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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.sf_poly.jacobi"></a><a class="link" href="jacobi.html" title="Jacobi Polynomials">Jacobi Polynomials</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.sf_poly.jacobi.h0"></a>
<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>
</h5>
<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>
</pre>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Jacobi polynomials are a family of orthogonal polynomials.
</p>
<p>
A basic usage is as follows:
</p>
<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>
<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.3</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">7.2</span><span class="special">;</span>
<span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
<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>
</pre>
<p>
All derivatives of the Jacobi polynomials are available. The <span class="emphasis"><em>k</em></span>-th
derivative of the <span class="emphasis"><em>n</em></span>-th Gegenbauer polynomial is given
by
</p>
<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>
<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.3</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">7.2</span><span class="special">;</span>
<span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>
<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>
</pre>
<p>
For consistency with the rest of the library, <code class="computeroutput"><span class="identifier">jacobi_prime</span></code>
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>
<span class="identifier">lambda</span><span class="special">,</span>
<span class="identifier">x</span><span class="special">,</span><span class="number">1</span><span class="special">)</span></code>.
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/jacobi.svg"></object></span>
</p>
<h4>
<a name="math_toolkit.sf_poly.jacobi.h1"></a>
<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>
</h4>
<p>
The implementation uses the 3-term recurrence for the Jacobi polynomials,
rising.
</p>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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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