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### Legendre-Stieltjes Polynomials

##### 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;
    }

    }}

##### 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 En+1 are defined by the property that they have n vanishing moments against the oscillatory measure Pn, i.e., ∫-11 En+1(x)Pn(x) xk dx = 0 for k = 0, 1, ..., n. The first few are

- E1(x) = P1(x)
- E2(x) = P2(x) - 2P0(x)/5
- E3(x) = P3(x) - 9P1(x)/14
- E4(x) = P4(x) - 20P2(x)/27 + 14P0(x)/891
- E5(x) = P5(x) - 35P3(x)/44 + 135P1(x)/12584

where Pi are the Legendre polynomials. The scaling follows 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 particulary 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());