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    25. 

  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h3 class="title">
    27. 

  27. <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
    28. 

  28.       arbitrary precision floating point types</a>
    29. 

  29. </h3></div></div></div>
    30. 

  30. <p>
    31. 

  31.         Sometimes one needs results with higher precision than provided by the standard
    32. 

  32.         floating point types. As odeint allows to configure the fundamental numerical
    33. 

  33.         type, it is well suited to be run with arbitrary precision types. Therefore,
    34. 

  34.         one only needs a library that provides a type representing values with arbitrary
    35. 

  35.         precision and the fundamental operations for those values. <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/" target="_top">Boost.Multiprecision</a>
    36. 

  36.         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>
    37. 

  37.         to solve odes with odeint is very simple, as the following example shows.
    38. 

  38.       </p>
    39. 

  39. <p>
    40. 

  40.         Here we use <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>
    41. 

  41.         as the fundamental value type, which ensures exact computations up to 50
    42. 

  42.         decimal digits.
    43. 

  43.       </p>
    44. 

  44. <p>
    45. 

  45. </p>
    46. 

  46. <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
    47. 

  47. <span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
    48. 

  48. 

  49. <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
    50. 

  50. <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>
    51. 

  51. 

  52. <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>
    53. 

  53. 

  54. <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="number">3</span> <span class="special">&gt;</span> <span class="identifier">state_type</span><span class="special">;</span>
    55. 

  55. </pre>
    56. 

  56. <p>
    57. 

  57.       </p>
    58. 

  58. <p>
    59. 

  59.         As exemplary ODE again the lorenz system is chosen, but here we have to make
    60. 

  60.         sure all constants are initialized as high precision values.
    61. 

  61.       </p>
    62. 

  62. <p>
    63. 

  63. </p>
    64. 

  64. <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span>
    65. 

  65. <span class="special">{</span>
    66. 

  66.     <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">&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="identifier">value_type</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
    67. 

  67.     <span class="special">{</span>
    68. 

  68.         <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>
    69. 

  69.         <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>
    70. 

  70.         <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>
    71. 

  71. 

  72.         <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>
    73. 

  73.         <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>
    74. 

  74.         <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>
    75. 

  75.     <span class="special">}</span>
    76. 

  76. <span class="special">};</span>
    77. 

  77. </pre>
    78. 

  78. <p>
    79. 

  79.       </p>
    80. 

  80. <p>
    81. 

  81.         The actual integration then is straight forward:
    82. 

  82.       </p>
    83. 

  83. <p>
    84. 

  84. </p>
    85. 

  85. <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>
    86. 

  86. 

  87. <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>
    88. 

  88. <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">runge_kutta4</span><span class="special">&lt;</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">&gt;()</span> <span class="special">,</span>
    89. 

  89.         <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>
    90. 

  90.         <span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">cout</span> <span class="special">)</span> <span class="special">);</span>
    91. 

  91. </pre>
    92. 

  92. <p>
    93. 

  93.       </p>
    94. 

  94. <p>
    95. 

  95.         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>.
    96. 

  96.         Another example that compares the accuracy of the high precision type with
    97. 

  97.         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>.
    98. 

  98.       </p>
    99. 

  99. <p>
    100. 

  100.         Furthermore, odeint can also be run with other multiprecision libraries,
    101. 

  101.         e.g. <a href="http://gmplib.org/" target="_top">gmp</a>. An example for this is
    102. 

  102.         given in <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/gmpxx/lorenz_gmpxx.cpp" target="_top">lorenz_gmpxx.cpp</a>.
    103. 

  103.       </p>
    104. 

  104. </div>
    105. 

  105. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    106. 

  106. <td align="left"></td>
    107. 

  107. <td align="right"><div class="copyright-footer">Copyright &#169; 2009-2015 Karsten Ahnert and Mario Mulansky<p>
    108. 

  108.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    109. 

  109.         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>)
    110. 

  110.       </p>
    111. 

  111. </div></td>
    112. 

  112. </tr></table>
    113. 

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