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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.cardinal_trigonometric"></a><a class="link" href="cardinal_trigonometric.html" title="Cardinal Trigonometric interpolation">Cardinal Trigonometric
  28     interpolation</a>
  29 </h2></div></div></div>
  30 <h4>
  31 <a name="math_toolkit.cardinal_trigonometric.h0"></a>
  32       <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>
  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_trigonometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  35
  36 <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">RandomAccessContainer</span><span class="special">&gt;</span>
  39 <span class="keyword">class</span> <span class="identifier">cardinal_trigonometric</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  42     <span class="identifier">cardinal_trigonometric</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</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>
  43
  44     <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>
  45
  46     <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>
  47
  48     <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>
  49
  50     <span class="identifier">Real</span> <span class="identifier">period</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  51
  52     <span class="identifier">Real</span> <span class="identifier">integrate</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  53
  54     <span class="identifier">Real</span> <span class="identifier">squared_l2</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  55 <span class="special">};</span>
  56 <span class="special">}}}</span>
  57 </pre>
  58 <h4>
  59 <a name="math_toolkit.cardinal_trigonometric.h1"></a>
  60       <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
  61       Trigonometric Interpolation</a>
  62     </h4>
  63 <p>
  64       The cardinal trigonometric interpolation problem takes uniformly spaced samples
  65       <span class="emphasis"><em>y</em></span><sub>j</sub> of a periodic function <span class="emphasis"><em>f</em></span> defined
  66       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> +
  67       <span class="emphasis"><em>j</em></span> <span class="emphasis"><em>h</em></span>) and represents them as a linear
  68       combination of sines and cosines. If the period of <span class="emphasis"><em>f</em></span> is
  69       <span class="emphasis"><em>T</em></span>, and the number of data points is <span class="emphasis"><em>n = 2m</em></span>
  70       or <span class="emphasis"><em>n = 2m+1</em></span>, we hope to have
  71     </p>
  72 <p>
  73       <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>t</em></span>) &#8776; <span class="emphasis"><em>a</em></span><sub>0</sub>/2
  74       + &#8721;<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(2&#960;
  75       <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)
  76       + <span class="emphasis"><em>b</em></span><sub><span class="emphasis"><em>k</em></span></sub> sin(2&#960; <span class="emphasis"><em>k</em></span>
  77       (<span class="emphasis"><em>t</em></span>-<span class="emphasis"><em>t</em></span><sub>0</sub>)/T)
  78     </p>
  79 <p>
  80       Convergence rates depend on the number of continuous derivatives of <span class="emphasis"><em>f</em></span>;
  81       see either Atkinson or Kress for details.
  82     </p>
  83 <p>
  84       A simple use of this interpolator is shown below:
  85     </p>
  86 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
  87 <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_trigonometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  88 <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>
  89 <span class="special">...</span>
  90 <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">17</span><span class="special">,</span> <span class="number">3.2</span><span class="special">);</span>
  91 <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>
  92 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ct(1.3) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ct</span><span class="special">(</span><span class="number">1.3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  93
  94 <span class="comment">// Derivative:</span>
  95 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  96 <span class="comment">// Second derivative:</span>
  97 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  98 </pre>
  99 <p>
 100       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,
 101       and hence if you wonder what this interpolator believes the period to be, you
 102       can query it with the <code class="computeroutput"><span class="special">.</span><span class="identifier">period</span><span class="special">()</span></code> member function.
 103     </p>
 104 <p>
 105       In addition, the transform into the trigonometric basis gives a trivial way
 106       to compute the integral of the function over a period; this is done via the
 107       <code class="computeroutput"><span class="special">.</span><span class="identifier">integrate</span><span class="special">()</span></code> member function. Evaluation of the square
 108       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.
 109     </p>
 110 <p>
 111       Below is a graph of a <span class="emphasis"><em>C</em></span><sup>&#8734;</sup> bump function approximated
 112       by trigonometric series. The graphs are visually indistinguishable at 20 samples.
 113     </p>
 114 <p>
 115       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/fourier_bump.svg"></object></span>
 116     </p>
 117 <h4>
 118 <a name="math_toolkit.cardinal_trigonometric.h2"></a>
 119       <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>
 120     </h4>
 121 <p>
 122       This routine depends on FFTW3, and hence will only compile in float, double,
 123       long double, and quad precision, unlike the large bulk of the library which
 124       is compatible with arbitrary precision arithmetic. The FFTW linker flags must
 125       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>
 126       for double precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lm</span>
 127       <span class="special">-</span><span class="identifier">lfftw3f</span></code>
 128       for float, so on.
 129     </p>
 130 <p>
 131       Evaluation of derivatives is done by differentiation of Horner's method. As
 132       always, differentiation amplifies noise; and because some rounding error is
 133       produced by computation of the Fourier coefficients, this error is amplified
 134       by differentiation.
 135     </p>
 136 <h4>
 137 <a name="math_toolkit.cardinal_trigonometric.h3"></a>
 138       <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>
 139     </h4>
 140 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 141 <li class="listitem">
 142           Atkinson, Kendall, and Weimin Han. <span class="emphasis"><em>Theoretical numerical analysis.</em></span>
 143           Vol. 39. Berlin: Springer, 2005.
 144         </li>
 145 <li class="listitem">
 146           Kress, Rainer. <span class="emphasis"><em>Numerical Analysis.</em></span> 1998. Academic
 147           Edition 1.
 148         </li>
 149 </ul></div>
 150 </div>
 151 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 152 <td align="left"></td>
 153 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 154       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 155       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 156       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 157       Daryle Walker and Xiaogang Zhang<p>
 158         Distributed under the Boost Software License, Version 1.0. (See accompanying
 159         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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