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

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

  27. <a name="math_toolkit.cardinal_quintic_b"></a><a class="link" href="cardinal_quintic_b.html" title="Cardinal Quintic B-spline interpolation">Cardinal Quintic B-spline
    28. 

  28.     interpolation</a>
    29. 

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

  30. <h4>
    31. 

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

  32.       <span class="phrase"><a name="math_toolkit.cardinal_quintic_b.synopsis"></a></span><a class="link" href="cardinal_quintic_b.html#math_toolkit.cardinal_quintic_b.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">cardinal_quintic_b_spline</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    35. 

  35. </pre>
    36. 

  36. <pre class="programlisting"><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> <span class="keyword">namespace</span> <span class="identifier">interpolators</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">Real</span><span class="special">&gt;</span>
    39. 

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

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

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

  42.     <span class="comment">// If you don't know the value of the derivative at the endpoints, leave them as NaNs and the routine will estimate them.</span>
    43. 

  43.     <span class="comment">// y[0] = y(a), y[n - 1] = y(b), step_size = (b - a)/(n -1).</span>
    44. 

  44.     <span class="identifier">cardinal_quintic_b_spline</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">y</span><span class="special">,</span>
    45. 

  45.                               <span class="identifier">size_t</span> <span class="identifier">n</span><span class="special">,</span>
    46. 

  46.                               <span class="identifier">Real</span> <span class="identifier">t0</span> <span class="comment">/* initial time, left endpoint */</span><span class="special">,</span>
    47. 

  47.                               <span class="identifier">Real</span> <span class="identifier">h</span>  <span class="comment">/*spacing, stepsize*/</span><span class="special">,</span>
    48. 

  48.                               <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">left_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">()},</span>
    49. 

  49.                               <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">right_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">()})</span>
    50. 

  50. 

  51.     <span class="identifier">cardinal_quintic_b_spline</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">y</span><span class="special">,</span>
    52. 

  52.                               <span class="identifier">Real</span> <span class="identifier">t0</span> <span class="comment">/* initial time, left endpoint */</span><span class="special">,</span>
    53. 

  53.                               <span class="identifier">Real</span> <span class="identifier">h</span>  <span class="comment">/*spacing, stepsize*/</span><span class="special">,</span>
    54. 

  54.                               <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">left_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">()},</span>
    55. 

  55.                               <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">right_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">()})</span>
    56. 

  56. 

  57.     <span class="identifier">Real</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>
    58. 

  58. 

  59.     <span class="identifier">Real</span> <span class="identifier">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>
    60. 

  60. 

  61.     <span class="identifier">Real</span> <span class="identifier">double_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>
    62. 

  62. 

  63. <span class="special">};</span>
    64. 

  64. <span class="special">}}}</span>
    65. 

  65. </pre>
    66. 

  66. <h4>
    67. 

  67. <a name="math_toolkit.cardinal_quintic_b.h1"></a>
    68. 

  68.       <span class="phrase"><a name="math_toolkit.cardinal_quintic_b.cardinal_quintic_b_spline_interp"></a></span><a class="link" href="cardinal_quintic_b.html#math_toolkit.cardinal_quintic_b.cardinal_quintic_b_spline_interp">Cardinal
    69. 

  69.       Quintic B-Spline Interpolation</a>
    70. 

  70.     </h4>
    71. 

  71. <p>
    72. 

  72.       The cardinal quintic B-spline interpolator is very nearly the same as the cubic
    73. 

  73.       B-spline interpolator, with the modification that the basis functions are constructed
    74. 

  74.       by convolving a box function with itself five times, rather than three times
    75. 

  75.       as is done with the cubic B-spline.
    76. 

  76.     </p>
    77. 

  77. <p>
    78. 

  78.       The basis functions of the quintic B-spline interpolator are more smooth than
    79. 

  79.       the cubic <span class="emphasis"><em>B</em></span>-spline interpolator, and hence this is very
    80. 

  80.       useful for computing second derivatives. For example, the second derivative
    81. 

  81.       of the cubic spline interpolator is a piecewise linear function, whereas the
    82. 

  82.       second derivative of the quintic <span class="emphasis"><em>B</em></span>-spline is a cubic spline.
    83. 

  83.       The graph of the second derivative of the quintic <span class="emphasis"><em>B</em></span>-spline
    84. 

  84.       is therefore more visually appealing, though whether it is in fact more accurate
    85. 

  85.       depends on the smoothness of your data.
    86. 

  86.     </p>
    87. 

  87. <p>
    88. 

  88.       And example usage is as follows:
    89. 

  89.     </p>
    90. 

  90. <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">cardinal_quintic_b_spline</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    91. 

  91. <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">interpolators</span><span class="special">::</span><span class="identifier">cardinal_quintic_b_spline</span><span class="special">;</span>
    92. 

  92. <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">v</span><span class="special">(</span><span class="number">512</span><span class="special">);</span>
    93. 

  93. <span class="comment">// fill v with data . . .</span>
    94. 

  94. <span class="keyword">double</span> <span class="identifier">t0</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="comment">// initial time</span>
    95. 

  95. <span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="number">0.125</span><span class="special">;</span> <span class="comment">// spacing</span>
    96. 

  96. <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</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> <span class="identifier">left_endpoint_derivatives</span><span class="special">{</span><span class="identifier">first_derivative_at_t0</span><span class="special">,</span> <span class="identifier">second_derivative_at_t0</span><span class="special">};</span>
    97. 

  97. <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</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> <span class="identifier">right_endpoint_derivatives</span><span class="special">{</span><span class="identifier">first_derivative_at_tf</span><span class="special">,</span> <span class="identifier">second_derivative_at_tf</span><span class="special">};</span>
    98. 

  98. <span class="keyword">auto</span> <span class="identifier">qs</span> <span class="special">=</span> <span class="identifier">cardinal_quintic_b_spline</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">t0</span><span class="special">,</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">left_endpoint_derivatives</span><span class="special">,</span> <span class="identifier">right_endpoint_derivatives</span><span class="special">);</span>
    99. 

  99. 

  100. <span class="comment">// Evaluate the interpolant at a point:</span>
    101. 

  101. <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">qs</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
    102. 

  102. <span class="comment">// Evaluate the derivative of the interpolant:</span>
    103. 

  103. <span class="keyword">double</span> <span class="identifier">yp</span> <span class="special">=</span> <span class="identifier">qs</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
    104. 

  104. <span class="comment">// Evaluate the second derivative of the interpolant:</span>
    105. 

  105. <span class="keyword">double</span> <span class="identifier">ypp</span> <span class="special">=</span> <span class="identifier">qs</span><span class="special">.</span><span class="identifier">double_prime</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
    106. 

  106. </pre>
    107. 

  107. <p>
    108. 

  108.       This routine will estimate the endpoint derivatives if they are not provided.
    109. 

  109.       <span class="emphasis"><em>Try to avoid this if possible.</em></span> The endpoint derivatives
    110. 

  110.       must be evaluated by finite differences and this is not robust again perturbations
    111. 

  111.       in the data. So if you have some way of knowing the endpoint derivatives, make
    112. 

  112.       sure to provide them.
    113. 

  113.     </p>
    114. 

  114. <h4>
    115. 

  115. <a name="math_toolkit.cardinal_quintic_b.h2"></a>
    116. 

  116.       <span class="phrase"><a name="math_toolkit.cardinal_quintic_b.references"></a></span><a class="link" href="cardinal_quintic_b.html#math_toolkit.cardinal_quintic_b.references">References</a>
    117. 

  117.     </h4>
    118. 

  118. <p>
    119. 

  119.       Cox, Maurice G. <span class="emphasis"><em>Numerical methods for the interpolation and approximation
    120. 

  120.       of data by spline functions.</em></span> Diss. City, University of London, 1975.
    121. 

  121.     </p>
    122. 

  122. </div>
    123. 

  123. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    124. 

  124. <td align="left"></td>
    125. 

  125. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    126. 

  126.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    127. 

  127.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    128. 

  128.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    129. 

  129.       Daryle Walker and Xiaogang Zhang<p>
    130. 

  130.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    131. 

  131.         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>)
    132. 

  132.       </p>
    133. 

  133. </div></td>
    134. 

  134. </tr></table>
    135. 

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    136. 

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