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boost/math/special_functions/chebyshev.hpp

//  (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 <type_traits>
#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/math/tools/promotion.hpp>
#include <boost/math/tools/throw_exception.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 tools::promote_args_t<T1, T2, T3> chebyshev_next(T1 const & x, T2 const & Tn, T3 const & Tn_1)
{
    return 2*x*Tn - Tn_1;
}

namespace detail {

// https://stackoverflow.com/questions/5625431/efficient-way-to-compute-pq-exponentiation-where-q-is-an-integer
template <typename T, typename std::enable_if<std::is_arithmetic<T>::value, bool>::type = true>
T expt(T p, unsigned q)
{
    T r = 1;

    while (q != 0) {
        if (q % 2 == 1) {    // q is odd
            r *= p;
            q--;
        }
        p *= p;
        q /= 2;
    }

    return r;
}

template <typename T, typename std::enable_if<!std::is_arithmetic<T>::value, bool>::type = true>
T expt(T p, unsigned q)
{
    using std::pow;
    return pow(p, static_cast<int>(q));
}

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;

    BOOST_MATH_IF_CONSTEXPR (second)
    {
        if (x > 1 || x < -1)
        {
            Real t = sqrt(x*x -1);
            return static_cast<Real>((expt(static_cast<Real>(x+t), n+1) - expt(static_cast<Real>(x-t), 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 = static_cast<Real>(boost::math::chebyshev_next(x, T0, T1));
       ++l;
    }
    return T1;
}
} // namespace detail

template <class Real, class Policy>
inline tools::promote_args_t<Real> chebyshev_t(unsigned n, Real const & x, const Policy&)
{
   using result_type = tools::promote_args_t<Real>;
   using value_type = typename policies::evaluation<result_type, Policy>::type;
   using forwarding_policy = typename policies::normalise<
                                                            Policy,
                                                            policies::promote_float<false>,
                                                            policies::promote_double<false>,
                                                            policies::discrete_quantile<>,
                                                            policies::assert_undefined<> >::type;

   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 tools::promote_args_t<Real> chebyshev_t(unsigned n, Real const & x)
{
    return chebyshev_t(n, x, policies::policy<>());
}

template <class Real, class Policy>
inline tools::promote_args_t<Real> chebyshev_u(unsigned n, Real const & x, const Policy&)
{
   using result_type = tools::promote_args_t<Real>;
   using value_type = typename policies::evaluation<result_type, Policy>::type;
   using forwarding_policy =  typename policies::normalise<
                                                            Policy,
                                                            policies::promote_float<false>,
                                                            policies::promote_double<false>,
                                                            policies::discrete_quantile<>,
                                                            policies::assert_undefined<> >::type;

   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 tools::promote_args_t<Real> chebyshev_u(unsigned n, Real const & x)
{
    return chebyshev_u(n, x, policies::policy<>());
}

template <class Real, class Policy>
inline tools::promote_args_t<Real> chebyshev_t_prime(unsigned n, Real const & x, const Policy&)
{
   using result_type = tools::promote_args_t<Real>;
   using value_type = typename policies::evaluation<result_type, Policy>::type;
   using forwarding_policy = typename policies::normalise<
                                                            Policy,
                                                            policies::promote_float<false>,
                                                            policies::promote_double<false>,
                                                            policies::discrete_quantile<>,
                                                            policies::assert_undefined<> >::type;
   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 tools::promote_args_t<Real> 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 auto cutoff = static_cast<Real>(0.6L);
    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 b1 = c[length - 1];
            Real d = b1;
            Real b2 = 0;
            for (size_t r = length - 2; r >= 1; --r)
            {
                d = 2*u*b1 - d + c[r];
                b2 = b1;
                b1 = d - b1;
            }
            return t*b1 - 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 b1 = c[length - 1];
            Real d = b1;
            Real b2 = 0;
            for (size_t r = length - 2; r >= 1; --r)
            {
                d = 2*u*b1 + d + c[r];
                b2 = b1;
                b1 = d + b1;
            }
            return t*b1 - 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)
    {
       BOOST_MATH_THROW_EXCEPTION(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);
}

}} // Namespace boost::math

#endif // BOOST_MATH_SPECIAL_CHEBYSHEV_HPP