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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2005, 2006 Klaus Spanderen
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file gaussianorthogonalpolynomial.hpp
\brief orthogonal polynomials for gaussian quadratures
*/
#ifndef quantlib_gaussian_orthogonal_polynomial_hpp
#define quantlib_gaussian_orthogonal_polynomial_hpp
#include <ql/types.hpp>
namespace QuantLib {
//! orthogonal polynomial for Gaussian quadratures
/*! References:
Gauss quadratures and orthogonal polynomials
G.H. Gloub and J.H. Welsch: Calculation of Gauss quadrature rule.
Math. Comput. 23 (1986), 221-230
"Numerical Recipes in C", 2nd edition,
Press, Teukolsky, Vetterling, Flannery,
The polynomials are defined by the three-term recurrence relation
\f[
P_{k+1}(x)=(x-\alpha_k) P_k(x) - \beta_k P_{k-1}(x)
\f]
and
\f[
\mu_0 = \int{w(x)dx}
\f]
*/
class GaussianOrthogonalPolynomial {
public:
virtual ~GaussianOrthogonalPolynomial() = default;
virtual Real mu_0() const = 0;
virtual Real alpha(Size i) const = 0;
virtual Real beta(Size i) const = 0;
virtual Real w(Real x) const = 0;
Real value(Size i, Real x) const;
Real weightedValue(Size i, Real x) const;
};
//! Gauss-Laguerre polynomial
class GaussLaguerrePolynomial : public GaussianOrthogonalPolynomial {
public:
explicit GaussLaguerrePolynomial(Real s = 0.0);
Real mu_0() const override;
Real alpha(Size i) const override;
Real beta(Size i) const override;
Real w(Real x) const override;
private:
const Real s_;
};
//! Gauss-Hermite polynomial
class GaussHermitePolynomial : public GaussianOrthogonalPolynomial {
public:
explicit GaussHermitePolynomial(Real mu = 0.0);
Real mu_0() const override;
Real alpha(Size i) const override;
Real beta(Size i) const override;
Real w(Real x) const override;
private:
const Real mu_;
};
//! Gauss-Jacobi polynomial
class GaussJacobiPolynomial : public GaussianOrthogonalPolynomial {
public:
explicit GaussJacobiPolynomial(Real alpha, Real beta);
Real mu_0() const override;
Real alpha(Size i) const override;
Real beta(Size i) const override;
Real w(Real x) const override;
private:
const Real alpha_;
const Real beta_;
};
//! Gauss-Legendre polynomial
class GaussLegendrePolynomial : public GaussJacobiPolynomial {
public:
GaussLegendrePolynomial();
};
//! Gauss-Chebyshev polynomial
class GaussChebyshevPolynomial : public GaussJacobiPolynomial {
public:
GaussChebyshevPolynomial();
};
//! Gauss-Chebyshev polynomial (second kind)
class GaussChebyshev2ndPolynomial : public GaussJacobiPolynomial {
public:
GaussChebyshev2ndPolynomial();
};
//! Gauss-Gegenbauer polynomial
class GaussGegenbauerPolynomial : public GaussJacobiPolynomial {
public:
explicit GaussGegenbauerPolynomial(Real lambda);
};
//! Gauss hyperbolic polynomial
class GaussHyperbolicPolynomial : public GaussianOrthogonalPolynomial {
public:
Real mu_0() const override;
Real alpha(Size i) const override;
Real beta(Size i) const override;
Real w(Real x) const override;
};
}
#endif
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