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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_poly.legendre_stieltjes"></a><a class="link" href="legendre_stieltjes.html" title="Legendre-Stieltjes Polynomials">Legendre-Stieltjes
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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.legendre_stieltjes.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.legendre_stieltjes.synopsis"></a></span><a class="link" href="legendre_stieltjes.html#math_toolkit.sf_poly.legendre_stieltjes.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting">
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#include <boost/math/special_functions/legendre_stieltjes.hpp>
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<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">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">legendre_stieltjes</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">legendre_stieltjes</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">m</span><span class="special">);</span>
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<span class="identifier">Real</span> <span class="identifier">norm_sq</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">zeros</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="special">}</span>
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<span class="special">}}</span>
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</pre>
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<h5>
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<a name="math_toolkit.sf_poly.legendre_stieltjes.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.legendre_stieltjes.description"></a></span><a class="link" href="legendre_stieltjes.html#math_toolkit.sf_poly.legendre_stieltjes.description">Description</a>
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</h5>
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<p>
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The Legendre-Stieltjes polynomials are a family of polynomials used to generate
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Gauss-Konrod quadrature formulas. Gauss-Konrod quadratures are algorithms
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which extend a Gaussian quadrature in such a way that all abscissas are reused
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when computed a higher-order estimate of the integral, allowing efficient
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calculation of an error estimate. The Legendre-Stieltjes polynomials assist
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with this task because their zeros <span class="emphasis"><em>interlace</em></span> the zeros
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of the Legendre polynomials, meaning that between any two zeros of a Legendre
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polynomial of degree n, there exists a zero of the Legendre-Stieltjes polynomial
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of degree n+1.
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</p>
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<p>
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The Legendre-Stieltjes polynomials <span class="emphasis"><em>E<sub>n+1</sub></em></span> are defined
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by the property that they have <span class="emphasis"><em>n</em></span> vanishing moments against
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the oscillatory measure <span class="emphasis"><em>P<sub>n</sub></em></span>, i.e.,
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">∫ <sub>-1</sub><sup>1</sup> E<sub>n+1</sub>(x)P<sub>n</sub>(x) x<sup>k</sup>dx = 0</span>
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</p></blockquote></div>
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<p>
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for <span class="emphasis"><em>k = 0, 1, ..., n</em></span>.
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</p>
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<p>
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The first few are
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">E<sub>1</sub>(x) = P<sub>1</sub>(x)</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">E<sub>2</sub>(x) = P<sub>2</sub>(x) - 2P<sub>0</sub>(x)/5</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">E<sub>3</sub>(x) = P<sub>3</sub>(x) - 9P<sub>1</sub>(x)/14</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">E<sub>4</sub>(x) = P<sub>4</sub>(x) - 20P<sub>2</sub>(x)/27 + 14P<sub>0</sub>(x)/891</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">E<sub>5</sub>(x) = P<sub>5</sub>(x) - 35P<sub>3</sub>(x)/44 + 135P<sub>1</sub>(x)/12584</span>
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</p></blockquote></div>
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<p>
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where <span class="emphasis"><em>P<sub>i</sub></em></span> are the Legendre polynomials. The scaling follows
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<a href="http://www.ams.org/journals/mcom/1968-22-104/S0025-5718-68-99866-9/S0025-5718-68-99866-9.pdf" target="_top">Patterson</a>,
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who expanded the Legendre-Stieltjes polynomials in a Legendre series and
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took the coefficient of the highest-order Legendre polynomial in the series
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to be unity.
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</p>
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<p>
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The Legendre-Stieltjes polynomials do not satisfy three-term recurrence relations
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or have a particulary simple representation. Hence the constructor call determines
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what, in fact, the polynomial is. Once the constructor comes back, the polynomial
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can be evaluated via the Legendre series.
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</p>
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<p>
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Example usage:
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</p>
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<pre class="programlisting"><span class="comment">// Call to the constructor determines the coefficients in the Legendre expansion</span>
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<span class="identifier">legendre_stieltjes</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">E</span><span class="special">(</span><span class="number">12</span><span class="special">);</span>
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<span class="comment">// Evaluate the polynomial at a point:</span>
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<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">E</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
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<span class="comment">// Evaluate the derivative at a point:</span>
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<span class="keyword">double</span> <span class="identifier">x_p</span> <span class="special">=</span> <span class="identifier">E</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
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<span class="comment">// Use the norm_sq to change between scalings, if desired:</span>
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<span class="keyword">double</span> <span class="identifier">norm</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">E</span><span class="special">.</span><span class="identifier">norm_sq</span><span class="special">());</span>
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</pre>
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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="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle 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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</p>
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