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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  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     interpolation</a>
  29 </h2></div></div></div>
  30 <h4>
  31 <a name="math_toolkit.cardinal_quintic_b.h0"></a>
  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     </h4>
  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 </pre>
  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
  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 <span class="keyword">class</span> <span class="identifier">cardinal_quintic_b_spline</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  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     <span class="comment">// y[0] = y(a), y[n - 1] = y(b), step_size = (b - a)/(n -1).</span>
  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                               <span class="identifier">size_t</span> <span class="identifier">n</span><span class="special">,</span>
  46                               <span class="identifier">Real</span> <span class="identifier">t0</span> <span class="comment">/* initial time, left endpoint */</span><span class="special">,</span>
  47                               <span class="identifier">Real</span> <span class="identifier">h</span>  <span class="comment">/*spacing, stepsize*/</span><span class="special">,</span>
  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                               <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
  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                               <span class="identifier">Real</span> <span class="identifier">t0</span> <span class="comment">/* initial time, left endpoint */</span><span class="special">,</span>
  53                               <span class="identifier">Real</span> <span class="identifier">h</span>  <span class="comment">/*spacing, stepsize*/</span><span class="special">,</span>
  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                               <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
  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
  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
  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
  63 <span class="special">};</span>
  64 <span class="special">}}}</span>
  65 </pre>
  66 <h4>
  67 <a name="math_toolkit.cardinal_quintic_b.h1"></a>
  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       Quintic B-Spline Interpolation</a>
  70     </h4>
  71 <p>
  72       The cardinal quintic B-spline interpolator is very nearly the same as the cubic
  73       B-spline interpolator, with the modification that the basis functions are constructed
  74       by convolving a box function with itself five times, rather than three times
  75       as is done with the cubic B-spline.
  76     </p>
  77 <p>
  78       The basis functions of the quintic B-spline interpolator are more smooth than
  79       the cubic <span class="emphasis"><em>B</em></span>-spline interpolator, and hence this is very
  80       useful for computing second derivatives. For example, the second derivative
  81       of the cubic spline interpolator is a piecewise linear function, whereas the
  82       second derivative of the quintic <span class="emphasis"><em>B</em></span>-spline is a cubic spline.
  83       The graph of the second derivative of the quintic <span class="emphasis"><em>B</em></span>-spline
  84       is therefore more visually appealing, though whether it is in fact more accurate
  85       depends on the smoothness of your data.
  86     </p>
  87 <p>
  88       And example usage is as follows:
  89     </p>
  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 <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 <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 <span class="comment">// fill v with data . . .</span>
  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 <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 <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 <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 <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
 100 <span class="comment">// Evaluate the interpolant at a point:</span>
 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 <span class="comment">// Evaluate the derivative of the interpolant:</span>
 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 <span class="comment">// Evaluate the second derivative of the interpolant:</span>
 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 </pre>
 107 <p>
 108       This routine will estimate the endpoint derivatives if they are not provided.
 109       <span class="emphasis"><em>Try to avoid this if possible.</em></span> The endpoint derivatives
 110       must be evaluated by finite differences and this is not robust again perturbations
 111       in the data. So if you have some way of knowing the endpoint derivatives, make
 112       sure to provide them.
 113     </p>
 114 <h4>
 115 <a name="math_toolkit.cardinal_quintic_b.h2"></a>
 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     </h4>
 118 <p>
 119       Cox, Maurice G. <span class="emphasis"><em>Numerical methods for the interpolation and approximation
 120       of data by spline functions.</em></span> Diss. City, University of London, 1975.
 121     </p>
 122 </div>
 123 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 124 <td align="left"></td>
 125 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 126       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 127       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 128       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 129       Daryle Walker and Xiaogang Zhang<p>
 130         Distributed under the Boost Software License, Version 1.0. (See accompanying
 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       </p>
 133 </div></td>
 134 </tr></table>
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