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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

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());

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