||||
vertical-align: middle;
|
||||
}
|
||||
|
||||
/*==============================================================================
|
||||
Super and Subscript: style so that line spacing isn't effected, see
|
||||
http://www.adobe.com/cfusion/communityengine/index.cfm?event=showdetails&productId=1&postId=5341
|
||||
==============================================================================*/
|
||||
|
||||
sup,
|
||||
sub {
|
||||
height: 0;
|
||||
line-height: 1;
|
||||
vertical-align: baseline;
|
||||
position: relative;
|
||||
|
||||
}
|
||||
|
||||
/* For internet explorer: */
|
||||
|
||||
* html sup,
|
||||
* html sub {
|
||||
vertical-align: bottom;
|
||||
}
|
||||
|
||||
sup {
|
||||
bottom: 1ex;
|
||||
}
|
||||
|
||||
sub {
|
||||
top: .5ex;
|
||||
}
|
||||
|
||||
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