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  4. <title>Using matrices as state types</title>
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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_matrices_as_state_types"></a><a class="link" href="using_matrices_as_state_types.html" title="Using matrices as state types">Using
    28. 

  28.       matrices as state types</a>
    29. 

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

  30. <p>
    31. 

  31.         odeint works well with a variety of different state types. It is not restricted
    32. 

  32.         to pure vector-wise types, like <code class="computeroutput"><span class="identifier">vector</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</span></code>, <code class="computeroutput"><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>,
    33. 

  33.         <code class="computeroutput"><span class="identifier">fusion</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span></code>,
    34. 

  34.         etc. but also works with types having a different topology then simple vectors.
    35. 

  35.         Here, we show how odeint can be used with matrices as states type, in the
    36. 

  36.         next section we will show how can be used to solve ODEs defined on complex
    37. 

  37.         networks.
    38. 

  38.       </p>
    39. 

  39. <p>
    40. 

  40.         By default, odeint can be used with <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special">&lt;</span> <span class="identifier">T</span> <span class="special">&gt;</span></code> as state type for matrices. A simple
    41. 

  41.         example is a two-dimensional lattice of coupled phase oscillators. Other
    42. 

  42.         matrix types like <code class="computeroutput"><span class="identifier">mtl</span><span class="special">::</span><span class="identifier">dense_matrix</span></code> or blitz arrays and matrices
    43. 

  43.         can used as well but need some kind of activation in order to work with odeint.
    44. 

  44.         This activation is described in following sections,
    45. 

  45.       </p>
    46. 

  46. <p>
    47. 

  47.         The definition of the system is
    48. 

  48.       </p>
    49. 

  49. <p>
    50. 

  50. </p>
    51. 

  51. <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">matrix</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>
    52. 

  52. 

  53. <span class="keyword">struct</span> <span class="identifier">two_dimensional_phase_lattice</span>
    54. 

  54. <span class="special">{</span>
    55. 

  55.     <span class="identifier">two_dimensional_phase_lattice</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">gamma</span> <span class="special">=</span> <span class="number">0.5</span> <span class="special">)</span>
    56. 

  56.     <span class="special">:</span> <span class="identifier">m_gamma</span><span class="special">(</span> <span class="identifier">gamma</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
    57. 

  57. 

  58.     <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="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span> <span class="keyword">const</span>
    59. 

  59.     <span class="special">{</span>
    60. 

  60.         <span class="identifier">size_t</span> <span class="identifier">size1</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size1</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">size2</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size2</span><span class="special">();</span>
    61. 

  61. 

  62.         <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">1</span> <span class="special">;</span> <span class="identifier">i</span><span class="special">&lt;</span><span class="identifier">size1</span><span class="special">-</span><span class="number">1</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
    63. 

  63.         <span class="special">{</span>
    64. 

  64.             <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">1</span> <span class="special">;</span> <span class="identifier">j</span><span class="special">&lt;</span><span class="identifier">size2</span><span class="special">-</span><span class="number">1</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">j</span> <span class="special">)</span>
    65. 

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

  66.                 <span class="identifier">dxdt</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">=</span>
    67. 

  67.                         <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</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="identifier">j</span> <span class="special">)</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">)</span> <span class="special">+</span>
    68. 

  68.                         <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</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="identifier">j</span> <span class="special">)</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">)</span> <span class="special">+</span>
    69. 

  69.                         <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</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="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">)</span> <span class="special">+</span>
    70. 

  70.                         <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</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="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">);</span>
    71. 

  71.             <span class="special">}</span>
    72. 

  72.         <span class="special">}</span>
    73. 

  73. 

  74.         <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">&lt;</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">size1</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">dxdt</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="identifier">dxdt</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">size2</span><span class="special">()</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>
    75. 

  75.         <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">&lt;</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">size2</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">j</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="identifier">j</span> <span class="special">)</span> <span class="special">=</span> <span class="identifier">dxdt</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size1</span><span class="special">()</span> <span class="special">-</span><span class="number">1</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
    76. 

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

  77. 

  78.     <span class="keyword">double</span> <span class="identifier">coupling_func</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">)</span> <span class="keyword">const</span>
    79. 

  79.     <span class="special">{</span>
    80. 

  80.         <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="identifier">m_gamma</span> <span class="special">*</span> <span class="special">(</span> <span class="number">1.0</span> <span class="special">-</span> <span class="identifier">cos</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">);</span>
    81. 

  81.     <span class="special">}</span>
    82. 

  82. 

  83.     <span class="keyword">double</span> <span class="identifier">m_gamma</span><span class="special">;</span>
    84. 

  84. <span class="special">};</span>
    85. 

  85. </pre>
    86. 

  86. <p>
    87. 

  87.       </p>
    88. 

  88. <p>
    89. 

  89.         In principle this is all. Please note, that the above code is far from being
    90. 

  90.         optimal. Better performance can be achieved if every interaction is only
    91. 

  91.         calculated once and iterators for columns and rows are used. Below are some
    92. 

  92.         visualizations of the evolution of this lattice equation.
    93. 

  93.       </p>
    94. 

  94. <p>
    95. 

  95.         <span class="inlinemediaobject"><img src="../../phase_lattice_2d_0000.jpg" alt="phase_lattice_2d_0000"></span> <span class="inlinemediaobject"><img src="../../phase_lattice_2d_0100.jpg" alt="phase_lattice_2d_0100"></span> <span class="inlinemediaobject"><img src="../../phase_lattice_2d_1000.jpg" alt="phase_lattice_2d_1000"></span>
    96. 

  96.       </p>
    97. 

  97. <p>
    98. 

  98.         The full cpp for this example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/two_dimensional_phase_lattice.cpp" target="_top">two_dimensional_phase_lattice.cpp</a>.
    99. 

  99.       </p>
    100. 

  100. </div>
    101. 

  101. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    102. 

  102. <td align="left"></td>
    103. 

  103. <td align="right"><div class="copyright-footer">Copyright &#169; 2009-2015 Karsten Ahnert and Mario Mulansky<p>
    104. 

  104.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    105. 

  105.         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>)
    106. 

  106.       </p>
    107. 

  107. </div></td>
    108. 

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    109. 

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