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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.whittaker_shannon"></a><a class="link" href="whittaker_shannon.html" title="Whittaker-Shannon interpolation">Whittaker-Shannon interpolation</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.whittaker_shannon.h0"></a>
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<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>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span>
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</pre>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">whittaker_shannon</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<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>
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<span class="identifier">whittaker_shannon</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&&</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>
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<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>
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<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>
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<span class="special">};</span>
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<span class="special">}}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h4>
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<a name="math_toolkit.whittaker_shannon.h1"></a>
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<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
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Interpolation</a>
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</h4>
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<p>
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The Whittaker-Shannon interpolator takes equispaced data and interpolates between
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them via a sum of sinc functions. This interpolation is stable and infinitely
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smooth, but has linear complexity in the data, making it slow relative to compactly-supported
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b-splines. In addition, we cannot pass an infinite amount of data into the
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class, and must truncate the (perhaps) infinite sinc series to a finite number
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of terms. Since the sinc function has slow <span class="emphasis"><em>1/x</em></span> decay,
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the truncation of the series can incur large error. Hence this interpolator
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works best when operating on samples of compactly-supported functions. Here
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is an example of interpolating a smooth "bump function":
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</p>
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<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">>=</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>
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<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>
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<span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">2049</span><span class="special">;</span>
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<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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">(</span><span class="identifier">n</span><span class="special">);</span>
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<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"><</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>
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<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>
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<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>
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<span class="special">}</span>
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<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>
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<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>
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</pre>
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<p>
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The derivative of the interpolant can also be evaluated, but the accuracy is
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not as high:
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</p>
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<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>
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</pre>
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<h4>
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<a name="math_toolkit.whittaker_shannon.h2"></a>
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<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
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and Performance</a>
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</h4>
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<p>
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The call to the constructor requires 𝑶(1) operations, simply moving data into
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the class. Each call to the interpolant is 𝑶(<span class="emphasis"><em>n</em></span>), where
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<span class="emphasis"><em>n</em></span> is the number of points to interpolate.
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</p>
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</div>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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