Initial commit
This commit is contained in:
Tibeuleu
235
code_ordonnee/boost/math/special_functions/legendre_stieltjes.hpp
Executable file
235
code_ordonnee/boost/math/special_functions/legendre_stieltjes.hpp
Executable file
@@ -0,0 +1,235 @@
|
||||
// Copyright Nick Thompson 2017.
|
||||
// Use, modification and distribution are subject to 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)
|
||||
|
||||
#ifndef BOOST_MATH_SPECIAL_LEGENDRE_STIELTJES_HPP
|
||||
#define BOOST_MATH_SPECIAL_LEGENDRE_STIELTJES_HPP
|
||||
|
||||
/*
|
||||
* Constructs the Legendre-Stieltjes polynomial of degree m.
|
||||
* The Legendre-Stieltjes polynomials are used to create extensions for Gaussian quadratures,
|
||||
* commonly called "Gauss-Konrod" quadratures.
|
||||
*
|
||||
* References:
|
||||
* Patterson, TNL. "The optimum addition of points to quadrature formulae." Mathematics of Computation 22.104 (1968): 847-856.
|
||||
*/
|
||||
|
||||
#include <iostream>
|
||||
#include <vector>
|
||||
#include <boost/math/tools/roots.hpp>
|
||||
#include <boost/math/special_functions/legendre.hpp>
|
||||
|
||||
namespace boost{
|
||||
namespace math{
|
||||
|
||||
template<class Real>
|
||||
class legendre_stieltjes
|
||||
{
|
||||
public:
|
||||
legendre_stieltjes(size_t m)
|
||||
{
|
||||
if (m == 0)
|
||||
{
|
||||
throw std::domain_error("The Legendre-Stieltjes polynomial is defined for order m > 0.\n");
|
||||
}
|
||||
m_m = static_cast<int>(m);
|
||||
std::ptrdiff_t n = m - 1;
|
||||
std::ptrdiff_t q;
|
||||
std::ptrdiff_t r;
|
||||
bool odd = n & 1;
|
||||
if (odd)
|
||||
{
|
||||
q = 1;
|
||||
r = (n-1)/2 + 2;
|
||||
}
|
||||
else
|
||||
{
|
||||
q = 0;
|
||||
r = n/2 + 1;
|
||||
}
|
||||
m_a.resize(r + 1);
|
||||
// We'll keep the ones-based indexing at the cost of storing a superfluous element
|
||||
// so that we can follow Patterson's notation exactly.
|
||||
m_a[r] = static_cast<Real>(1);
|
||||
// Make sure using the zero index is a bug:
|
||||
m_a[0] = std::numeric_limits<Real>::quiet_NaN();
|
||||
|
||||
for (std::ptrdiff_t k = 1; k < r; ++k)
|
||||
{
|
||||
Real ratio = 1;
|
||||
m_a[r - k] = 0;
|
||||
for (std::ptrdiff_t i = r + 1 - k; i <= r; ++i)
|
||||
{
|
||||
// See Patterson, equation 12
|
||||
std::ptrdiff_t num = (n - q + 2*(i + k - 1))*(n + q + 2*(k - i + 1))*(n-1-q+2*(i-k))*(2*(k+i-1) -1 -q -n);
|
||||
std::ptrdiff_t den = (n - q + 2*(i - k))*(2*(k + i - 1) - q - n)*(n + 1 + q + 2*(k - i))*(n - 1 - q + 2*(i + k));
|
||||
ratio *= static_cast<Real>(num)/static_cast<Real>(den);
|
||||
m_a[r - k] -= ratio*m_a[i];
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
Real norm_sq() const
|
||||
{
|
||||
Real t = 0;
|
||||
bool odd = m_m & 1;
|
||||
for (size_t i = 1; i < m_a.size(); ++i)
|
||||
{
|
||||
if(odd)
|
||||
{
|
||||
t += 2*m_a[i]*m_a[i]/static_cast<Real>(4*i-1);
|
||||
}
|
||||
else
|
||||
{
|
||||
t += 2*m_a[i]*m_a[i]/static_cast<Real>(4*i-3);
|
||||
}
|
||||
}
|
||||
return t;
|
||||
}
|
||||
|
||||
|
||||
Real operator()(Real x) const
|
||||
{
|
||||
// Trivial implementation:
|
||||
// Em += m_a[i]*legendre_p(2*i - 1, x); m odd
|
||||
// Em += m_a[i]*legendre_p(2*i - 2, x); m even
|
||||
size_t r = m_a.size() - 1;
|
||||
Real p0 = 1;
|
||||
Real p1 = x;
|
||||
|
||||
Real Em;
|
||||
bool odd = m_m & 1;
|
||||
if (odd)
|
||||
{
|
||||
Em = m_a[1]*p1;
|
||||
}
|
||||
else
|
||||
{
|
||||
Em = m_a[1]*p0;
|
||||
}
|
||||
|
||||
unsigned n = 1;
|
||||
for (size_t i = 2; i <= r; ++i)
|
||||
{
|
||||
std::swap(p0, p1);
|
||||
p1 = boost::math::legendre_next(n, x, p0, p1);
|
||||
++n;
|
||||
if (!odd)
|
||||
{
|
||||
Em += m_a[i]*p1;
|
||||
}
|
||||
std::swap(p0, p1);
|
||||
p1 = boost::math::legendre_next(n, x, p0, p1);
|
||||
++n;
|
||||
if(odd)
|
||||
{
|
||||
Em += m_a[i]*p1;
|
||||
}
|
||||
}
|
||||
return Em;
|
||||
}
|
||||
|
||||
|
||||
Real prime(Real x) const
|
||||
{
|
||||
Real Em_prime = 0;
|
||||
|
||||
for (size_t i = 1; i < m_a.size(); ++i)
|
||||
{
|
||||
if(m_m & 1)
|
||||
{
|
||||
Em_prime += m_a[i]*detail::legendre_p_prime_imp(static_cast<unsigned>(2*i - 1), x, policies::policy<>());
|
||||
}
|
||||
else
|
||||
{
|
||||
Em_prime += m_a[i]*detail::legendre_p_prime_imp(static_cast<unsigned>(2*i - 2), x, policies::policy<>());
|
||||
}
|
||||
}
|
||||
return Em_prime;
|
||||
}
|
||||
|
||||
std::vector<Real> zeros() const
|
||||
{
|
||||
using boost::math::constants::half;
|
||||
|
||||
std::vector<Real> stieltjes_zeros;
|
||||
std::vector<Real> legendre_zeros = legendre_p_zeros<Real>(m_m - 1);
|
||||
int k;
|
||||
if (m_m & 1)
|
||||
{
|
||||
stieltjes_zeros.resize(legendre_zeros.size() + 1, std::numeric_limits<Real>::quiet_NaN());
|
||||
stieltjes_zeros[0] = 0;
|
||||
k = 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
stieltjes_zeros.resize(legendre_zeros.size(), std::numeric_limits<Real>::quiet_NaN());
|
||||
k = 0;
|
||||
}
|
||||
|
||||
while (k < (int)stieltjes_zeros.size())
|
||||
{
|
||||
Real lower_bound;
|
||||
Real upper_bound;
|
||||
if (m_m & 1)
|
||||
{
|
||||
lower_bound = legendre_zeros[k - 1];
|
||||
if (k == (int)legendre_zeros.size())
|
||||
{
|
||||
upper_bound = 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
upper_bound = legendre_zeros[k];
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
lower_bound = legendre_zeros[k];
|
||||
if (k == (int)legendre_zeros.size() - 1)
|
||||
{
|
||||
upper_bound = 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
upper_bound = legendre_zeros[k+1];
|
||||
}
|
||||
}
|
||||
|
||||
// The root bracketing is not very tight; to keep weird stuff from happening
|
||||
// in the Newton's method, let's tighten up the tolerance using a few bisections.
|
||||
boost::math::tools::eps_tolerance<Real> tol(6);
|
||||
auto g = [&](Real t) { return this->operator()(t); };
|
||||
auto p = boost::math::tools::bisect(g, lower_bound, upper_bound, tol);
|
||||
|
||||
Real x_nk_guess = p.first + (p.second - p.first)*half<Real>();
|
||||
boost::uintmax_t number_of_iterations = 500;
|
||||
|
||||
auto f = [&] (Real x) { Real Pn = this->operator()(x);
|
||||
Real Pn_prime = this->prime(x);
|
||||
return std::pair<Real, Real>(Pn, Pn_prime); };
|
||||
|
||||
const Real x_nk = boost::math::tools::newton_raphson_iterate(f, x_nk_guess,
|
||||
p.first, p.second,
|
||||
2*std::numeric_limits<Real>::digits10,
|
||||
number_of_iterations);
|
||||
|
||||
BOOST_ASSERT(p.first < x_nk);
|
||||
BOOST_ASSERT(x_nk < p.second);
|
||||
stieltjes_zeros[k] = x_nk;
|
||||
++k;
|
||||
}
|
||||
return stieltjes_zeros;
|
||||
}
|
||||
|
||||
private:
|
||||
// Coefficients of Legendre expansion
|
||||
std::vector<Real> m_a;
|
||||
int m_m;
|
||||
};
|
||||
|
||||
}}
|
||||
#endif
|
||||
Reference in New Issue
Block a user