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[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).
]
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