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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2005 Klaus Spanderen
Copyright (C) 2005 Gary Kennedy
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 gaussianquadratures.hpp
\brief Integral of a 1-dimensional function using the Gauss quadratures
*/
#include <ql/utilities/null.hpp>
#include <ql/math/integrals/gaussianquadratures.hpp>
#include <ql/math/matrixutilities/tqreigendecomposition.hpp>
#include <ql/math/matrixutilities/symmetricschurdecomposition.hpp>
#include <map>
namespace QuantLib {
GaussianQuadrature::GaussianQuadrature(
Size n,
const GaussianOrthogonalPolynomial& orthPoly)
: x_(n), w_(n) {
// set-up matrix to compute the roots and the weights
Array e(n-1);
Size i;
for (i=1; i < n; ++i) {
x_[i] = orthPoly.alpha(i);
e[i-1] = std::sqrt(orthPoly.beta(i));
}
x_[0] = orthPoly.alpha(0);
TqrEigenDecomposition tqr(
x_, e,
TqrEigenDecomposition::OnlyFirstRowEigenVector,
TqrEigenDecomposition::Overrelaxation);
x_ = tqr.eigenvalues();
const Matrix& ev = tqr.eigenvectors();
Real mu_0 = orthPoly.mu_0();
for (i=0; i<n; ++i) {
w_[i] = mu_0*ev[0][i]*ev[0][i] / orthPoly.w(x_[i]);
}
}
MultiDimGaussianIntegration::MultiDimGaussianIntegration(
const std::vector<Size>& ns,
const std::function<ext::shared_ptr<GaussianQuadrature>(Size)>& genQuad)
: weights_(std::accumulate(ns.begin(), ns.end(), Size(1), std::multiplies<>()), 1.0),
x_(weights_.size(), Array(ns.size())) {
const Size m = ns.size();
const Size n = x_.size();
std::vector<Size> spacing(m);
spacing[0] = 1;
std::partial_sum(ns.begin(), ns.end()-1, spacing.begin()+1, std::multiplies<>());
std::map<Size, Array> n2weights, n2x;
for (auto order: ns) {
if (n2x.find(order) == n2x.end()) {
const ext::shared_ptr<GaussianQuadrature> quad = genQuad(order);
n2x[order] = quad->x();
n2weights[order] = quad->weights();
}
}
for (Size i=0; i < n; ++i) {
for (Size j=0; j < m; ++j) {
const Size order = ns[j];
const Size nx = (i / spacing[j]) % ns[j];
weights_[i] *= n2weights[order][nx];
x_[i][j] = n2x[order][nx];
}
}
}
Real MultiDimGaussianIntegration::operator()(
const std::function<Real(Array)>& f) const {
Real s = 0.0;
const Size n = x_.size();
for (Size i=0; i < n; ++i)
s += weights_[i]*f(x_[i]);
return s;
}
namespace detail {
template <class Integration>
GaussianQuadratureIntegrator<Integration>::GaussianQuadratureIntegrator(
Size n)
: Integrator(Null<Real>(), n),
integration_(ext::make_shared<Integration>(n))
{ }
template <class Integration>
Real GaussianQuadratureIntegrator<Integration>::integrate(
const std::function<Real (Real)>& f, Real a, Real b) const {
const Real c1 = 0.5*(b-a);
const Real c2 = 0.5*(a+b);
return c1*integration_->operator()(
[c1, c2, f](Real x) { return f(c1*x+c2);});
}
template class GaussianQuadratureIntegrator<GaussLegendreIntegration>;
template class GaussianQuadratureIntegrator<GaussChebyshevIntegration>;
template class GaussianQuadratureIntegrator<GaussChebyshev2ndIntegration>;
}
void TabulatedGaussLegendre::order(Size order) {
switch(order) {
case(6):
order_=order; x_=x6; w_=w6; n_=n6;
break;
case(7):
order_=order; x_=x7; w_=w7; n_=n7;
break;
case(12):
order_=order; x_=x12; w_=w12; n_=n12;
break;
case(20):
order_=order; x_=x20; w_=w20; n_=n20;
break;
default:
QL_FAIL("order " << order << " not supported");
}
}
// Abscissas and Weights from Abramowitz and Stegun
/* order 6 */
const Real TabulatedGaussLegendre::x6[3] = { 0.238619186083197,
0.661209386466265,
0.932469514203152 };
const Real TabulatedGaussLegendre::w6[3] = { 0.467913934572691,
0.360761573048139,
0.171324492379170 };
const Size TabulatedGaussLegendre::n6 = 3;
/* order 7 */
const Real TabulatedGaussLegendre::x7[4] = { 0.000000000000000,
0.405845151377397,
0.741531185599394,
0.949107912342759 };
const Real TabulatedGaussLegendre::w7[4] = { 0.417959183673469,
0.381830050505119,
0.279705391489277,
0.129484966168870 };
const Size TabulatedGaussLegendre::n7 = 4;
/* order 12 */
const Real TabulatedGaussLegendre::x12[6] = { 0.125233408511469,
0.367831498998180,
0.587317954286617,
0.769902674194305,
0.904117256370475,
0.981560634246719 };
const Real TabulatedGaussLegendre::w12[6] = { 0.249147045813403,
0.233492536538355,
0.203167426723066,
0.160078328543346,
0.106939325995318,
0.047175336386512 };
const Size TabulatedGaussLegendre::n12 = 6;
/* order 20 */
const Real TabulatedGaussLegendre::x20[10] = { 0.076526521133497,
0.227785851141645,
0.373706088715420,
0.510867001950827,
0.636053680726515,
0.746331906460151,
0.839116971822219,
0.912234428251326,
0.963971927277914,
0.993128599185095 };
const Real TabulatedGaussLegendre::w20[10] = { 0.152753387130726,
0.149172986472604,
0.142096109318382,
0.131688638449177,
0.118194531961518,
0.101930119817240,
0.083276741576704,
0.062672048334109,
0.040601429800387,
0.017614007139152 };
const Size TabulatedGaussLegendre::n20 = 10;
}
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