236 lines
6.7 KiB
C++
236 lines
6.7 KiB
C++
// Copyright Nick Thompson 2017.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SPECIAL_LEGENDRE_STIELTJES_HPP
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#define BOOST_MATH_SPECIAL_LEGENDRE_STIELTJES_HPP
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/*
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* Constructs the Legendre-Stieltjes polynomial of degree m.
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* The Legendre-Stieltjes polynomials are used to create extensions for Gaussian quadratures,
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* commonly called "Gauss-Konrod" quadratures.
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*
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* References:
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* Patterson, TNL. "The optimum addition of points to quadrature formulae." Mathematics of Computation 22.104 (1968): 847-856.
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*/
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#include <iostream>
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#include <vector>
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#include <boost/math/tools/roots.hpp>
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#include <boost/math/special_functions/legendre.hpp>
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namespace boost{
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namespace math{
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template<class Real>
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class legendre_stieltjes
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{
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public:
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legendre_stieltjes(size_t m)
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{
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if (m == 0)
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{
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throw std::domain_error("The Legendre-Stieltjes polynomial is defined for order m > 0.\n");
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}
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m_m = static_cast<int>(m);
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std::ptrdiff_t n = m - 1;
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std::ptrdiff_t q;
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std::ptrdiff_t r;
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bool odd = n & 1;
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if (odd)
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{
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q = 1;
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r = (n-1)/2 + 2;
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}
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else
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{
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q = 0;
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r = n/2 + 1;
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}
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m_a.resize(r + 1);
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// We'll keep the ones-based indexing at the cost of storing a superfluous element
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// so that we can follow Patterson's notation exactly.
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m_a[r] = static_cast<Real>(1);
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// Make sure using the zero index is a bug:
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m_a[0] = std::numeric_limits<Real>::quiet_NaN();
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for (std::ptrdiff_t k = 1; k < r; ++k)
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{
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Real ratio = 1;
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m_a[r - k] = 0;
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for (std::ptrdiff_t i = r + 1 - k; i <= r; ++i)
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{
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// See Patterson, equation 12
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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);
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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));
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ratio *= static_cast<Real>(num)/static_cast<Real>(den);
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m_a[r - k] -= ratio*m_a[i];
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}
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}
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}
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Real norm_sq() const
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{
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Real t = 0;
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bool odd = m_m & 1;
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for (size_t i = 1; i < m_a.size(); ++i)
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{
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if(odd)
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{
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t += 2*m_a[i]*m_a[i]/static_cast<Real>(4*i-1);
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}
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else
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{
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t += 2*m_a[i]*m_a[i]/static_cast<Real>(4*i-3);
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}
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}
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return t;
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}
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Real operator()(Real x) const
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{
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// Trivial implementation:
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// Em += m_a[i]*legendre_p(2*i - 1, x); m odd
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// Em += m_a[i]*legendre_p(2*i - 2, x); m even
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size_t r = m_a.size() - 1;
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Real p0 = 1;
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Real p1 = x;
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Real Em;
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bool odd = m_m & 1;
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if (odd)
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{
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Em = m_a[1]*p1;
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}
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else
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{
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Em = m_a[1]*p0;
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}
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unsigned n = 1;
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for (size_t i = 2; i <= r; ++i)
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{
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std::swap(p0, p1);
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p1 = boost::math::legendre_next(n, x, p0, p1);
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++n;
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if (!odd)
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{
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Em += m_a[i]*p1;
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}
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std::swap(p0, p1);
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p1 = boost::math::legendre_next(n, x, p0, p1);
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++n;
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if(odd)
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{
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Em += m_a[i]*p1;
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}
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}
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return Em;
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}
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Real prime(Real x) const
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{
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Real Em_prime = 0;
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for (size_t i = 1; i < m_a.size(); ++i)
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{
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if(m_m & 1)
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{
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Em_prime += m_a[i]*detail::legendre_p_prime_imp(static_cast<unsigned>(2*i - 1), x, policies::policy<>());
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}
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else
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{
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Em_prime += m_a[i]*detail::legendre_p_prime_imp(static_cast<unsigned>(2*i - 2), x, policies::policy<>());
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}
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}
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return Em_prime;
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}
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std::vector<Real> zeros() const
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{
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using boost::math::constants::half;
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std::vector<Real> stieltjes_zeros;
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std::vector<Real> legendre_zeros = legendre_p_zeros<Real>(m_m - 1);
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int k;
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if (m_m & 1)
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{
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stieltjes_zeros.resize(legendre_zeros.size() + 1, std::numeric_limits<Real>::quiet_NaN());
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stieltjes_zeros[0] = 0;
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k = 1;
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}
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else
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{
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stieltjes_zeros.resize(legendre_zeros.size(), std::numeric_limits<Real>::quiet_NaN());
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k = 0;
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}
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while (k < (int)stieltjes_zeros.size())
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{
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Real lower_bound;
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Real upper_bound;
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if (m_m & 1)
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{
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lower_bound = legendre_zeros[k - 1];
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if (k == (int)legendre_zeros.size())
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{
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upper_bound = 1;
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}
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else
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{
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upper_bound = legendre_zeros[k];
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}
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}
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else
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{
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lower_bound = legendre_zeros[k];
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if (k == (int)legendre_zeros.size() - 1)
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{
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upper_bound = 1;
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}
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else
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{
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upper_bound = legendre_zeros[k+1];
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}
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}
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// The root bracketing is not very tight; to keep weird stuff from happening
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// in the Newton's method, let's tighten up the tolerance using a few bisections.
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boost::math::tools::eps_tolerance<Real> tol(6);
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auto g = [&](Real t) { return this->operator()(t); };
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auto p = boost::math::tools::bisect(g, lower_bound, upper_bound, tol);
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Real x_nk_guess = p.first + (p.second - p.first)*half<Real>();
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std::uintmax_t number_of_iterations = 500;
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auto f = [&] (Real x) { Real Pn = this->operator()(x);
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Real Pn_prime = this->prime(x);
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return std::pair<Real, Real>(Pn, Pn_prime); };
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const Real x_nk = boost::math::tools::newton_raphson_iterate(f, x_nk_guess,
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p.first, p.second,
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tools::digits<Real>(),
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number_of_iterations);
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BOOST_MATH_ASSERT(p.first < x_nk);
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BOOST_MATH_ASSERT(x_nk < p.second);
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stieltjes_zeros[k] = x_nk;
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++k;
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}
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return stieltjes_zeros;
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}
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private:
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// Coefficients of Legendre expansion
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std::vector<Real> m_a;
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int m_m;
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};
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}}
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#endif
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