math/special_functions/sinc.hpp

0001 //  boost sinc.hpp header file
0002
0003 //  (C) Copyright Hubert Holin 2001.
0004 //  Distributed under the Boost Software License, Version 1.0. (See
0005 //  accompanying file LICENSE_1_0.txt or copy at
0006 //  http://www.boost.org/LICENSE_1_0.txt)
0007
0008 // See http://www.boost.org for updates, documentation, and revision history.
0009
0010 #ifndef BOOST_SINC_HPP
0011 #define BOOST_SINC_HPP
0012
0013
0014 #ifdef _MSC_VER
0015 #pragma once
0016 #endif
0017
0018 #include <boost/math/tools/config.hpp>
0019 #include <boost/math/tools/precision.hpp>
0020 #include <boost/math/policies/policy.hpp>
0021 #include <boost/math/special_functions/math_fwd.hpp>
0022 #include <limits>
0023 #include <string>
0024 #include <stdexcept>
0025 #include <cmath>
0026
0027 // These are the the "Sinus Cardinal" functions.
0028
0029 namespace boost
0030 {
0031     namespace math
0032     {
0033        namespace detail
0034        {
0035         // This is the "Sinus Cardinal" of index Pi.
0036
0037         template<typename T>
0038         inline T    sinc_pi_imp(const T x)
0039         {
0040             BOOST_MATH_STD_USING
0041
0042             if    (abs(x) >= 3.3 * tools::forth_root_epsilon<T>())
0043             {
0044                 return(sin(x)/x);
0045             }
0046             else
0047             {
0048                 // |x| < (eps*120)^(1/4)
0049                 return 1 - x * x / 6;
0050             }
0051         }
0052
0053        } // namespace detail
0054
0055        template <class T>
0056        inline typename tools::promote_args<T>::type sinc_pi(T x)
0057        {
0058           typedef typename tools::promote_args<T>::type result_type;
0059           return detail::sinc_pi_imp(static_cast<result_type>(x));
0060        }
0061
0062        template <class T, class Policy>
0063        inline typename tools::promote_args<T>::type sinc_pi(T x, const Policy&)
0064        {
0065           typedef typename tools::promote_args<T>::type result_type;
0066           return detail::sinc_pi_imp(static_cast<result_type>(x));
0067        }
0068
0069         template<typename T, template<typename> class U>
0070         inline U<T>    sinc_pi(const U<T> x)
0071         {
0072             BOOST_MATH_STD_USING
0073             using    ::std::numeric_limits;
0074
0075             T const    taylor_0_bound = tools::epsilon<T>();
0076             T const    taylor_2_bound = tools::root_epsilon<T>();
0077             T const    taylor_n_bound = tools::forth_root_epsilon<T>();
0078
0079             if    (abs(x) >= taylor_n_bound)
0080             {
0081                 return(sin(x)/x);
0082             }
0083             else
0084             {
0085                 // approximation by taylor series in x at 0 up to order 0
0086 #ifdef __MWERKS__
0087                 U<T>    result = static_cast<U<T> >(1);
0088 #else
0089                 U<T>    result = U<T>(1);
0090 #endif
0091
0092                 if    (abs(x) >= taylor_0_bound)
0093                 {
0094                     U<T>    x2 = x*x;
0095
0096                     // approximation by taylor series in x at 0 up to order 2
0097                     result -= x2/static_cast<T>(6);
0098
0099                     if    (abs(x) >= taylor_2_bound)
0100                     {
0101                         // approximation by taylor series in x at 0 up to order 4
0102                         result += (x2*x2)/static_cast<T>(120);
0103                     }
0104                 }
0105
0106                 return(result);
0107             }
0108         }
0109
0110         template<typename T, template<typename> class U, class Policy>
0111         inline U<T>    sinc_pi(const U<T> x, const Policy&)
0112         {
0113            return sinc_pi(x);
0114         }
0115     }
0116 }
0117
0118 #endif /* BOOST_SINC_HPP */
0119