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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_numeric_odeint.tutorial.using_arbitrary_precision_floating_point_types"></a><a class="link" href="using_arbitrary_precision_floating_point_types.html" title="Using arbitrary precision floating point types">Using |
| arbitrary precision floating point types</a> |
| </h3></div></div></div> |
| <p> |
| Sometimes one needs results with higher precision than provided by the standard |
| floating point types. As odeint allows to configure the fundamental numerical |
| type, it is well suited to be run with arbitrary precision types. Therefore, |
| one only needs a library that provides a type representing values with arbitrary |
| precision and the fundamental operations for those values. <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/" target="_top">Boost.Multiprecision</a> |
| is a boost library that does exactly this. Making use of <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/" target="_top">Boost.Multiprecision</a> |
| to solve odes with odeint is very simple, as the following example shows. |
| </p> |
| <p> |
| Here we use <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code> |
| as the fundamental value type, which ensures exact computations up to 50 |
| decimal digits. |
| </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> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| |
| <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> |
| <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> |
| |
| <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">value_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">value_type</span> <span class="special">,</span> <span class="number">3</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| As exemplary ODE again the lorenz system is chosen, but here we have to make |
| sure all constants are initialized as high precision values. |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</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="identifier">state_type</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="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">sigma</span><span class="special">(</span> <span class="number">10</span> <span class="special">);</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">R</span><span class="special">(</span> <span class="number">28</span> <span class="special">);</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">b</span><span class="special">(</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">8</span> <span class="special">)</span> <span class="special">/</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">3</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> |
| </pre> |
| <p> |
| </p> |
| <p> |
| The actual integration then is straight forward: |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{{</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">}};</span> |
| |
| <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span> <span class="number">50</span> <span class="special">);</span> |
| <span class="identifier">integrate_const</span><span class="special">(</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="special">,</span> |
| <span class="identifier">lorenz</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</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="number">1.0</span> <span class="special">)</span> <span class="special">/</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">)</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> |
| The full example can be found at <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/multiprecision/lorenz_mp.cpp" target="_top">lorenz_mp.cpp</a>. |
| Another example that compares the accuracy of the high precision type with |
| standard double can be found at <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/multiprecision/cmp_precision.cpp" target="_top">cmp_precision.cpp</a>. |
| </p> |
| <p> |
| Furthermore, odeint can also be run with other multiprecision libraries, |
| e.g. <a href="http://gmplib.org/" target="_top">gmp</a>. An example for this is |
| given in <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/gmpxx/lorenz_gmpxx.cpp" target="_top">lorenz_gmpxx.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 © 2009-2012 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> |
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