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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_poly.legendre_stieltjes"></a><a class="link" href="legendre_stieltjes.html" title="Legendre-Stieltjes Polynomials">Legendre-Stieltjes
  28       Polynomials</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.sf_poly.legendre_stieltjes.h0"></a>
  32         <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>
  33       </h5>
  34 <pre class="programlisting">
  35 #include &lt;boost/math/special_functions/legendre_stieltjes.hpp&gt;
  36
  37
  38 <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>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  41 <span class="keyword">class</span> <span class="identifier">legendre_stieltjes</span>
  42 <span class="special">{</span>
  43 <span class="keyword">public</span><span class="special">:</span>
  44     <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>
  45
  46     <span class="identifier">Real</span> <span class="identifier">norm_sq</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  47
  48     <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>
  49
  50     <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>
  51
  52     <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">zeros</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  53 <span class="special">}</span>
  54
  55 <span class="special">}}</span>
  56 </pre>
  57 <h5>
  58 <a name="math_toolkit.sf_poly.legendre_stieltjes.h1"></a>
  59         <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>
  60       </h5>
  61 <p>
  62         The Legendre-Stieltjes polynomials are a family of polynomials used to generate
  63         Gauss-Konrod quadrature formulas. Gauss-Konrod quadratures are algorithms
  64         which extend a Gaussian quadrature in such a way that all abscissas are reused
  65         when computed a higher-order estimate of the integral, allowing efficient
  66         calculation of an error estimate. The Legendre-Stieltjes polynomials assist
  67         with this task because their zeros <span class="emphasis"><em>interlace</em></span> the zeros
  68         of the Legendre polynomials, meaning that between any two zeros of a Legendre
  69         polynomial of degree n, there exists a zero of the Legendre-Stieltjes polynomial
  70         of degree n+1.
  71       </p>
  72 <p>
  73         The Legendre-Stieltjes polynomials <span class="emphasis"><em>E<sub>n+1</sub></em></span> are defined
  74         by the property that they have <span class="emphasis"><em>n</em></span> vanishing moments against
  75         the oscillatory measure <span class="emphasis"><em>P<sub>n</sub></em></span>, i.e.,
  76       </p>
  77 <div class="blockquote"><blockquote class="blockquote"><p>
  78           <span class="serif_italic">&#8747; <sub>-1</sub><sup>1</sup> E<sub>n+1</sub>(x)P<sub>n</sub>(x) x<sup>k</sup>dx = 0</span>
  79         </p></blockquote></div>
  80 <p>
  81         for <span class="emphasis"><em>k = 0, 1, ..., n</em></span>.
  82       </p>
  83 <p>
  84         The first few are
  85       </p>
  86 <div class="blockquote"><blockquote class="blockquote"><p>
  87           <span class="serif_italic">E<sub>1</sub>(x) = P<sub>1</sub>(x)</span>
  88         </p></blockquote></div>
  89 <div class="blockquote"><blockquote class="blockquote"><p>
  90           <span class="serif_italic">E<sub>2</sub>(x) = P<sub>2</sub>(x) - 2P<sub>0</sub>(x)/5</span>
  91         </p></blockquote></div>
  92 <div class="blockquote"><blockquote class="blockquote"><p>
  93           <span class="serif_italic">E<sub>3</sub>(x) = P<sub>3</sub>(x) - 9P<sub>1</sub>(x)/14</span>
  94         </p></blockquote></div>
  95 <div class="blockquote"><blockquote class="blockquote"><p>
  96           <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>
  97         </p></blockquote></div>
  98 <div class="blockquote"><blockquote class="blockquote"><p>
  99           <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>
 100         </p></blockquote></div>
 101 <p>
 102         where <span class="emphasis"><em>P<sub>i</sub></em></span> are the Legendre polynomials. The scaling follows
 103         <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>,
 104         who expanded the Legendre-Stieltjes polynomials in a Legendre series and
 105         took the coefficient of the highest-order Legendre polynomial in the series
 106         to be unity.
 107       </p>
 108 <p>
 109         The Legendre-Stieltjes polynomials do not satisfy three-term recurrence relations
 110         or have a particulary simple representation. Hence the constructor call determines
 111         what, in fact, the polynomial is. Once the constructor comes back, the polynomial
 112         can be evaluated via the Legendre series.
 113       </p>
 114 <p>
 115         Example usage:
 116       </p>
 117 <pre class="programlisting"><span class="comment">// Call to the constructor determines the coefficients in the Legendre expansion</span>
 118 <span class="identifier">legendre_stieltjes</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">E</span><span class="special">(</span><span class="number">12</span><span class="special">);</span>
 119 <span class="comment">// Evaluate the polynomial at a point:</span>
 120 <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>
 121 <span class="comment">// Evaluate the derivative at a point:</span>
 122 <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>
 123 <span class="comment">// Use the norm_sq to change between scalings, if desired:</span>
 124 <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>
 125 </pre>
 126 </div>
 127 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 128 <td align="left"></td>
 129 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 130       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 131       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 132       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 133       Daryle Walker and Xiaogang Zhang<p>
 134         Distributed under the Boost Software License, Version 1.0. (See accompanying
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