[section:legendre_stieltjes Legendre-Stieltjes Polynomials]

[h4 Synopsis]

    ``
    #include <boost/math/special_functions/legendre_stieltjes.hpp>
    ``

    namespace boost{ namespace math{

    template <class T>
    class legendre_stieltjes
    {
    public:
        legendre_stieltjes(size_t m);

        Real norm_sq() const;

        Real operator()(Real x) const;

        Real prime(Real x) const;

        std::vector<Real> zeros() const;
    }

    }}

[h4 Description]

The Legendre-Stieltjes polynomials are a family of polynomials used to generate Gauss-Konrod quadrature formulas.
Gauss-Konrod quadratures are algorithms which extend a Gaussian quadrature in such a way that all abscissas
are reused when computed a higher-order estimate of the integral, allowing efficient calculation of an error estimate.
The Legendre-Stieltjes polynomials assist with this task because their zeros /interlace/ the zeros of the Legendre polynomials,
meaning that between any two zeros of a Legendre polynomial of degree n, there exists a zero of the Legendre-Stieltjes polynomial
of degree n+1.

The Legendre-Stieltjes polynomials ['E[sub n+1]] are defined by the property that they have /n/ vanishing moments against the oscillatory measure ['P[sub n]], i.e., 

[expression [int] [sub -1][super 1] E[sub n+1](x)P[sub n](x) x[super k]dx = 0] for /k = 0, 1, ..., n/.

The first few are

[expression E[sub 1](x) = P[sub 1](x)]

[expression E[sub 2](x) = P[sub 2](x) - 2P[sub 0](x)/5]

[expression  E[sub 3](x) = P[sub 3](x) - 9P[sub 1](x)/14]

[expression  E[sub 4](x) = P[sub 4](x) - 20P[sub 2](x)/27 + 14P[sub 0](x)/891]

[expression  E[sub 5](x) = P[sub 5](x) - 35P[sub 3](x)/44 + 135P[sub 1](x)/12584]

where ['P[sub i]] are the Legendre polynomials.
The scaling follows [@http://www.ams.org/journals/mcom/1968-22-104/S0025-5718-68-99866-9/S0025-5718-68-99866-9.pdf Patterson],
who expanded the Legendre-Stieltjes polynomials in a Legendre series and took the coefficient of the highest-order Legendre polynomial in the series to be unity.

The Legendre-Stieltjes polynomials do not satisfy three-term recurrence relations or have a particularly simple representation.
Hence the constructor call determines what, in fact, the polynomial is.
Once the constructor comes back, the polynomial can be evaluated via the Legendre series.

Example usage:

    // Call to the constructor determines the coefficients in the Legendre expansion
    legendre_stieltjes<double> E(12);
    // Evaluate the polynomial at a point:
    double x = E(0.3);
    // Evaluate the derivative at a point:
    double x_p = E.prime(0.3);
    // Use the norm_sq to change between scalings, if desired:
    double norm = std::sqrt(E.norm_sq());

[endsect] [/section:legendre_stieltjes Legendre-Stieltjes Polynomials]

[/
  Copyright 2017 Nick Thompson
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]