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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_poly.gegenbauer"></a><a class="link" href="gegenbauer.html" title="Gegenbauer Polynomials">Gegenbauer Polynomials</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_poly.gegenbauer.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_poly.gegenbauer.synopsis"></a></span><a class="link" href="gegenbauer.html#math_toolkit.sf_poly.gegenbauer.synopsis">Synopsis</a>
  32       </h5>
  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">special_functions</span><span class="special">/</span><span class="identifier">gegenbauer</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>
  36
  37 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  38 <span class="identifier">Real</span> <span class="identifier">gegenbauer</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  41 <span class="identifier">Real</span> <span class="identifier">gegenbauer_prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
  42
  43 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  44 <span class="identifier">Real</span> <span class="identifier">gegenbauer_derivative</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">k</span><span class="special">);</span>
  45
  46 <span class="special">}}</span> <span class="comment">// namespaces</span>
  47 </pre>
  48 <p>
  49         Gegenbauer polynomials are a family of orthogonal polynomials.
  50       </p>
  51 <p>
  52         A basic usage is as follows:
  53       </p>
  54 <pre class="programlisting"><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">gegenbauer</span><span class="special">;</span>
  55 <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
  56 <span class="keyword">double</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
  57 <span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
  58 <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">gegenbauer</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>
  59 </pre>
  60 <p>
  61         All derivatives of the Gegenbauer polynomials are available. The <span class="emphasis"><em>k</em></span>-th
  62         derivative of the <span class="emphasis"><em>n</em></span>-th Gegenbauer polynomial is given
  63         by
  64       </p>
  65 <pre class="programlisting"><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">gegenbauer_derivative</span><span class="special">;</span>
  66 <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
  67 <span class="keyword">double</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
  68 <span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
  69 <span class="keyword">unsigned</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>
  70 <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">gegenbauer_derivative</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">k</span><span class="special">);</span>
  71 </pre>
  72 <p>
  73         For consistency with the rest of the library, <code class="computeroutput"><span class="identifier">gegenbauer_prime</span></code>
  74         is provided which simply returns <code class="computeroutput"><span class="identifier">gegenbauer_derivative</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span>
  75         <span class="identifier">lambda</span><span class="special">,</span>
  76         <span class="identifier">x</span><span class="special">,</span><span class="number">1</span> <span class="special">)</span></code>.
  77       </p>
  78 <p>
  79         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/gegenbauer.svg"></object></span>
  80       </p>
  81 <h4>
  82 <a name="math_toolkit.sf_poly.gegenbauer.h1"></a>
  83         <span class="phrase"><a name="math_toolkit.sf_poly.gegenbauer.implementation"></a></span><a class="link" href="gegenbauer.html#math_toolkit.sf_poly.gegenbauer.implementation">Implementation</a>
  84       </h4>
  85 <p>
  86         The implementation uses the 3-term recurrence for the Gegenbauer polynomials,
  87         rising.
  88       </p>
  89 <h4>
  90 <a name="math_toolkit.sf_poly.gegenbauer.h2"></a>
  91         <span class="phrase"><a name="math_toolkit.sf_poly.gegenbauer.performance"></a></span><a class="link" href="gegenbauer.html#math_toolkit.sf_poly.gegenbauer.performance">Performance</a>
  92       </h4>
  93 <p>
  94         Double precision timing on a consumer x86 laptop is shown below. Included
  95         is the time to generate a random number argument in the interval [-1, 1]
  96         (which takes 11.5ns).
  97       </p>
  98 <pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="special">(</span><span class="number">16</span> <span class="identifier">X</span> <span class="number">4300</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span> <span class="identifier">s</span><span class="special">)</span>
  99 <span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
 100   <span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
 101   <span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
 102   <span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">1024</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
 103   <span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">11264</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x1</span><span class="special">)</span>
 104 <span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">0.21</span><span class="special">,</span> <span class="number">0.33</span><span class="special">,</span> <span class="number">0.29</span>
 105 <span class="special">-----------------------------------------</span>
 106 <span class="identifier">Benchmark</span>                            <span class="identifier">Time</span>
 107 <span class="special">-----------------------------------------</span>
 108 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1</span>              <span class="number">12.5</span> <span class="identifier">ns</span>
 109 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2</span>              <span class="number">13.5</span> <span class="identifier">ns</span>
 110 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">3</span>              <span class="number">14.6</span> <span class="identifier">ns</span>
 111 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4</span>              <span class="number">16.0</span> <span class="identifier">ns</span>
 112 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">5</span>              <span class="number">17.5</span> <span class="identifier">ns</span>
 113 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">6</span>              <span class="number">19.2</span> <span class="identifier">ns</span>
 114 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">7</span>              <span class="number">20.7</span> <span class="identifier">ns</span>
 115 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>              <span class="number">22.2</span> <span class="identifier">ns</span>
 116 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">9</span>              <span class="number">23.6</span> <span class="identifier">ns</span>
 117 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">10</span>             <span class="number">25.2</span> <span class="identifier">ns</span>
 118 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">11</span>             <span class="number">26.9</span> <span class="identifier">ns</span>
 119 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">12</span>             <span class="number">28.7</span> <span class="identifier">ns</span>
 120 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">13</span>             <span class="number">30.5</span> <span class="identifier">ns</span>
 121 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">14</span>             <span class="number">32.5</span> <span class="identifier">ns</span>
 122 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">15</span>             <span class="number">34.3</span> <span class="identifier">ns</span>
 123 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>             <span class="number">36.3</span> <span class="identifier">ns</span>
 124 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">17</span>             <span class="number">38.0</span> <span class="identifier">ns</span>
 125 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">18</span>             <span class="number">39.9</span> <span class="identifier">ns</span>
 126 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">19</span>             <span class="number">41.8</span> <span class="identifier">ns</span>
 127 <span class="identifier">Gegenbauer</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">20</span>             <span class="number">43.8</span> <span class="identifier">ns</span>
 128 <span class="identifier">UniformReal</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span>               <span class="number">11.5</span> <span class="identifier">ns</span>
 129 </pre>
 130 <h4>
 131 <a name="math_toolkit.sf_poly.gegenbauer.h3"></a>
 132         <span class="phrase"><a name="math_toolkit.sf_poly.gegenbauer.accuracy"></a></span><a class="link" href="gegenbauer.html#math_toolkit.sf_poly.gegenbauer.accuracy">Accuracy</a>
 133       </h4>
 134 <p>
 135         Some representative ULP plots are shown below. The relative accuracy cannot
 136         be controlled at the roots of the polynomial, as is to be expected.
 137       </p>
 138 <p>
 139         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/gegenbauer_ulp_3.svg"></object></span> <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/gegenbauer_ulp_5.svg"></object></span>
 140         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/gegenbauer_ulp_9.svg"></object></span>
 141       </p>
 142 <h4>
 143 <a name="math_toolkit.sf_poly.gegenbauer.h4"></a>
 144         <span class="phrase"><a name="math_toolkit.sf_poly.gegenbauer.caveats"></a></span><a class="link" href="gegenbauer.html#math_toolkit.sf_poly.gegenbauer.caveats">Caveats</a>
 145       </h4>
 146 <p>
 147         Some programs define the Gegenbauer polynomial with &#955; = 0 via renormalization
 148         (which makes them Chebyshev polynomials). We do not follow this convention:
 149         In this case, only the zeroth Gegenbauer polynomial is nonzero.
 150       </p>
 151 </div>
 152 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 153 <td align="left"></td>
 154 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 155       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 156       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 157       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 158       Daryle Walker and Xiaogang Zhang<p>
 159         Distributed under the Boost Software License, Version 1.0. (See accompanying
 160         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>)
 161       </p>
 162 </div></td>
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