// (C) 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_CHEBYSHEV_HPP
|
#define BOOST_MATH_SPECIAL_CHEBYSHEV_HPP
|
#include <cmath>
|
#include <boost/math/policies/error_handling.hpp>
|
#include <boost/math/constants/constants.hpp>
|
|
#if (__cplusplus > 201103) || (defined(_CPPLIB_VER) && (_CPPLIB_VER >= 610))
|
# define BOOST_MATH_CHEB_USE_STD_ACOSH
|
#endif
|
|
#ifndef BOOST_MATH_CHEB_USE_STD_ACOSH
|
# include <boost/math/special_functions/acosh.hpp>
|
#endif
|
|
namespace boost { namespace math {
|
|
template<class T1, class T2, class T3>
|
inline typename tools::promote_args<T1, T2, T3>::type chebyshev_next(T1 const & x, T2 const & Tn, T3 const & Tn_1)
|
{
|
return 2*x*Tn - Tn_1;
|
}
|
|
namespace detail {
|
|
template<class Real, bool second, class Policy>
|
inline Real chebyshev_imp(unsigned n, Real const & x, const Policy&)
|
{
|
#ifdef BOOST_MATH_CHEB_USE_STD_ACOSH
|
using std::acosh;
|
#define BOOST_MATH_ACOSH_POLICY
|
#else
|
using boost::math::acosh;
|
#define BOOST_MATH_ACOSH_POLICY , Policy()
|
#endif
|
using std::cosh;
|
using std::pow;
|
using std::sqrt;
|
Real T0 = 1;
|
Real T1;
|
if (second)
|
{
|
if (x > 1 || x < -1)
|
{
|
Real t = sqrt(x*x -1);
|
return static_cast<Real>((pow(x+t, (int)(n+1)) - pow(x-t, (int)(n+1)))/(2*t));
|
}
|
T1 = 2*x;
|
}
|
else
|
{
|
if (x > 1)
|
{
|
return cosh(n*acosh(x BOOST_MATH_ACOSH_POLICY));
|
}
|
if (x < -1)
|
{
|
if (n & 1)
|
{
|
return -cosh(n*acosh(-x BOOST_MATH_ACOSH_POLICY));
|
}
|
else
|
{
|
return cosh(n*acosh(-x BOOST_MATH_ACOSH_POLICY));
|
}
|
}
|
T1 = x;
|
}
|
|
if (n == 0)
|
{
|
return T0;
|
}
|
|
unsigned l = 1;
|
while(l < n)
|
{
|
std::swap(T0, T1);
|
T1 = boost::math::chebyshev_next(x, T0, T1);
|
++l;
|
}
|
return T1;
|
}
|
} // namespace detail
|
|
template <class Real, class Policy>
|
inline typename tools::promote_args<Real>::type
|
chebyshev_t(unsigned n, Real const & x, const Policy&)
|
{
|
typedef typename tools::promote_args<Real>::type result_type;
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
typedef typename policies::normalise<
|
Policy,
|
policies::promote_float<false>,
|
policies::promote_double<false>,
|
policies::discrete_quantile<>,
|
policies::assert_undefined<> >::type forwarding_policy;
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::chebyshev_imp<value_type, false>(n, static_cast<value_type>(x), forwarding_policy()), "boost::math::chebyshev_t<%1%>(unsigned, %1%)");
|
}
|
|
template<class Real>
|
inline typename tools::promote_args<Real>::type chebyshev_t(unsigned n, Real const & x)
|
{
|
return chebyshev_t(n, x, policies::policy<>());
|
}
|
|
template <class Real, class Policy>
|
inline typename tools::promote_args<Real>::type
|
chebyshev_u(unsigned n, Real const & x, const Policy&)
|
{
|
typedef typename tools::promote_args<Real>::type result_type;
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
typedef typename policies::normalise<
|
Policy,
|
policies::promote_float<false>,
|
policies::promote_double<false>,
|
policies::discrete_quantile<>,
|
policies::assert_undefined<> >::type forwarding_policy;
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::chebyshev_imp<value_type, true>(n, static_cast<value_type>(x), forwarding_policy()), "boost::math::chebyshev_u<%1%>(unsigned, %1%)");
|
}
|
|
template<class Real>
|
inline typename tools::promote_args<Real>::type chebyshev_u(unsigned n, Real const & x)
|
{
|
return chebyshev_u(n, x, policies::policy<>());
|
}
|
|
template <class Real, class Policy>
|
inline typename tools::promote_args<Real>::type
|
chebyshev_t_prime(unsigned n, Real const & x, const Policy&)
|
{
|
typedef typename tools::promote_args<Real>::type result_type;
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
typedef typename policies::normalise<
|
Policy,
|
policies::promote_float<false>,
|
policies::promote_double<false>,
|
policies::discrete_quantile<>,
|
policies::assert_undefined<> >::type forwarding_policy;
|
if (n == 0)
|
{
|
return result_type(0);
|
}
|
return policies::checked_narrowing_cast<result_type, Policy>(n * detail::chebyshev_imp<value_type, true>(n - 1, static_cast<value_type>(x), forwarding_policy()), "boost::math::chebyshev_t_prime<%1%>(unsigned, %1%)");
|
}
|
|
template<class Real>
|
inline typename tools::promote_args<Real>::type chebyshev_t_prime(unsigned n, Real const & x)
|
{
|
return chebyshev_t_prime(n, x, policies::policy<>());
|
}
|
|
/*
|
* This is Algorithm 3.1 of
|
* Gil, Amparo, Javier Segura, and Nico M. Temme.
|
* Numerical methods for special functions.
|
* Society for Industrial and Applied Mathematics, 2007.
|
* https://www.siam.org/books/ot99/OT99SampleChapter.pdf
|
* However, our definition of c0 differs by a factor of 1/2, as stated in the docs. . .
|
*/
|
template<class Real, class T2>
|
inline Real chebyshev_clenshaw_recurrence(const Real* const c, size_t length, const T2& x)
|
{
|
using boost::math::constants::half;
|
if (length < 2)
|
{
|
if (length == 0)
|
{
|
return 0;
|
}
|
return c[0]/2;
|
}
|
Real b2 = 0;
|
Real b1 = c[length -1];
|
for(size_t j = length - 2; j >= 1; --j)
|
{
|
Real tmp = 2*x*b1 - b2 + c[j];
|
b2 = b1;
|
b1 = tmp;
|
}
|
return x*b1 - b2 + half<Real>()*c[0];
|
}
|
|
|
|
namespace detail {
|
template<class Real>
|
inline Real unchecked_chebyshev_clenshaw_recurrence(const Real* const c, size_t length, const Real & a, const Real & b, const Real& x)
|
{
|
Real t;
|
Real u;
|
// This cutoff is not super well defined, but it's a good estimate.
|
// See "An Error Analysis of the Modified Clenshaw Method for Evaluating Chebyshev and Fourier Series"
|
// J. OLIVER, IMA Journal of Applied Mathematics, Volume 20, Issue 3, November 1977, Pages 379–391
|
// https://doi.org/10.1093/imamat/20.3.379
|
const Real cutoff = 0.6;
|
if (x - a < b - x)
|
{
|
u = 2*(x-a)/(b-a);
|
t = u - 1;
|
if (t > -cutoff)
|
{
|
Real b2 = 0;
|
Real b1 = c[length -1];
|
for(size_t j = length - 2; j >= 1; --j)
|
{
|
Real tmp = 2*t*b1 - b2 + c[j];
|
b2 = b1;
|
b1 = tmp;
|
}
|
return t*b1 - b2 + c[0]/2;
|
}
|
else
|
{
|
Real b = c[length -1];
|
Real d = b;
|
Real b2 = 0;
|
for (size_t r = length - 2; r >= 1; --r)
|
{
|
d = 2*u*b - d + c[r];
|
b2 = b;
|
b = d - b;
|
}
|
return t*b - b2 + c[0]/2;
|
}
|
}
|
else
|
{
|
u = -2*(b-x)/(b-a);
|
t = u + 1;
|
if (t < cutoff)
|
{
|
Real b2 = 0;
|
Real b1 = c[length -1];
|
for(size_t j = length - 2; j >= 1; --j)
|
{
|
Real tmp = 2*t*b1 - b2 + c[j];
|
b2 = b1;
|
b1 = tmp;
|
}
|
return t*b1 - b2 + c[0]/2;
|
}
|
else
|
{
|
Real b = c[length -1];
|
Real d = b;
|
Real b2 = 0;
|
for (size_t r = length - 2; r >= 1; --r)
|
{
|
d = 2*u*b + d + c[r];
|
b2 = b;
|
b = d + b;
|
}
|
return t*b - b2 + c[0]/2;
|
}
|
}
|
}
|
|
} // namespace detail
|
|
template<class Real>
|
inline Real chebyshev_clenshaw_recurrence(const Real* const c, size_t length, const Real & a, const Real & b, const Real& x)
|
{
|
if (x < a || x > b)
|
{
|
throw std::domain_error("x in [a, b] is required.");
|
}
|
if (length < 2)
|
{
|
if (length == 0)
|
{
|
return 0;
|
}
|
return c[0]/2;
|
}
|
return detail::unchecked_chebyshev_clenshaw_recurrence(c, length, a, b, x);
|
}
|
|
|
}}
|
#endif
|
