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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_poly.gegenbauer"></a><a class="link" href="gegenbauer.html" title="Gegenbauer Polynomials">Gegenbauer Polynomials</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_poly.gegenbauer.h0"></a>
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<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>
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</h5>
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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">special_functions</span><span class="special">/</span><span class="identifier">gegenbauer</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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<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>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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Gegenbauer polynomials are a family of orthogonal polynomials.
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</p>
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<p>
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A basic usage is as follows:
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</p>
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<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>
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<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
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<span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</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">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>
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</pre>
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<p>
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All derivatives of the Gegenbauer polynomials are available. The <span class="emphasis"><em>k</em></span>-th
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derivative of the <span class="emphasis"><em>n</em></span>-th Gegenbauer polynomial is given
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by
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</p>
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<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>
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<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
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<span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
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<span class="keyword">unsigned</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">2</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">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>
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</pre>
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<p>
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For consistency with the rest of the library, <code class="computeroutput"><span class="identifier">gegenbauer_prime</span></code>
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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>
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<span class="identifier">lambda</span><span class="special">,</span>
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<span class="identifier">x</span><span class="special">,</span><span class="number">1</span> <span class="special">)</span></code>.
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/gegenbauer.svg"></object></span>
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.gegenbauer.h1"></a>
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<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>
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</h4>
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<p>
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The implementation uses the 3-term recurrence for the Gegenbauer polynomials,
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rising.
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.gegenbauer.h2"></a>
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<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>
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</h4>
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<p>
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Double precision timing on a consumer x86 laptop is shown below. Included
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is the time to generate a random number argument in the interval [-1, 1]
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(which takes 11.5ns).
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</p>
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<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>
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<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
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<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>
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<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>
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<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>
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<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>
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<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>
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<span class="special">-----------------------------------------</span>
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<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
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<span class="special">-----------------------------------------</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1</span> <span class="number">12.5</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2</span> <span class="number">13.5</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">3</span> <span class="number">14.6</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4</span> <span class="number">16.0</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">5</span> <span class="number">17.5</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">6</span> <span class="number">19.2</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">7</span> <span class="number">20.7</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8</span> <span class="number">22.2</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">9</span> <span class="number">23.6</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">10</span> <span class="number">25.2</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">11</span> <span class="number">26.9</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">12</span> <span class="number">28.7</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">13</span> <span class="number">30.5</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">14</span> <span class="number">32.5</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">15</span> <span class="number">34.3</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16</span> <span class="number">36.3</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">17</span> <span class="number">38.0</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">18</span> <span class="number">39.9</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">19</span> <span class="number">41.8</span> <span class="identifier">ns</span>
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<span class="identifier">Gegenbauer</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">20</span> <span class="number">43.8</span> <span class="identifier">ns</span>
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<span class="identifier">UniformReal</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="number">11.5</span> <span class="identifier">ns</span>
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</pre>
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<h4>
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<a name="math_toolkit.sf_poly.gegenbauer.h3"></a>
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<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>
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</h4>
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<p>
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Some representative ULP plots are shown below. The relative accuracy cannot
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be controlled at the roots of the polynomial, as is to be expected.
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</p>
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<p>
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<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>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/gegenbauer_ulp_9.svg"></object></span>
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.gegenbauer.h4"></a>
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<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>
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</h4>
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<p>
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Some programs define the Gegenbauer polynomial with λ = 0 via renormalization
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(which makes them Chebyshev polynomials). We do not follow this convention:
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In this case, only the zeroth Gegenbauer polynomial is nonzero.
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</p>
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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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