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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2005 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
<http://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 gaussianquadratures.hpp
\brief Integral of a 1-dimensional function using the Gauss quadratures
*/
#include <ql/math/integrals/gaussianorthogonalpolynomial.hpp>
#include <ql/math/distributions/gammadistribution.hpp>
#include <ql/math/comparison.hpp>
#include <ql/errors.hpp>
#include <cmath>
namespace QuantLib {
Real GaussianOrthogonalPolynomial::value(Size n, Real x) const {
if (n > 1) {
return (x-alpha(n-1)) * value(n-1, x)
- beta(n-1) * value(n-2, x);
}
else if (n == 1) {
return x-alpha(0);
}
return 1;
}
Real GaussianOrthogonalPolynomial::weightedValue(Size n, Real x) const {
return std::sqrt(w(x))*value(n, x);
}
GaussLaguerrePolynomial::GaussLaguerrePolynomial(Real s)
: s_(s) {
QL_REQUIRE(s > -1.0, "s must be bigger than -1");
}
Real GaussLaguerrePolynomial::mu_0() const {
return std::exp(GammaFunction().logValue(s_+1));
}
Real GaussLaguerrePolynomial::alpha(Size i) const {
return 2*i+1+s_;
}
Real GaussLaguerrePolynomial::beta(Size i) const {
return i*(i+s_);
}
Real GaussLaguerrePolynomial::w(Real x) const {
return std::pow(x, s_)*std::exp(-x);
}
GaussHermitePolynomial::GaussHermitePolynomial(Real mu)
: mu_(mu) {
QL_REQUIRE(mu > -0.5, "mu must be bigger than -0.5");
}
Real GaussHermitePolynomial::mu_0() const {
return std::exp(GammaFunction().logValue(mu_+0.5));
}
Real GaussHermitePolynomial::alpha(Size) const {
return 0.0;
}
Real GaussHermitePolynomial::beta(Size i) const {
return (i % 2) != 0U ? Real(i / 2.0 + mu_) : Real(i / 2.0);
}
Real GaussHermitePolynomial::w(Real x) const {
return std::pow(std::fabs(x), 2*mu_)*std::exp(-x*x);
}
GaussJacobiPolynomial::GaussJacobiPolynomial(Real alpha, Real beta)
: alpha_(alpha), beta_ (beta) {
QL_REQUIRE(alpha_+beta_ > -2.0,"alpha+beta must be bigger than -2");
QL_REQUIRE(alpha_ > -1.0,"alpha must be bigger than -1");
QL_REQUIRE(beta_ > -1.0,"beta must be bigger than -1");
}
Real GaussJacobiPolynomial::mu_0() const {
return std::pow(2.0, alpha_+beta_+1)
* std::exp( GammaFunction().logValue(alpha_+1)
+GammaFunction().logValue(beta_ +1)
-GammaFunction().logValue(alpha_+beta_+2));
}
Real GaussJacobiPolynomial::alpha(Size i) const {
Real num = beta_*beta_ - alpha_*alpha_;
Real denom = (2.0*i+alpha_+beta_)*(2.0*i+alpha_+beta_+2);
if (close_enough(denom,0.0)) {
if (!close_enough(num,0.0)) {
QL_FAIL("can't compute a_k for jacobi integration\n");
}
else {
// l'Hospital
num = 2*beta_;
denom= 2*(2.0*i+alpha_+beta_+1);
QL_ASSERT(!close_enough(denom,0.0), "can't compute a_k for jacobi integration\n");
}
}
return num / denom;
}
Real GaussJacobiPolynomial::beta(Size i) const {
Real num = 4.0*i*(i+alpha_)*(i+beta_)*(i+alpha_+beta_);
Real denom = (2.0*i+alpha_+beta_)*(2.0*i+alpha_+beta_)
* ((2.0*i+alpha_+beta_)*(2.0*i+alpha_+beta_)-1);
if (close_enough(denom,0.0)) {
if (!close_enough(num,0.0)) {
QL_FAIL("can't compute b_k for jacobi integration\n");
} else {
// l'Hospital
num = 4.0*i*(i+beta_)* (2.0*i+2*alpha_+beta_);
denom= 2.0*(2.0*i+alpha_+beta_);
denom*=denom-1;
QL_ASSERT(!close_enough(denom,0.0), "can't compute b_k for jacobi integration\n");
}
}
return num / denom;
}
Real GaussJacobiPolynomial::w(Real x) const {
return std::pow(1-x, alpha_)*std::pow(1+x, beta_);
}
GaussLegendrePolynomial::GaussLegendrePolynomial()
: GaussJacobiPolynomial(0.0, 0.0) {
}
GaussChebyshev2ndPolynomial::GaussChebyshev2ndPolynomial()
: GaussJacobiPolynomial(0.5, 0.5) {
}
GaussChebyshevPolynomial::GaussChebyshevPolynomial()
: GaussJacobiPolynomial(-0.5, -0.5) {
}
GaussGegenbauerPolynomial::GaussGegenbauerPolynomial(Real lambda)
: GaussJacobiPolynomial(lambda-0.5, lambda-0.5){
}
Real GaussHyperbolicPolynomial::mu_0() const {
return M_PI;
}
Real GaussHyperbolicPolynomial::alpha(Size) const {
return 0.0;
}
Real GaussHyperbolicPolynomial::beta(Size i) const {
return i != 0U ? M_PI_2 * M_PI_2 * i * i : M_PI;
}
Real GaussHyperbolicPolynomial::w(Real x) const {
return 1/std::cosh(x);
}
}
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