...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#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; } }}
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_{n+1} are defined by the property that they have n vanishing moments against the oscillatory measure P_{n}, i.e., ∫_{-1}^{1} E_{n+1}(x)P_{n}(x) x^{k} dx = 0 for k = 0, 1, ..., n. The first few are
where P_{i} 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());