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<title>Harmonic oscillator</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.tutorial.harmonic_oscillator"></a><a class="link" href="harmonic_oscillator.html" title="Harmonic oscillator">Harmonic
oscillator</a>
</h3></div></div></div>
<div class="toc"><dl>
<dt><span class="section"><a href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.define_the_ode">Define
the ODE</a></span></dt>
<dt><span class="section"><a href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.stepper_types">Stepper
Types</a></span></dt>
<dt><span class="section"><a href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.integration_with_constant_step_size">Integration
with Constant Step Size</a></span></dt>
<dt><span class="section"><a href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.integration_with_adaptive_step_size">Integration
with Adaptive Step Size</a></span></dt>
</dl></div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.tutorial.harmonic_oscillator.define_the_ode"></a><a class="link" href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.define_the_ode" title="Define the ODE">Define
the ODE</a>
</h4></div></div></div>
<p>
First of all, you have to specify the data type that represents a state
<span class="emphasis"><em>x</em></span> of your system. Mathematically, this usually is
an n-dimensional vector with real numbers or complex numbers as scalar
objects. For odeint the most natural way is to use <code class="computeroutput"><span class="identifier">vector</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</span></code> or <code class="computeroutput"><span class="identifier">vector</span><span class="special">&lt;</span> <span class="identifier">complex</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="special">&gt;</span></code>
to represent the system state. However, odeint can deal with other container
types as well, e.g. <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">&gt;</span></code>, as long as it is fulfills some requirements
defined below.
</p>
<p>
To integrate a differential equation numerically, one also has to define
the rhs of the equation <span class="emphasis"><em>x' = f(x)</em></span>. In odeint you supply
this function in terms of an object that implements the ()-operator with
a certain parameter structure. Hence, the straight forward way would be
to just define a function, e.g:
</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">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</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">&amp;</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&amp;</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>
The parameters of the function must follow the example above where <code class="computeroutput"><span class="identifier">x</span></code> is the current state, here a two-component
vector containing position <span class="emphasis"><em>q</em></span> and momentum <span class="emphasis"><em>p</em></span>
of the oscillator, <code class="computeroutput"><span class="identifier">dxdt</span></code>
is the derivative <span class="emphasis"><em>x'</em></span> and should be filled by the function
with <span class="emphasis"><em>f(x)</em></span>, and <code class="computeroutput"><span class="identifier">t</span></code>
is the current time. Note that in this example <span class="emphasis"><em>t</em></span> is
not required to calculate <span class="emphasis"><em>f</em></span>, however odeint expects
the function signature to have exactly three parameters (there are exception,
discussed later).
</p>
<p>
A more sophisticated approach is to implement the system as a class where
the rhs function is defined as the ()-operator of the class with the same
parameter structure as above:
</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">&amp;</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&amp;</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>
odeint can deal with instances of such classes instead of pure functions
which allows for cleaner code.
</p>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.tutorial.harmonic_oscillator.stepper_types"></a><a class="link" href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.stepper_types" title="Stepper Types">Stepper
Types</a>
</h4></div></div></div>
<p>
Numerical integration works iteratively, that means you start at a state
<span class="emphasis"><em>x(t)</em></span> and perform a time-step of length <span class="emphasis"><em>dt</em></span>
to obtain the approximate state <span class="emphasis"><em>x(t+dt)</em></span>. There exist
many different methods to perform such a time-step each of which has a
certain order <span class="emphasis"><em>q</em></span>. If the order of a method is <span class="emphasis"><em>q</em></span>
than it is accurate up to term <span class="emphasis"><em>~dt<sup>q</sup></em></span> that means the
error in <span class="emphasis"><em>x</em></span> made by such a step is <span class="emphasis"><em>~dt<sup>q+1</sup></em></span>.
odeint provides several steppers of different orders from which you can
choose:
</p>
<div class="table">
<a name="boost_numeric_odeint.tutorial.harmonic_oscillator.stepper_types.stepper_algorithms"></a><p class="title"><b>Table&#160;1.2.&#160;Stepper Algorithms</b></p>
<div class="table-contents"><table class="table" summary="Stepper Algorithms">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Algorithm
</p>
</th>
<th>
<p>
Class
</p>
</th>
<th>
<p>
Concept
</p>
</th>
<th>
<p>
System Concept
</p>
</th>
<th>
<p>
Order
</p>
</th>
<th>
<p>
Error Estimation
</p>
</th>
<th>
<p>
Dense Output
</p>
</th>
<th>
<p>
Internal state
</p>
</th>
<th>
<p>
Remarks
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Explicit Euler
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">euler</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
Output Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
1
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Very simple, only for demonstrating purpose
</p>
</td>
</tr>
<tr>
<td>
<p>
Modified Midpoint
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">modified_midpoint</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
configurable (2)
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Used in Bulirsch-Stoer implementation
</p>
</td>
</tr>
<tr>
<td>
<p>
Runge-Kutta 4
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">runge_kutta4</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
4
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
The classical Runge Kutta scheme, good general scheme without
error control
</p>
</td>
</tr>
<tr>
<td>
<p>
Cash-Karp
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">runge_kutta_cash_karp54</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
5
</p>
</td>
<td>
<p>
Yes (4)
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Good general scheme with error estimation, to be used in controlled_error_stepper
</p>
</td>
</tr>
<tr>
<td>
<p>
Dormand-Prince 5
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
5
</p>
</td>
<td>
<p>
Yes (4)
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Standard method with error control and dense output, to be used
in controlled_error_stepper and in dense_output_controlled_explicit_fsal.
</p>
</td>
</tr>
<tr>
<td>
<p>
Fehlberg 78
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">runge_kutta_fehlberg78</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
8
</p>
</td>
<td>
<p>
Yes (7)
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Good high order method with error estimation, to be used in controlled_error_stepper.
</p>
</td>
</tr>
<tr>
<td>
<p>
Adams Bashforth
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">adams_bashforth</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
configurable
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Multistep method
</p>
</td>
</tr>
<tr>
<td>
<p>
Adams Moulton
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">adams_moulton</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
configurable
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Multistep method
</p>
</td>
</tr>
<tr>
<td>
<p>
Adams Bashforth Moulton
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">adams_bashforth_moulton</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
configurable
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Combined multistep method
</p>
</td>
</tr>
<tr>
<td>
<p>
Controlled Runge Kutta
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">controlled_runge_kutta</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
depends
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
depends
</p>
</td>
<td>
<p>
Error control for <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
Stepper</a>. Requires an <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
Stepper</a> from above. Order depends on the given ErrorStepper
</p>
</td>
</tr>
<tr>
<td>
<p>
Dense Output Runge Kutta
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">dense_output_runge_kutta</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
Output Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
depends
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Dense output for <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
and <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
Stepper</a> from above if they provide dense output functionality
(like <code class="computeroutput"><span class="identifier">euler</span></code> and
<code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>).
Order depends on the given stepper.
</p>
</td>
</tr>
<tr>
<td>
<p>
Bulirsch-Stoer
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">bulirsch_stoer</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
variable
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Stepper with step size and order control. Very good if high precision
is required.
</p>
</td>
</tr>
<tr>
<td>
<p>
Bulirsch-Stoer Dense Output
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">bulirsch_stoer_dense_out</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
Output Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/system.html" title="System">System</a>
</p>
</td>
<td>
<p>
variable
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Stepper with step size and order control as well as dense output.
Very good if high precision and dense output is required.
</p>
</td>
</tr>
<tr>
<td>
<p>
Implicit Euler
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">implicit_euler</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
System</a>
</p>
</td>
<td>
<p>
1
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Basic implicit routine. Requires the Jacobian. Works only with
<a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.UBlas</a>
vectors as state types.
</p>
</td>
</tr>
<tr>
<td>
<p>
Rosenbrock 4
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">rosenbrock4</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
System</a>
</p>
</td>
<td>
<p>
4
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Good for stiff systems. Works only with <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.UBlas</a>
vectors as state types.
</p>
</td>
</tr>
<tr>
<td>
<p>
Controlled Rosenbrock 4
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">rosenbrock4_controller</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
System</a>
</p>
</td>
<td>
<p>
4
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Rosenbrock 4 with error control. Works only with <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.UBlas</a>
vectors as state types.
</p>
</td>
</tr>
<tr>
<td>
<p>
Dense Output Rosenbrock 4
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">rosenbrock4_dense_output</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
Output Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
System</a>
</p>
</td>
<td>
<p>
4
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
Yes
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Controlled Rosenbrock 4 with dense output. Works only with <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.UBlas</a>
vectors as state types.
</p>
</td>
</tr>
<tr>
<td>
<p>
Symplectic Euler
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">symplectic_euler</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/symplectic_system.html" title="Symplectic System">Symplectic
System</a> <a class="link" href="../concepts/simple_symplectic_system.html" title="Simple Symplectic System">Simple
Symplectic System</a>
</p>
</td>
<td>
<p>
1
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Basic symplectic solver for separable Hamiltonian system
</p>
</td>
</tr>
<tr>
<td>
<p>
Symplectic RKN McLachlan
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
</p>
</td>
<td>
<p>
<a class="link" href="../concepts/symplectic_system.html" title="Symplectic System">Symplectic
System</a> <a class="link" href="../concepts/simple_symplectic_system.html" title="Simple Symplectic System">Simple
Symplectic System</a>
</p>
</td>
<td>
<p>
6
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
No
</p>
</td>
<td>
<p>
Symplectic solver for separable Hamiltonian system with order
6
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
Some of steppers in the table above are special: Some need the Jacobian
of the ODE, others are constructed for special ODE-systems like Hamiltonian
systems. We will show typical examples and use-cases in this tutorial and
which kind of steppers should be applied.
</p>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.tutorial.harmonic_oscillator.integration_with_constant_step_size"></a><a class="link" href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.integration_with_constant_step_size" title="Integration with Constant Step Size">Integration
with Constant Step Size</a>
</h4></div></div></div>
<p>
The basic stepper just performs one time-step and doesn't give you any
information about the error that was made (except that you know it is of
order <span class="emphasis"><em>q+1</em></span>). Such steppers are used with constant step
size that should be chosen small enough to have reasonable small errors.
However, you should apply some sort of validity check of your results (like
observing conserved quantities) because you have no other control of the
error. The following example defines a basic stepper based on the classical
Runge-Kutta scheme of 4th order. The declaration of the stepper requires
the state type as template parameter. The integration can now be done by
using the <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">state</span><span class="special">,</span> <span class="identifier">start_time</span><span class="special">,</span> <span class="identifier">end_time</span><span class="special">,</span> <span class="identifier">step_size</span>
<span class="special">)</span></code> function from odeint:
</p>
<p>
</p>
<pre class="programlisting"><span class="identifier">runge_kutta4</span><span class="special">&lt;</span> <span class="identifier">state_type</span> <span class="special">&gt;</span> <span class="identifier">stepper</span><span class="special">;</span>
<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">stepper</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">0.01</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
This call integrates the system defined by <code class="computeroutput"><span class="identifier">harmonic_oscillator</span></code>
using the RK4 method from <span class="emphasis"><em>t=0</em></span> to <span class="emphasis"><em>10</em></span>
with a step-size <span class="emphasis"><em>dt=0.01</em></span> and the initial condition
given in <code class="computeroutput"><span class="identifier">x</span></code>. The result,
<span class="emphasis"><em>x(t=10)</em></span> is stored in <code class="computeroutput"><span class="identifier">x</span></code>
(in-place). Each stepper defines a <code class="computeroutput"><span class="identifier">do_step</span></code>
method which can used directly. So, you write down the above example as
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span>
<span class="keyword">for</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">t</span><span class="special">&lt;</span><span class="number">10.0</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">stepper</span><span class="special">.</span><span class="identifier">do_step</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="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
</pre>
<p>
</p>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.tutorial.harmonic_oscillator.integration_with_adaptive_step_size"></a><a class="link" href="harmonic_oscillator.html#boost_numeric_odeint.tutorial.harmonic_oscillator.integration_with_adaptive_step_size" title="Integration with Adaptive Step Size">Integration
with Adaptive Step Size</a>
</h4></div></div></div>
<p>
To improve the numerical results and additionally minimize the computational
effort, the application of a step size control is advisable. Step size
control is realized via stepper algorithms that additionally provide an
error estimation of the applied step. odeint provides a number of such
<span class="bold"><strong>ErrorSteppers</strong></span> and we will show their usage
on the example of <code class="computeroutput"><span class="identifier">explicit_error_rk54_ck</span></code>
-- a 5th order Runge-Kutta method with 4th order error estimation and coefficients
introduced by Cash and Karp.
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_cash_karp54</span><span class="special">&lt;</span> <span class="identifier">state_type</span> <span class="special">&gt;</span> <span class="identifier">error_stepper_type</span><span class="special">;</span>
</pre>
<p>
</p>
<p>
Given the error stepper, one still needs an instance that checks the error
and adjusts the step size accordingly. In odeint, this is done by <span class="bold"><strong>ControlledSteppers</strong></span>. For the <code class="computeroutput"><span class="identifier">runge_kutta_cash_karp54</span></code>
stepper a <code class="computeroutput"><span class="identifier">controlled_runge_kutta</span></code>
stepper exists which can be used via
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">controlled_runge_kutta</span><span class="special">&lt;</span> <span class="identifier">error_stepper_type</span> <span class="special">&gt;</span> <span class="identifier">controlled_stepper_type</span><span class="special">;</span>
<span class="identifier">controlled_stepper_type</span> <span class="identifier">controlled_stepper</span><span class="special">;</span>
<span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">controlled_stepper</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">0.01</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
As above, this integrates the system defined by <code class="computeroutput"><span class="identifier">harmonic_oscillator</span></code>,
but now using an adaptive step size method based on the Runge-Kutta Cash-Karp
54 scheme from <span class="emphasis"><em>t=0</em></span> to <span class="emphasis"><em>10</em></span> with
an initial step size of <span class="emphasis"><em>dt=0.01</em></span> (will be adjusted)
and the initial condition given in x. The result, <span class="emphasis"><em>x(t=10)</em></span>,
will also be stored in x (in-place).
</p>
<p>
In the above example an error stepper is nested in a controlled stepper.
This is a nice technique; however one drawback is that one always needs
to define both steppers. Of course, one could also write the instantiation
of the controlled stepper into the call of the integrate function but a
complete knowledge of the underlying stepper types is still necessary.
Another point is, that the error tolerances for the step size control are
not easily included into the controlled stepper. Both issues can be solved
by using <code class="computeroutput"><span class="identifier">make_controlled</span></code>:
</p>
<p>
</p>
<pre class="programlisting"><span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">make_controlled</span><span class="special">&lt;</span> <span class="identifier">error_stepper_type</span> <span class="special">&gt;(</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">)</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">0.01</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
<code class="computeroutput"><span class="identifier">make_controlled</span></code> can be
used with many of the steppers of odeint. The first parameter is the absolute
error tolerance <span class="emphasis"><em>eps_abs</em></span> and the second is the relative
error tolerance <span class="emphasis"><em>eps_rel</em></span> which is used during the integration.
The template parameter determines from which error stepper a controlled
stepper should be instantiated. An alternative syntax of <code class="computeroutput"><span class="identifier">make_controlled</span></code> is
</p>
<p>
</p>
<pre class="programlisting"><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-10</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">error_stepper_type</span><span class="special">()</span> <span class="special">)</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">0.01</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
For the Runge Kutta controller the error made during one step is compared
with <span class="emphasis"><em>eps_abs + eps_rel * ( a<sub>x</sub> * |x| + a<sub>dxdt</sub> * dt * |dxdt| )</em></span>.
If the error is smaller than this value the current step is accept otherwise
it is rejected and the step size is decreased. Note, that the step size
is also increased if the error gets too small compared to the rhs of the
above relation. The full instantiation of the <code class="computeroutput"><span class="identifier">controlled_runge_kutta</span></code>
with all parameters is therefore
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">abs_err</span> <span class="special">=</span> <span class="number">1.0e-10</span> <span class="special">,</span> <span class="identifier">rel_err</span> <span class="special">=</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">a_x</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">,</span> <span class="identifier">a_dxdt</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
<span class="identifier">controlled_stepper_type</span> <span class="identifier">controlled_stepper</span><span class="special">(</span>
<span class="identifier">default_error_checker</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;(</span> <span class="identifier">abs_err</span> <span class="special">,</span> <span class="identifier">rel_err</span> <span class="special">,</span> <span class="identifier">a_x</span> <span class="special">,</span> <span class="identifier">a_dxdt</span> <span class="special">)</span> <span class="special">);</span>
<span class="identifier">integrate_adaptive</span><span class="special">(</span> <span class="identifier">controlled_stepper</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">0.01</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
When using <code class="computeroutput"><span class="identifier">make_controlled</span></code>
the parameter <span class="emphasis"><em>a<sub>x</sub></em></span> and <span class="emphasis"><em>a<sub>dxdt</sub></em></span> are
used with their standard values of 1.
</p>
<p>
In the tables below, one can find all steppers which are working with
<code class="computeroutput"><span class="identifier">make_controlled</span></code> and <code class="computeroutput"><span class="identifier">make_dense_output</span></code> which is the analog
for the dense output steppers.
</p>
<div class="table">
<a name="boost_numeric_odeint.tutorial.harmonic_oscillator.integration_with_adaptive_step_size.generation_functions_make_controlled__abs_error___rel_error___stepper__"></a><p class="title"><b>Table&#160;1.3.&#160;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">&lt;</span> <span class="identifier">runge_kutta_cash_karp54</span>
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special">&lt;...&gt;</span> <span class="special">&gt;</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">&lt;</span> <span class="identifier">runge_kutta_fehlberg78</span>
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special">&lt;...&gt;</span> <span class="special">&gt;</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">&lt;</span> <span class="identifier">runge_kutta_dopri5</span>
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special">&lt;...&gt;</span> <span class="special">&gt;</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">&lt;</span> <span class="identifier">rosenbrock4</span>
<span class="special">&gt;</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.tutorial.harmonic_oscillator.integration_with_adaptive_step_size.generation_functions_make_dense_output__abs_error___rel_error___stepper__"></a><p class="title"><b>Table&#160;1.4.&#160;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">&lt;</span> <span class="identifier">controlled_runge_kutta</span><span class="special">&lt;</span> <span class="identifier">runge_kutta_dopri5</span>
<span class="special">,</span> <span class="identifier">default_error_checker</span><span class="special">&lt;...&gt;</span> <span class="special">&gt;</span>
<span class="special">&gt;</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">&lt;</span> <span class="identifier">rosenbrock4_controller</span><span class="special">&lt;</span> <span class="identifier">rosenbrock4</span>
<span class="special">&gt;</span> <span class="special">&gt;</span></code>
</p>
</td>
<td>
<p>
-
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
When using <code class="computeroutput"><span class="identifier">make_controlled</span></code>
or <code class="computeroutput"><span class="identifier">make_dense_output</span></code> one
should be aware which exact type is used and how the step size control
works.
</p>
</div>
<p>
The full cpp file for this example can be found here: <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/harmonic_oscillator.cpp" target="_top">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 &#169; 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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