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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_poly.jacobi"></a><a class="link" href="jacobi.html" title="Jacobi Polynomials">Jacobi Polynomials</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_poly.jacobi.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_poly.jacobi.synopsis"></a></span><a class="link" href="jacobi.html#math_toolkit.sf_poly.jacobi.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">jacobi</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">jacobi</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">alpha</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">beta</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">jacobi_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">alpha</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">beta</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>
  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">jacobi_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">alpha</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
  45
  46 <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>
  47 <span class="identifier">Real</span> <span class="identifier">jacobi_double_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">alpha</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
  48
  49 <span class="special">}}</span> <span class="comment">// namespaces</span>
  50 </pre>
  51 <p>
  52         Jacobi polynomials are a family of orthogonal polynomials.
  53       </p>
  54 <p>
  55         A basic usage is as follows:
  56       </p>
  57 <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">jacobi</span><span class="special">;</span>
  58 <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
  59 <span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.3</span><span class="special">;</span>
  60 <span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">7.2</span><span class="special">;</span>
  61 <span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
  62 <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">jacobi</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>
  63 </pre>
  64 <p>
  65         All derivatives of the Jacobi polynomials are available. The <span class="emphasis"><em>k</em></span>-th
  66         derivative of the <span class="emphasis"><em>n</em></span>-th Gegenbauer polynomial is given
  67         by
  68       </p>
  69 <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">jacobi_derivative</span><span class="special">;</span>
  70 <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
  71 <span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.3</span><span class="special">;</span>
  72 <span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">7.2</span><span class="special">;</span>
  73 <span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
  74 <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">jacobi_derivative</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">k</span><span class="special">);</span>
  75 </pre>
  76 <p>
  77         For consistency with the rest of the library, <code class="computeroutput"><span class="identifier">jacobi_prime</span></code>
  78         is provided which simply returns <code class="computeroutput"><span class="identifier">jacobi_derivative</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span>
  79         <span class="identifier">lambda</span><span class="special">,</span>
  80         <span class="identifier">x</span><span class="special">,</span><span class="number">1</span><span class="special">)</span></code>.
  81       </p>
  82 <p>
  83         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/jacobi.svg"></object></span>
  84       </p>
  85 <h4>
  86 <a name="math_toolkit.sf_poly.jacobi.h1"></a>
  87         <span class="phrase"><a name="math_toolkit.sf_poly.jacobi.implementation"></a></span><a class="link" href="jacobi.html#math_toolkit.sf_poly.jacobi.implementation">Implementation</a>
  88       </h4>
  89 <p>
  90         The implementation uses the 3-term recurrence for the Jacobi polynomials,
  91         rising.
  92       </p>
  93 </div>
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  95 <td align="left"></td>
  96 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
  97       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
  98       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
  99       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 100       Daryle Walker and Xiaogang Zhang<p>
 101         Distributed under the Boost Software License, Version 1.0. (See accompanying
 102         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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