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<div class="section" lang="en">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_sandbox_numeric_odeint.getting_started.overview"></a><a class="link" href="overview.html" title="Overview">Overview</a>
</h3></div></div></div>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
Boost.Odeint is not an official boost library!
</p></td></tr>
</table></div>
<p>
Odeint is a library for solving initial value problems (IVP) of ordinary
differential equations. Mathematically, these problems are formulated as
follows: <span class="emphasis"><em>x'(t) = f(x,t)</em></span>, <span class="emphasis"><em>x(0) = x0</em></span>.
<span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>f</em></span> can be vectors and the
solution is some function <span class="emphasis"><em>x(t)</em></span> fullfilling both equations
above. In the following we will refer to <span class="emphasis"><em>x'(t)</em></span> also
<code class="computeroutput"><span class="identifier">dxdt</span></code> which is also our notation
for the derivative in the source code.
</p>
<p>
Numerical approximations for the solution <span class="emphasis"><em>x(t)</em></span> are calculated
iteratively. The easiest algorithm is the Euler-Scheme, where starting at
<span class="emphasis"><em>x(0)</em></span> one finds <span class="emphasis"><em>x(dt) = x(0) + dt f(x(0),0)</em></span>.
Now one can use <span class="emphasis"><em>x(dt)</em></span> and obtain <span class="emphasis"><em>x(2dt)</em></span>
in a similar way and so on. The Euler method is of order 1, that means the
error at each step is <span class="emphasis"><em>~ dt<sup>2</sup></em></span>. This is, of course, not
very satisfying, which is why the Euler method is merely used for real life
problems and serves just as illustrative example.
</p>
<p>
The main focus of odeint is to provide numerical methods implemented in a
way where the algorithm is completely independent on the data structure used
to represent the state <span class="emphasis"><em>x</em></span>. In doing so, odeint is applicable
for a broad variety of situations and it can be used with many other libraries.
Besides the usual case where the state is defined as a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
or a <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>, we provide native support for the
following libraries:
</p>
<div class="itemizedlist"><ul type="disc">
<li>
<a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.UBlas</a>
</li>
<li>
<a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>, making
odeint naturally running on CUDA devices
</li>
<li>
gsl_vector for compatibility with the many numerical function in the
GSL
</li>
<li>
<a href="http://www.boost.org/doc/libs/release/libs/range/index.html" target="_top">Boost.Range</a>
</li>
<li>
<a href="http://www.boost.org/doc/libs/release/libs/fusion/index.html" target="_top">Boost.Fusion</a>
(the state type can be a fusion vector)
</li>
<li>
<a href="http://www.boost.org/doc/libs/release/libs/units/index.html" target="_top">Boost.Units</a>
</li>
<li>
<a href="http://software.intel.com/en-us/articles/intel-mkl/" target="_top">Intel
Math Kernel Library</a> for maximum performance
</li>
<li>
<a href="http://www.boost.org/doc/libs/1_47_0/libs/graph/doc/table_of_contents.html" target="_top">Boost.Graph</a>
(still experimentally)
</li>
</ul></div>
<p>
In odeint, the following algorithms are implemented:
</p>
<div class="table">
<a name="boost_sandbox_numeric_odeint.getting_started.overview.stepper_algorithms"></a><p class="title"><b>Table 1.1. Stepper Algorithms</b></p>
<div class="table-contents"><table class="table" summary="Stepper Algorithms">
<colgroup>
<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>
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>
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>
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>
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>
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>
Standard method with error control and dense ouput, 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>
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>
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>
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>
Combined multistep method
</p>
</td>
</tr>
<tr>
<td>
<p>
Controlled Error Stepper
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">controlled_error_stepper</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>
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 Stepper
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">dense_output_explicit</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>
Dense ouput 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 ouput 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>
Dense Output Controlled Stepper
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">dense_output_controlled_explicit</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>
Dense ouput for <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
Stepper</a> from above if they provide dense ouput functionality
(like <code class="computeroutput"><span class="identifier">controlled_error_stepper</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span>
<span class="special">></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>
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>
Stepper with step size and order control as well as dense ouput.
Very good if high precision and dense ouput 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>
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>
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>
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 Ouput Rosenbrock 4
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">rosenbrock4_dense_ouput</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>
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>
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>
Symplectic solver for separable Hamiltonian system with order 6
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
Ordinary differential equations occur nearly everywhere in natural sciences.
For example, the whole Newtonian mechanics are described by second order
differential equations. Be sure, you will find them in every discipline.
They also occur if partial differential equations (PDEs) are discretized
in one coordinate. Then, a system of coupled ordinary differential occurs,
sometimes also refered as lattices ODEs.
</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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