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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_poly.jacobi"></a><a class="link" href="jacobi.html" title="Jacobi Polynomials">Jacobi 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.jacobi.h0"></a>
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<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>
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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">jacobi</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">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>
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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">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>
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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">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>
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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">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>
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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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Jacobi 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">jacobi</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">alpha</span> <span class="special">=</span> <span class="number">0.3</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">7.2</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">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>
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</pre>
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<p>
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All derivatives of the Jacobi 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">jacobi_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">alpha</span> <span class="special">=</span> <span class="number">0.3</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">7.2</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">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>
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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">jacobi_prime</span></code>
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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>
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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/jacobi.svg"></object></span>
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.jacobi.h1"></a>
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<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>
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</h4>
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<p>
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The implementation uses the 3-term recurrence for the Jacobi polynomials,
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rising.
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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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