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    25. 

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

  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
    27. 

  27. <a name="math_toolkit.vector_barycentric"></a><a class="link" href="vector_barycentric.html" title="Vector-valued Barycentric Rational Interpolation">Vector-valued Barycentric
    28. 

  28.     Rational Interpolation</a>
    29. 

  29. </h2></div></div></div>
    30. 

  30. <h4>
    31. 

  31. <a name="math_toolkit.vector_barycentric.h0"></a>
    32. 

  32.       <span class="phrase"><a name="math_toolkit.vector_barycentric.synopsis"></a></span><a class="link" href="vector_barycentric.html#math_toolkit.vector_barycentric.synopsis">Synopsis</a>
    33. 

  33.     </h4>
    34. 

  34. <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">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">vector_barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    35. 

  35. 

  36. <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
    37. 

  37. 

  38. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">TimeContainer</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">SpaceContainer</span><span class="special">&gt;</span>
    39. 

  39. <span class="keyword">class</span> <span class="identifier">vector_barycentric_rational</span>
    40. 

  40. <span class="special">{</span>
    41. 

  41. <span class="keyword">public</span><span class="special">:</span>
    42. 

  42.     <span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">TimeContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
    43. 

  43.     <span class="keyword">using</span> <span class="identifier">Point</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">SpaceContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
    44. 

  44.     <span class="identifier">vector_barycentric_rational</span><span class="special">(</span><span class="identifier">TimeContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">times</span><span class="special">,</span> <span class="identifier">SpaceContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">points</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">approximation_order</span> <span class="special">=</span> <span class="number">3</span><span class="special">);</span>
    45. 

  45. 

  46.     <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Point</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
    47. 

  47. 

  48.     <span class="identifier">Point</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
    49. 

  49. 

  50.     <span class="keyword">void</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Point</span><span class="special">&amp;</span> <span class="identifier">dxdt</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
    51. 

  51. 

  52.     <span class="identifier">Point</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">);</span>
    53. 

  53. 

  54.     <span class="keyword">void</span> <span class="identifier">eval_with_prime</span><span class="special">(</span><span class="identifier">Point</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Point</span><span class="special">&amp;</span> <span class="identifier">dxdt</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
    55. 

  55. 

  56.     <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Point</span><span class="special">,</span> <span class="identifier">Point</span><span class="special">&gt;</span> <span class="identifier">eval_with_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
    57. 

  57. <span class="special">};</span>
    58. 

  58. 

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

  60. </pre>
    61. 

  61. <h4>
    62. 

  62. <a name="math_toolkit.vector_barycentric.h1"></a>
    63. 

  63.       <span class="phrase"><a name="math_toolkit.vector_barycentric.description"></a></span><a class="link" href="vector_barycentric.html#math_toolkit.vector_barycentric.description">Description</a>
    64. 

  64.     </h4>
    65. 

  65. <p>
    66. 

  66.       The <span class="emphasis"><em>n</em></span> dimensional vector-valued barycentric rational interpolator
    67. 

  67.       is exactly the same as <span class="emphasis"><em>n</em></span> scalar-valued barycentric rational
    68. 

  68.       interpolators. This is provided primarily for convenience and a slight improvement
    69. 

  69.       in efficiency over using <span class="emphasis"><em>n</em></span> different rational interpolators
    70. 

  70.       and combining their results.
    71. 

  71.     </p>
    72. 

  72. <p>
    73. 

  73.       Use of the class requires a <code class="computeroutput"><span class="identifier">Point</span></code>-type
    74. 

  74.       which has size known at compile-time. These requirements are satisfied by (for
    75. 

  75.       example) <code class="computeroutput"><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span></code>s and <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">N</span><span class="special">&gt;</span></code> classes. The call to the constructor computes
    76. 

  76.       the weights:
    77. 

  77.     </p>
    78. 

  78. <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">vector_barycentric_rational</span><span class="special">;</span>
    79. 

  79. <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">t</span><span class="special">(</span><span class="number">100</span><span class="special">);</span>
    80. 

  80. <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span><span class="special">&gt;</span> <span class="identifier">y</span><span class="special">(</span><span class="number">100</span><span class="special">);</span>
    81. 

  81. <span class="comment">// initialize t and y . . .</span>
    82. 

  82. <span class="identifier">vector_barycentric_rational</span><span class="special">&lt;</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">t</span><span class="special">),</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">y</span><span class="special">)&gt;</span> <span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">t</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">y</span><span class="special">));</span>
    83. 

  83. </pre>
    84. 

  84. <p>
    85. 

  85.       To evaluate the interpolant, use
    86. 

  86.     </p>
    87. 

  87. <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">2.3</span><span class="special">;</span>
    88. 

  88. <span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span>
    89. 

  89. </pre>
    90. 

  90. <p>
    91. 

  91.       If you want to populate a vector passed into the interpolant, rather than get
    92. 

  92.       it returned, that syntax is supported:
    93. 

  93.     </p>
    94. 

  94. <pre class="programlisting"><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span> <span class="identifier">y</span><span class="special">;</span>
    95. 

  95. <span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">y</span><span class="special">,</span> <span class="identifier">t</span><span class="special">);</span>
    96. 

  96. </pre>
    97. 

  97. <p>
    98. 

  98.       We tested this with <code class="computeroutput"><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector</span></code>s and found no performance benefit,
    99. 

  99.       but other <code class="computeroutput"><span class="identifier">Point</span></code>-types might
    100. 

  100.       not be the same.
    101. 

  101.     </p>
    102. 

  102. <p>
    103. 

  103.       To evaluate the derivative of the interpolant use
    104. 

  104.     </p>
    105. 

  105. <pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">y</span><span class="special">,</span> <span class="identifier">y_prime</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">.</span><span class="identifier">eval_with_prime</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
    106. 

  106. </pre>
    107. 

  107. <p>
    108. 

  108.       Computation of the derivative requires evaluation, so if you can try to use
    109. 

  109.       both values at once.
    110. 

  110.     </p>
    111. 

  111. </div>
    112. 

  112. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    113. 

  113. <td align="left"></td>
    114. 

  114. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    115. 

  115.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    116. 

  116.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    117. 

  117.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    118. 

  118.       Daryle Walker and Xiaogang Zhang<p>
    119. 

  119.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    120. 

  120.         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>)
    121. 

  121.       </p>
    122. 

  122. </div></td>
    123. 

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    124. 

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