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  27 <a name="math_toolkit.whittaker_shannon"></a><a class="link" href="whittaker_shannon.html" title="Whittaker-Shannon interpolation">Whittaker-Shannon interpolation</a>
  28 </h2></div></div></div>
  29 <h4>
  30 <a name="math_toolkit.whittaker_shannon.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.whittaker_shannon.synopsis"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.synopsis">Synopsis</a>
  32     </h4>
  33 <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">whittaker_shannon</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <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>
  36
  37   <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  38   <span class="keyword">class</span> <span class="identifier">whittaker_shannon</span>
  39   <span class="special">{</span>
  40   <span class="keyword">public</span><span class="special">:</span>
  41
  42       <span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
  43
  44       <span class="identifier">whittaker_shannon</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">left_endpoint</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">step_size</span><span class="special">);</span>
  45
  46       <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  47
  48       <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  49   <span class="special">};</span>
  50
  51 <span class="special">}}}</span> <span class="comment">// namespaces</span>
  52 </pre>
  53 <h4>
  54 <a name="math_toolkit.whittaker_shannon.h1"></a>
  55       <span class="phrase"><a name="math_toolkit.whittaker_shannon.whittaker_shannon_interpolation"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.whittaker_shannon_interpolation">Whittaker-Shannon
  56       Interpolation</a>
  57     </h4>
  58 <p>
  59       The Whittaker-Shannon interpolator takes equispaced data and interpolates between
  60       them via a sum of sinc functions. This interpolation is stable and infinitely
  61       smooth, but has linear complexity in the data, making it slow relative to compactly-supported
  62       b-splines. In addition, we cannot pass an infinite amount of data into the
  63       class, and must truncate the (perhaps) infinite sinc series to a finite number
  64       of terms. Since the sinc function has slow <span class="emphasis"><em>1/x</em></span> decay,
  65       the truncation of the series can incur large error. Hence this interpolator
  66       works best when operating on samples of compactly-supported functions. Here
  67       is an example of interpolating a smooth "bump function":
  68     </p>
  69 <pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">bump</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">if</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&gt;=</span> <span class="number">1</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="number">0.0</span><span class="special">;</span> <span class="special">}</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special">(-</span><span class="number">1.0</span><span class="special">/(</span><span class="number">1.0</span><span class="special">-</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">));</span> <span class="special">};</span>
  70
  71 <span class="keyword">double</span> <span class="identifier">t0</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">;</span>
  72 <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">2049</span><span class="special">;</span>
  73 <span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="number">2.0</span><span class="special">/(</span><span class="identifier">n</span><span class="special">-</span><span class="number">1.0</span><span class="special">);</span>
  74
  75 <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="identifier">n</span><span class="special">);</span>
  76 <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">n</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> <span class="special">{</span>
  77     <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">t0</span> <span class="special">+</span> <span class="identifier">i</span><span class="special">*</span><span class="identifier">h</span><span class="special">;</span>
  78     <span class="identifier">v</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">bump</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span>
  79 <span class="special">}</span>
  80
  81 <span class="keyword">auto</span> <span class="identifier">ws</span> <span class="special">=</span> <span class="identifier">whittaker_shannon</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">v</span><span class="special">),</span> <span class="identifier">t0</span><span class="special">,</span> <span class="identifier">h</span><span class="special">);</span>
  82
  83 <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">ws</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
  84 </pre>
  85 <p>
  86       The derivative of the interpolant can also be evaluated, but the accuracy is
  87       not as high:
  88     </p>
  89 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">yp</span> <span class="special">=</span> <span class="identifier">ws</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
  90 </pre>
  91 <h4>
  92 <a name="math_toolkit.whittaker_shannon.h2"></a>
  93       <span class="phrase"><a name="math_toolkit.whittaker_shannon.complexity_and_performance"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.complexity_and_performance">Complexity
  94       and Performance</a>
  95     </h4>
  96 <p>
  97       The call to the constructor requires &#119926;(1) operations, simply moving data into
  98       the class. Each call to the interpolant is &#119926;(<span class="emphasis"><em>n</em></span>), where
  99       <span class="emphasis"><em>n</em></span> is the number of points to interpolate.
 100     </p>
 101 </div>
 102 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 103 <td align="left"></td>
 104 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 105       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 106       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 107       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 108       Daryle Walker and Xiaogang Zhang<p>
 109         Distributed under the Boost Software License, Version 1.0. (See accompanying
 110         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>)
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