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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.cardinal_trigonometric"></a><a class="link" href="cardinal_trigonometric.html" title="Cardinal Trigonometric interpolation">Cardinal Trigonometric
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interpolation</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.cardinal_trigonometric.h0"></a>
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<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.synopsis"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.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">cardinal_trigonometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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">cardinal_trigonometric</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="identifier">cardinal_trigonometric</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">h</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">t</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">t</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">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>
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<span class="identifier">Real</span> <span class="identifier">period</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">integrate</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">squared_l2</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>
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</pre>
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<h4>
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<a name="math_toolkit.cardinal_trigonometric.h1"></a>
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<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.cardinal_trigonometric_interpola"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.cardinal_trigonometric_interpola">Cardinal
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Trigonometric Interpolation</a>
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</h4>
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<p>
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The cardinal trigonometric interpolation problem takes uniformly spaced samples
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<span class="emphasis"><em>y</em></span><sub>j</sub> of a periodic function <span class="emphasis"><em>f</em></span> defined
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via <span class="emphasis"><em>y</em></span><sub><span class="emphasis"><em>j</em></span></sub> = <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>t</em></span><sub>0</sub> +
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<span class="emphasis"><em>j</em></span> <span class="emphasis"><em>h</em></span>) and represents them as a linear
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combination of sines and cosines. If the period of <span class="emphasis"><em>f</em></span> is
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<span class="emphasis"><em>T</em></span>, and the number of data points is <span class="emphasis"><em>n = 2m</em></span>
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or <span class="emphasis"><em>n = 2m+1</em></span>, we hope to have
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</p>
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<p>
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<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>t</em></span>) ≈ <span class="emphasis"><em>a</em></span><sub>0</sub>/2
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+ ∑<sub><span class="emphasis"><em>k</em></span> = 1</sub><sup><span class="emphasis"><em>m</em></span></sup> <span class="emphasis"><em>a</em></span><sub><span class="emphasis"><em>k</em></span></sub> cos(2π
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<span class="emphasis"><em>k</em></span> (<span class="emphasis"><em>t</em></span>-<span class="emphasis"><em>t</em></span><sub>0</sub>) /T)
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+ <span class="emphasis"><em>b</em></span><sub><span class="emphasis"><em>k</em></span></sub> sin(2π <span class="emphasis"><em>k</em></span>
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(<span class="emphasis"><em>t</em></span>-<span class="emphasis"><em>t</em></span><sub>0</sub>)/T)
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</p>
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<p>
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Convergence rates depend on the number of continuous derivatives of <span class="emphasis"><em>f</em></span>;
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see either Atkinson or Kress for details.
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</p>
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<p>
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A simple use of this interpolator is shown below:
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</p>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span>
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<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">cardinal_trigonometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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_trigonometric</span><span class="special">;</span>
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<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="number">17</span><span class="special">,</span> <span class="number">3.2</span><span class="special">);</span>
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<span class="keyword">auto</span> <span class="identifier">ct</span> <span class="special">=</span> <span class="identifier">cardinal_trigonometric</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="comment">/*t0 = */</span> <span class="number">0.0</span><span class="special">,</span> <span class="comment">/* h = */</span> <span class="number">0.125</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ct(1.3) = "</span> <span class="special"><<</span> <span class="identifier">ct</span><span class="special">(</span><span class="number">1.3</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="comment">// Derivative:</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">ct</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">1.2</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="comment">// Second derivative:</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">ct</span><span class="special">.</span><span class="identifier">double_prime</span><span class="special">(</span><span class="number">1.2</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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</pre>
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<p>
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The period is always given by <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()*</span><span class="identifier">h</span></code>. Off-by-one errors are common in programming,
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and hence if you wonder what this interpolator believes the period to be, you
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can query it with the <code class="computeroutput"><span class="special">.</span><span class="identifier">period</span><span class="special">()</span></code> member function.
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</p>
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<p>
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In addition, the transform into the trigonometric basis gives a trivial way
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to compute the integral of the function over a period; this is done via the
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<code class="computeroutput"><span class="special">.</span><span class="identifier">integrate</span><span class="special">()</span></code> member function. Evaluation of the square
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of the L<sup>2</sup> norm is trivial in this basis; it is computed by the <code class="computeroutput"><span class="special">.</span><span class="identifier">squared_l2</span><span class="special">()</span></code> member function.
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</p>
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<p>
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Below is a graph of a <span class="emphasis"><em>C</em></span><sup>∞</sup> bump function approximated
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by trigonometric series. The graphs are visually indistinguishable at 20 samples.
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/fourier_bump.svg"></object></span>
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</p>
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<h4>
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<a name="math_toolkit.cardinal_trigonometric.h2"></a>
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<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.caveats"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.caveats">Caveats</a>
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</h4>
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<p>
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This routine depends on FFTW3, and hence will only compile in float, double,
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long double, and quad precision, unlike the large bulk of the library which
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is compatible with arbitrary precision arithmetic. The FFTW linker flags must
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be added to the compile step, i.e., <code class="computeroutput"><span class="special">-</span><span class="identifier">lm</span> <span class="special">-</span><span class="identifier">lfftw3</span></code>
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for double precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lm</span>
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<span class="special">-</span><span class="identifier">lfftw3f</span></code>
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for float, so on.
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</p>
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<p>
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Evaluation of derivatives is done by differentiation of Horner's method. As
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always, differentiation amplifies noise; and because some rounding error is
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produced by computation of the Fourier coefficients, this error is amplified
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by differentiation.
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</p>
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<h4>
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<a name="math_toolkit.cardinal_trigonometric.h3"></a>
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<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.references"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.references">References</a>
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</h4>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Atkinson, Kendall, and Weimin Han. <span class="emphasis"><em>Theoretical numerical analysis.</em></span>
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Vol. 39. Berlin: Springer, 2005.
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</li>
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<li class="listitem">
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Kress, Rainer. <span class="emphasis"><em>Numerical Analysis.</em></span> 1998. Academic
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Edition 1.
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</li>
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</ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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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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