FM / math

/usr/include/boost/math

Name	Size	Mode	Actions
bindings/	-	0755	rm
complex/	-	0755	rm
concepts/	-	0755	rm
constants/	-	0755	rm
cstdfloat/	-	0755	rm
differentiation/	-	0755	rm
distributions/	-	0755	rm
filters/	-	0755	rm
interpolators/	-	0755	rm
policies/	-	0755	rm
quadrature/	-	0755	rm
special_functions/	-	0755	rm
statistics/	-	0755	rm
tools/	-	0755	rm
common_factor.hpp	631	0644	edit dl rm
common_factor_ct.hpp	840	0644	edit dl rm
common_factor_rt.hpp	789	0644	edit dl rm
complex.hpp	971	0644	edit dl rm
distributions.hpp	2407	0644	edit dl rm
octonion.hpp	246465	0644	edit dl rm
quaternion.hpp	49132	0644	edit dl rm
special_functions.hpp	4239	0644	edit dl rm
tr1.hpp	78049	0644	edit dl rm
tr1_c_macros.ipp	14195	0644	edit dl rm

Edit: /usr/include/boost/math/quaternion.hpp (49132B)

//  boost quaternion.hpp header file

//  (C) Copyright Hubert Holin 2001.
//  Distributed under 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)

// See http://www.boost.org for updates, documentation, and revision history.

#ifndef BOOST_QUATERNION_HPP
#define BOOST_QUATERNION_HPP

#include <boost/config.hpp> // for BOOST_NO_STD_LOCALE
#include <boost/math_fwd.hpp>
#include <boost/detail/workaround.hpp>
#include <boost/type_traits/is_convertible.hpp>
#include <boost/utility/enable_if.hpp>
#ifndef    BOOST_NO_STD_LOCALE
#  include <locale>                                    // for the "<<" operator
#endif /* BOOST_NO_STD_LOCALE */

#include <complex>
#include <iosfwd>                                    // for the "<<" and ">>" operators
#include <sstream>                                    // for the "<<" operator

#include <boost/math/special_functions/sinc.hpp>    // for the Sinus cardinal
#include <boost/math/special_functions/sinhc.hpp>    // for the Hyperbolic Sinus cardinal
#include <boost/math/tools/cxx03_warn.hpp>

#if defined(BOOST_NO_CXX11_NOEXCEPT) || defined(BOOST_NO_CXX11_RVALUE_REFERENCES) || defined(BOOST_NO_SFINAE_EXPR)
#include <boost/type_traits/is_pod.hpp>
#endif

namespace boost
{
   namespace math
   {

namespace detail {

#if !defined(BOOST_NO_CXX11_NOEXCEPT) && !defined(BOOST_NO_CXX11_RVALUE_REFERENCES) && !defined(BOOST_NO_SFINAE_EXPR)

template <class T>
         struct is_trivial_arithmetic_type_imp
         {
            typedef boost::integral_constant<bool,
               noexcept(std::declval<T&>() += std::declval<T>())
               && noexcept(std::declval<T&>() -= std::declval<T>())
               && noexcept(std::declval<T&>() *= std::declval<T>())
               && noexcept(std::declval<T&>() /= std::declval<T>())
            > type;
         };

template <class T>
         struct is_trivial_arithmetic_type : public is_trivial_arithmetic_type_imp<T>::type {};
#else

template <class T>
         struct is_trivial_arithmetic_type : public boost::is_pod<T> {};

#endif

}

#ifndef BOOST_NO_CXX14_CONSTEXPR
      namespace constexpr_detail
      {
         template <class T>
         constexpr void swap(T& a, T& b)
         {
            T t(a);
            a = b;
            b = t;
         }
       }
#endif

template<typename T>
        class quaternion
        {
        public:
            
            typedef T value_type;
            
            
            // constructor for H seen as R^4
            // (also default constructor)
            
            BOOST_CONSTEXPR explicit            quaternion( T const & requested_a = T(),
                                            T const & requested_b = T(),
                                            T const & requested_c = T(),
                                            T const & requested_d = T())
            :   a(requested_a),
                b(requested_b),
                c(requested_c),
                d(requested_d)
            {
                // nothing to do!
            }
            
            
            // constructor for H seen as C^2
                
            BOOST_CONSTEXPR explicit            quaternion( ::std::complex<T> const & z0,
                                            ::std::complex<T> const & z1 = ::std::complex<T>())
            :   a(z0.real()),
                b(z0.imag()),
                c(z1.real()),
                d(z1.imag())
            {
                // nothing to do!
            }
            
            
            // UNtemplated copy constructor
            BOOST_CONSTEXPR quaternion(quaternion const & a_recopier)
               : a(a_recopier.R_component_1()),
               b(a_recopier.R_component_2()),
               c(a_recopier.R_component_3()),
               d(a_recopier.R_component_4()) {}
#ifndef BOOST_NO_CXX11_RVALUE_REFERENCES
            BOOST_CONSTEXPR quaternion(quaternion && a_recopier)
               : a(std::move(a_recopier.R_component_1())),
               b(std::move(a_recopier.R_component_2())),
               c(std::move(a_recopier.R_component_3())),
               d(std::move(a_recopier.R_component_4())) {}
#endif
            
            // templated copy constructor
            
            template<typename X>
            BOOST_CONSTEXPR explicit            quaternion(quaternion<X> const & a_recopier)
            :   a(static_cast<T>(a_recopier.R_component_1())),
                b(static_cast<T>(a_recopier.R_component_2())),
                c(static_cast<T>(a_recopier.R_component_3())),
                d(static_cast<T>(a_recopier.R_component_4()))
            {
                // nothing to do!
            }
            
            
            // destructor
            // (this is taken care of by the compiler itself)
            
            
            // accessors
            //
            // Note:    Like complex number, quaternions do have a meaningful notion of "real part",
            //            but unlike them there is no meaningful notion of "imaginary part".
            //            Instead there is an "unreal part" which itself is a quaternion, and usually
            //            nothing simpler (as opposed to the complex number case).
            //            However, for practicality, there are accessors for the other components
            //            (these are necessary for the templated copy constructor, for instance).
            
            BOOST_CONSTEXPR T real() const
            {
               return(a);
            }

BOOST_CONSTEXPR quaternion<T> unreal() const
            {
               return(quaternion<T>(static_cast<T>(0), b, c, d));
            }

BOOST_CONSTEXPR T R_component_1() const
            {
               return(a);
            }

BOOST_CONSTEXPR T R_component_2() const
            {
               return(b);
            }

BOOST_CONSTEXPR T R_component_3() const
            {
               return(c);
            }

BOOST_CONSTEXPR T R_component_4() const
            {
               return(d);
            }

BOOST_CONSTEXPR ::std::complex<T> C_component_1() const
            {
               return(::std::complex<T>(a, b));
            }

BOOST_CONSTEXPR ::std::complex<T> C_component_2() const
            {
               return(::std::complex<T>(c, d));
            }

BOOST_CXX14_CONSTEXPR void swap(quaternion& o)
            {
#ifndef BOOST_NO_CXX14_CONSTEXPR
               using constexpr_detail::swap;
#else
               using std::swap;
#endif
               swap(a, o.a);
               swap(b, o.b);
               swap(c, o.c);
               swap(d, o.d);
            }

// assignment operators
            
            template<typename X>
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator = (quaternion<X> const  & a_affecter)
            {
               a = static_cast<T>(a_affecter.R_component_1());
               b = static_cast<T>(a_affecter.R_component_2());
               c = static_cast<T>(a_affecter.R_component_3());
               d = static_cast<T>(a_affecter.R_component_4());

return(*this);
            }

BOOST_CXX14_CONSTEXPR quaternion<T> &        operator = (quaternion<T> const & a_affecter)
            {
               a = a_affecter.a;
               b = a_affecter.b;
               c = a_affecter.c;
               d = a_affecter.d;

return(*this);
            }
#ifndef BOOST_NO_CXX11_RVALUE_REFERENCES
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator = (quaternion<T> && a_affecter)
            {
               a = std::move(a_affecter.a);
               b = std::move(a_affecter.b);
               c = std::move(a_affecter.c);
               d = std::move(a_affecter.d);

return(*this);
            }
#endif
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator = (T const & a_affecter)
            {
               a = a_affecter;

b = c = d = static_cast<T>(0);

return(*this);
            }

BOOST_CXX14_CONSTEXPR quaternion<T> &        operator = (::std::complex<T> const & a_affecter)
            {
               a = a_affecter.real();
               b = a_affecter.imag();

c = d = static_cast<T>(0);

return(*this);
            }

// other assignment-related operators
            //
            // NOTE:    Quaternion multiplication is *NOT* commutative;
            //            symbolically, "q *= rhs;" means "q = q * rhs;"
            //            and "q /= rhs;" means "q = q * inverse_of(rhs);"
            //
            // Note2:   Each operator comes in 2 forms - one for the simple case where
            //          type T throws no exceptions, and one exception-safe version
            //          for the case where it might.
         private:
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_add(T const & rhs, const boost::true_type&)
            {
               a += rhs;
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_add(T const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a + rhs, b, c, d); // exception guard
               swap(result);
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_add(std::complex<T> const & rhs, const boost::true_type&)
            {
               a += std::real(rhs);
               b += std::imag(rhs);
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_add(std::complex<T> const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a + std::real(rhs), b + std::imag(rhs), c, d); // exception guard
               swap(result);
               return *this;
            }
            template <class X>
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_add(quaternion<X> const & rhs, const boost::true_type&)
            {
               a += rhs.R_component_1();
               b += rhs.R_component_2();
               c += rhs.R_component_3();
               d += rhs.R_component_4();
               return *this;
            }
            template <class X>
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_add(quaternion<X> const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a + rhs.R_component_1(), b + rhs.R_component_2(), c + rhs.R_component_3(), d + rhs.R_component_4()); // exception guard
               swap(result);
               return *this;
            }

BOOST_CXX14_CONSTEXPR quaternion<T> &        do_subtract(T const & rhs, const boost::true_type&)
            {
               a -= rhs;
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_subtract(T const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a - rhs, b, c, d); // exception guard
               swap(result);
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_subtract(std::complex<T> const & rhs, const boost::true_type&)
            {
               a -= std::real(rhs);
               b -= std::imag(rhs);
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_subtract(std::complex<T> const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a - std::real(rhs), b - std::imag(rhs), c, d); // exception guard
               swap(result);
               return *this;
            }
            template <class X>
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_subtract(quaternion<X> const & rhs, const boost::true_type&)
            {
               a -= rhs.R_component_1();
               b -= rhs.R_component_2();
               c -= rhs.R_component_3();
               d -= rhs.R_component_4();
               return *this;
            }
            template <class X>
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_subtract(quaternion<X> const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a - rhs.R_component_1(), b - rhs.R_component_2(), c - rhs.R_component_3(), d - rhs.R_component_4()); // exception guard
               swap(result);
               return *this;
            }

BOOST_CXX14_CONSTEXPR quaternion<T> &        do_multiply(T const & rhs, const boost::true_type&)
            {
               a *= rhs;
               b *= rhs;
               c *= rhs;
               d *= rhs;
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_multiply(T const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a * rhs, b * rhs, c * rhs, d * rhs); // exception guard
               swap(result);
               return *this;
            }

BOOST_CXX14_CONSTEXPR quaternion<T> &        do_divide(T const & rhs, const boost::true_type&)
            {
               a /= rhs;
               b /= rhs;
               c /= rhs;
               d /= rhs;
               return *this;
            }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        do_divide(T const & rhs, const boost::false_type&)
            {
               quaternion<T> result(a / rhs, b / rhs, c / rhs, d / rhs); // exception guard
               swap(result);
               return *this;
            }
         public:

BOOST_CXX14_CONSTEXPR quaternion<T> &        operator += (T const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator += (::std::complex<T> const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
            template<typename X> BOOST_CXX14_CONSTEXPR quaternion<T> & operator += (quaternion<X> const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }

BOOST_CXX14_CONSTEXPR quaternion<T> &        operator -= (T const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator -= (::std::complex<T> const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
            template<typename X> BOOST_CXX14_CONSTEXPR quaternion<T> & operator -= (quaternion<X> const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
            
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator *= (T const & rhs) { return do_multiply(rhs, detail::is_trivial_arithmetic_type<T>()); }
            
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator *= (::std::complex<T> const & rhs)
            {
                T    ar = rhs.real();
                T    br = rhs.imag();
                quaternion<T> result(a*ar - b*br, a*br + b*ar, c*ar + d*br, -c*br+d*ar);
                swap(result);
                return(*this);
            }
            
            template<typename X>
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator *= (quaternion<X> const & rhs)
            {
                T    ar = static_cast<T>(rhs.R_component_1());
                T    br = static_cast<T>(rhs.R_component_2());
                T    cr = static_cast<T>(rhs.R_component_3());
                T    dr = static_cast<T>(rhs.R_component_4());
                
                quaternion<T> result(a*ar - b*br - c*cr - d*dr, a*br + b*ar + c*dr - d*cr, a*cr - b*dr + c*ar + d*br, a*dr + b*cr - c*br + d*ar);
                swap(result);
                return(*this);
            }
            
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator /= (T const & rhs) { return do_divide(rhs, detail::is_trivial_arithmetic_type<T>()); }
            
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator /= (::std::complex<T> const & rhs)
            {
                T    ar = rhs.real();
                T    br = rhs.imag();
                T    denominator = ar*ar+br*br;
                quaternion<T> result((+a*ar + b*br) / denominator, (-a*br + b*ar) / denominator, (+c*ar - d*br) / denominator, (+c*br + d*ar) / denominator);
                swap(result);
                return(*this);
            }
            
            template<typename X>
            BOOST_CXX14_CONSTEXPR quaternion<T> &        operator /= (quaternion<X> const & rhs)
            {
                T    ar = static_cast<T>(rhs.R_component_1());
                T    br = static_cast<T>(rhs.R_component_2());
                T    cr = static_cast<T>(rhs.R_component_3());
                T    dr = static_cast<T>(rhs.R_component_4());
                
                T    denominator = ar*ar+br*br+cr*cr+dr*dr;
                quaternion<T> result((+a*ar+b*br+c*cr+d*dr)/denominator, (-a*br+b*ar-c*dr+d*cr)/denominator, (-a*cr+b*dr+c*ar-d*br)/denominator, (-a*dr-b*cr+c*br+d*ar)/denominator);
                swap(result);
                return(*this);
            }
        private:
           T a, b, c, d;
            
        };

// swap:
template <class T>
BOOST_CXX14_CONSTEXPR void swap(quaternion<T>& a, quaternion<T>& b) { a.swap(b); }
        
// operator+
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator + (const quaternion<T1>& a, const T2& b)
{
   return quaternion<T1>(static_cast<T1>(a.R_component_1() + b), a.R_component_2(), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator + (const T1& a, const quaternion<T2>& b)
{
   return quaternion<T2>(static_cast<T2>(b.R_component_1() + a), b.R_component_2(), b.R_component_3(), b.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator + (const quaternion<T1>& a, const std::complex<T2>& b)
{
   return quaternion<T1>(a.R_component_1() + std::real(b), a.R_component_2() + std::imag(b), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator + (const std::complex<T1>& a, const quaternion<T2>& b)
{
   return quaternion<T1>(b.R_component_1() + real(a), b.R_component_2() + imag(a), b.R_component_3(), b.R_component_4());
}
template <class T>
inline BOOST_CONSTEXPR quaternion<T> operator + (const quaternion<T>& a, const quaternion<T>& b)
{
   return quaternion<T>(a.R_component_1() + b.R_component_1(), a.R_component_2() + b.R_component_2(), a.R_component_3() + b.R_component_3(), a.R_component_4() + b.R_component_4());
}
// operator-
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator - (const quaternion<T1>& a, const T2& b)
{
   return quaternion<T1>(static_cast<T1>(a.R_component_1() - b), a.R_component_2(), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator - (const T1& a, const quaternion<T2>& b)
{
   return quaternion<T2>(static_cast<T2>(a - b.R_component_1()), -b.R_component_2(), -b.R_component_3(), -b.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator - (const quaternion<T1>& a, const std::complex<T2>& b)
{
   return quaternion<T1>(a.R_component_1() - std::real(b), a.R_component_2() - std::imag(b), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator - (const std::complex<T1>& a, const quaternion<T2>& b)
{
   return quaternion<T1>(real(a) - b.R_component_1(), imag(a) - b.R_component_2(), -b.R_component_3(), -b.R_component_4());
}
template <class T>
inline BOOST_CONSTEXPR quaternion<T> operator - (const quaternion<T>& a, const quaternion<T>& b)
{
   return quaternion<T>(a.R_component_1() - b.R_component_1(), a.R_component_2() - b.R_component_2(), a.R_component_3() - b.R_component_3(), a.R_component_4() - b.R_component_4());
}

// operator*
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator * (const quaternion<T1>& a, const T2& b)
{
   return quaternion<T1>(static_cast<T1>(a.R_component_1() * b), a.R_component_2() * b, a.R_component_3() * b, a.R_component_4() * b);
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator * (const T1& a, const quaternion<T2>& b)
{
   return quaternion<T2>(static_cast<T2>(a * b.R_component_1()), a * b.R_component_2(), a * b.R_component_3(), a * b.R_component_4());
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator * (const quaternion<T1>& a, const std::complex<T2>& b)
{
   quaternion<T1> result(a);
   result *= b;
   return result;
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator * (const std::complex<T1>& a, const quaternion<T2>& b)
{
   quaternion<T1> result(a);
   result *= b;
   return result;
}
template <class T>
inline BOOST_CXX14_CONSTEXPR quaternion<T> operator * (const quaternion<T>& a, const quaternion<T>& b)
{
   quaternion<T> result(a);
   result *= b;
   return result;
}

// operator/
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator / (const quaternion<T1>& a, const T2& b)
{
   return quaternion<T1>(a.R_component_1() / b, a.R_component_2() / b, a.R_component_3() / b, a.R_component_4() / b);
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator / (const T1& a, const quaternion<T2>& b)
{
   quaternion<T2> result(a);
   result /= b;
   return result;
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator / (const quaternion<T1>& a, const std::complex<T2>& b)
{
   quaternion<T1> result(a);
   result /= b;
   return result;
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator / (const std::complex<T1>& a, const quaternion<T2>& b)
{
   quaternion<T2> result(a);
   result /= b;
   return result;
}
template <class T>
inline BOOST_CXX14_CONSTEXPR quaternion<T> operator / (const quaternion<T>& a, const quaternion<T>& b)
{
   quaternion<T> result(a);
   result /= b;
   return result;
}
        
        
        template<typename T>
        inline BOOST_CONSTEXPR const quaternion<T>&             operator + (quaternion<T> const & q)
        {
            return q;
        }
        
        
        template<typename T>
        inline BOOST_CONSTEXPR quaternion<T>                    operator - (quaternion<T> const & q)
        {
            return(quaternion<T>(-q.R_component_1(),-q.R_component_2(),-q.R_component_3(),-q.R_component_4()));
        }
        
        
        template<typename R, typename T>
        inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<R, T>::value, bool>::type operator == (R const & lhs, quaternion<T> const & rhs)
        {
            return    (
                        (rhs.R_component_1() == lhs)&&
                        (rhs.R_component_2() == static_cast<T>(0))&&
                        (rhs.R_component_3() == static_cast<T>(0))&&
                        (rhs.R_component_4() == static_cast<T>(0))
                    );
        }
        
        
        template<typename T, typename R>
        inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<R, T>::value, bool>::type operator == (quaternion<T> const & lhs, R const & rhs)
        {
           return rhs == lhs;
        }
        
        
        template<typename T>
        inline BOOST_CONSTEXPR bool                                operator == (::std::complex<T> const & lhs, quaternion<T> const & rhs)
        {
            return    (
                        (rhs.R_component_1() == lhs.real())&&
                        (rhs.R_component_2() == lhs.imag())&&
                        (rhs.R_component_3() == static_cast<T>(0))&&
                        (rhs.R_component_4() == static_cast<T>(0))
                    );
        }
        
        
        template<typename T>
        inline BOOST_CONSTEXPR bool                                operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs)
        {
           return rhs == lhs;
        }
        
        
        template<typename T>
        inline BOOST_CONSTEXPR bool                                operator == (quaternion<T> const & lhs, quaternion<T> const & rhs)
        {
            return    (
                        (rhs.R_component_1() == lhs.R_component_1())&&
                        (rhs.R_component_2() == lhs.R_component_2())&&
                        (rhs.R_component_3() == lhs.R_component_3())&&
                        (rhs.R_component_4() == lhs.R_component_4())
                    );
        }
                
        template<typename R, typename T> inline BOOST_CONSTEXPR bool operator != (R const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
        template<typename T, typename R> inline BOOST_CONSTEXPR bool operator != (quaternion<T> const & lhs, R const & rhs) { return !(lhs == rhs); }
        template<typename T> inline BOOST_CONSTEXPR bool operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
        template<typename T> inline BOOST_CONSTEXPR bool operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs) { return !(lhs == rhs); }
        template<typename T> inline BOOST_CONSTEXPR bool operator != (quaternion<T> const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
        
        
        // Note:    we allow the following formats, with a, b, c, and d reals
        //            a
        //            (a), (a,b), (a,b,c), (a,b,c,d)
        //            (a,(c)), (a,(c,d)), ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b),(c,d))
        template<typename T, typename charT, class traits>
        ::std::basic_istream<charT,traits> &    operator >> (    ::std::basic_istream<charT,traits> & is,
                                                                quaternion<T> & q)
        {
            
#ifdef    BOOST_NO_STD_LOCALE
#else
            const ::std::ctype<charT> & ct = ::std::use_facet< ::std::ctype<charT> >(is.getloc());
#endif /* BOOST_NO_STD_LOCALE */
            
            T    a = T();
            T    b = T();
            T    c = T();
            T    d = T();
            
            ::std::complex<T>    u = ::std::complex<T>();
            ::std::complex<T>    v = ::std::complex<T>();
            
            charT    ch = charT();
            char    cc;
            
            is >> ch;                                        // get the first lexeme
            
            if    (!is.good())    goto finish;
            
#ifdef    BOOST_NO_STD_LOCALE
            cc = ch;
#else
            cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
            
            if    (cc == '(')                            // read "(", possible: (a), (a,b), (a,b,c), (a,b,c,d), (a,(c)), (a,(c,d)), ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b,),(c,d,))
            {
                is >> ch;                                    // get the second lexeme
                
                if    (!is.good())    goto finish;
                
#ifdef    BOOST_NO_STD_LOCALE
                cc = ch;
#else
                cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                
                if    (cc == '(')                        // read "((", possible: ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b,),(c,d,))
                {
                    is.putback(ch);
                    
                    is >> u;                                // we extract the first and second components
                    a = u.real();
                    b = u.imag();
                    
                    if    (!is.good())    goto finish;
                    
                    is >> ch;                                // get the next lexeme
                    
                    if    (!is.good())    goto finish;
                    
#ifdef    BOOST_NO_STD_LOCALE
                    cc = ch;
#else
                    cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                    
                    if        (cc == ')')                    // format: ((a)) or ((a,b))
                    {
                        q = quaternion<T>(a,b);
                    }
                    else if    (cc == ',')                // read "((a)," or "((a,b),", possible: ((a),c), ((a),(c)), ((a),(c,d)), ((a,b),c), ((a,b),(c)), ((a,b,),(c,d,))
                    {
                        is >> v;                            // we extract the third and fourth components
                        c = v.real();
                        d = v.imag();
                        
                        if    (!is.good())    goto finish;
                        
                        is >> ch;                                // get the last lexeme
                        
                        if    (!is.good())    goto finish;
                        
#ifdef    BOOST_NO_STD_LOCALE
                        cc = ch;
#else
                        cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                        
                        if    (cc == ')')                    // format: ((a),c), ((a),(c)), ((a),(c,d)), ((a,b),c), ((a,b),(c)) or ((a,b,),(c,d,))
                        {
                            q = quaternion<T>(a,b,c,d);
                        }
                        else                            // error
                        {
                            is.setstate(::std::ios_base::failbit);
                        }
                    }
                    else                                // error
                    {
                        is.setstate(::std::ios_base::failbit);
                    }
                }
                else                                // read "(a", possible: (a), (a,b), (a,b,c), (a,b,c,d), (a,(c)), (a,(c,d))
                {
                    is.putback(ch);
                    
                    is >> a;                                // we extract the first component
                    
                    if    (!is.good())    goto finish;
                    
                    is >> ch;                                // get the third lexeme
                    
                    if    (!is.good())    goto finish;
                    
#ifdef    BOOST_NO_STD_LOCALE
                    cc = ch;
#else
                    cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                    
                    if        (cc == ')')                    // format: (a)
                    {
                        q = quaternion<T>(a);
                    }
                    else if    (cc == ',')                // read "(a,", possible: (a,b), (a,b,c), (a,b,c,d), (a,(c)), (a,(c,d))
                    {
                        is >> ch;                            // get the fourth lexeme
                        
                        if    (!is.good())    goto finish;
                        
#ifdef    BOOST_NO_STD_LOCALE
                        cc = ch;
#else
                        cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                        
                        if    (cc == '(')                // read "(a,(", possible: (a,(c)), (a,(c,d))
                        {
                            is.putback(ch);
                            
                            is >> v;                        // we extract the third and fourth component
                            
                            c = v.real();
                            d = v.imag();
                            
                            if    (!is.good())    goto finish;
                            
                            is >> ch;                        // get the ninth lexeme
                            
                            if    (!is.good())    goto finish;
                            
#ifdef    BOOST_NO_STD_LOCALE
                            cc = ch;
#else
                            cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                            
                            if    (cc == ')')                // format: (a,(c)) or (a,(c,d))
                            {
                                q = quaternion<T>(a,b,c,d);
                            }
                            else                        // error
                            {
                                is.setstate(::std::ios_base::failbit);
                            }
                        }
                        else                        // read "(a,b", possible: (a,b), (a,b,c), (a,b,c,d)
                        {
                            is.putback(ch);
                            
                            is >> b;                        // we extract the second component
                            
                            if    (!is.good())    goto finish;
                            
                            is >> ch;                        // get the fifth lexeme
                            
                            if    (!is.good())    goto finish;
                            
#ifdef    BOOST_NO_STD_LOCALE
                            cc = ch;
#else
                            cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                            
                            if    (cc == ')')                // format: (a,b)
                            {
                                q = quaternion<T>(a,b);
                            }
                            else if    (cc == ',')        // read "(a,b,", possible: (a,b,c), (a,b,c,d)
                            {
                                is >> c;                    // we extract the third component
                                
                                if    (!is.good())    goto finish;
                                
                                is >> ch;                    // get the seventh lexeme
                                
                                if    (!is.good())    goto finish;
                                
#ifdef    BOOST_NO_STD_LOCALE
                                cc = ch;
#else
                                cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                                
                                if        (cc == ')')        // format: (a,b,c)
                                {
                                    q = quaternion<T>(a,b,c);
                                }
                                else if    (cc == ',')    // read "(a,b,c,", possible: (a,b,c,d)
                                {
                                    is >> d;                // we extract the fourth component
                                    
                                    if    (!is.good())    goto finish;
                                    
                                    is >> ch;                // get the ninth lexeme
                                    
                                    if    (!is.good())    goto finish;
                                    
#ifdef    BOOST_NO_STD_LOCALE
                                    cc = ch;
#else
                                    cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
                                    
                                    if    (cc == ')')        // format: (a,b,c,d)
                                    {
                                        q = quaternion<T>(a,b,c,d);
                                    }
                                    else                // error
                                    {
                                        is.setstate(::std::ios_base::failbit);
                                    }
                                }
                                else                    // error
                                {
                                    is.setstate(::std::ios_base::failbit);
                                }
                            }
                            else                        // error
                            {
                                is.setstate(::std::ios_base::failbit);
                            }
                        }
                    }
                    else                                // error
                    {
                        is.setstate(::std::ios_base::failbit);
                    }
                }
            }
            else                                        // format:    a
            {
                is.putback(ch);
                
                is >> a;                                    // we extract the first component
                
                if    (!is.good())    goto finish;
                
                q = quaternion<T>(a);
            }
            
            finish:
            return(is);
        }
        
        
        template<typename T, typename charT, class traits>
        ::std::basic_ostream<charT,traits> &    operator << (    ::std::basic_ostream<charT,traits> & os,
                                                                quaternion<T> const & q)
        {
            ::std::basic_ostringstream<charT,traits>    s;

s.flags(os.flags());
#ifdef    BOOST_NO_STD_LOCALE
#else
            s.imbue(os.getloc());
#endif /* BOOST_NO_STD_LOCALE */
            s.precision(os.precision());
            
            s << '('    << q.R_component_1() << ','
                        << q.R_component_2() << ','
                        << q.R_component_3() << ','
                        << q.R_component_4() << ')';
            
            return os << s.str();
        }
        
        
        // values
        
        template<typename T>
        inline BOOST_CONSTEXPR T real(quaternion<T> const & q)
        {
            return(q.real());
        }
        
        
        template<typename T>
        inline BOOST_CONSTEXPR quaternion<T> unreal(quaternion<T> const & q)
        {
            return(q.unreal());
        }
                
        template<typename T>
        inline T sup(quaternion<T> const & q)
        {
            using    ::std::abs;
            return (std::max)((std::max)(abs(q.R_component_1()), abs(q.R_component_2())), (std::max)(abs(q.R_component_3()), abs(q.R_component_4())));
        }
        
        
        template<typename T>
        inline T l1(quaternion<T> const & q)
        {
           using    ::std::abs;
           return abs(q.R_component_1()) + abs(q.R_component_2()) + abs(q.R_component_3()) + abs(q.R_component_4());
        }
        
        
        template<typename T>
        inline T abs(quaternion<T> const & q)
        {
            using    ::std::abs;
            using    ::std::sqrt;
            
            T            maxim = sup(q);    // overflow protection
            
            if    (maxim == static_cast<T>(0))
            {
                return(maxim);
            }
            else
            {
                T    mixam = static_cast<T>(1)/maxim;    // prefer multiplications over divisions
                
                T a = q.R_component_1() * mixam;
                T b = q.R_component_2() * mixam;
                T c = q.R_component_3() * mixam;
                T d = q.R_component_4() * mixam;

a *= a;
                b *= b;
                c *= c;
                d *= d;
                
                return(maxim * sqrt(a + b + c + d));
            }
            
            //return(sqrt(norm(q)));
        }
        
        
        // Note:    This is the Cayley norm, not the Euclidean norm...
        
        template<typename T>
        inline BOOST_CXX14_CONSTEXPR T norm(quaternion<T>const  & q)
        {
            return(real(q*conj(q)));
        }
        
        
        template<typename T>
        inline BOOST_CONSTEXPR quaternion<T> conj(quaternion<T> const & q)
        {
            return(quaternion<T>(   +q.R_component_1(),
                                    -q.R_component_2(),
                                    -q.R_component_3(),
                                    -q.R_component_4()));
        }
        
        
        template<typename T>
        inline quaternion<T>                    spherical(  T const & rho,
                                                            T const & theta,
                                                            T const & phi1,
                                                            T const & phi2)
        {
            using ::std::cos;
            using ::std::sin;
            
            //T    a = cos(theta)*cos(phi1)*cos(phi2);
            //T    b = sin(theta)*cos(phi1)*cos(phi2);
            //T    c = sin(phi1)*cos(phi2);
            //T    d = sin(phi2);
            
            T    courrant = static_cast<T>(1);
            
            T    d = sin(phi2);
            
            courrant *= cos(phi2);
            
            T    c = sin(phi1)*courrant;
            
            courrant *= cos(phi1);
            
            T    b = sin(theta)*courrant;
            T    a = cos(theta)*courrant;
            
            return(rho*quaternion<T>(a,b,c,d));
        }
        
        
        template<typename T>
        inline quaternion<T>                    semipolar(  T const & rho,
                                                            T const & alpha,
                                                            T const & theta1,
                                                            T const & theta2)
        {
            using ::std::cos;
            using ::std::sin;
            
            T    a = cos(alpha)*cos(theta1);
            T    b = cos(alpha)*sin(theta1);
            T    c = sin(alpha)*cos(theta2);
            T    d = sin(alpha)*sin(theta2);
            
            return(rho*quaternion<T>(a,b,c,d));
        }
        
        
        template<typename T>
        inline quaternion<T>                    multipolar( T const & rho1,
                                                            T const & theta1,
                                                            T const & rho2,
                                                            T const & theta2)
        {
            using ::std::cos;
            using ::std::sin;
            
            T    a = rho1*cos(theta1);
            T    b = rho1*sin(theta1);
            T    c = rho2*cos(theta2);
            T    d = rho2*sin(theta2);
            
            return(quaternion<T>(a,b,c,d));
        }
        
        
        template<typename T>
        inline quaternion<T>                    cylindrospherical(  T const & t,
                                                                    T const & radius,
                                                                    T const & longitude,
                                                                    T const & latitude)
        {
            using ::std::cos;
            using ::std::sin;
            
            
            
            T    b = radius*cos(longitude)*cos(latitude);
            T    c = radius*sin(longitude)*cos(latitude);
            T    d = radius*sin(latitude);
            
            return(quaternion<T>(t,b,c,d));
        }
        
        
        template<typename T>
        inline quaternion<T>                    cylindrical(T const & r,
                                                            T const & angle,
                                                            T const & h1,
                                                            T const & h2)
        {
            using ::std::cos;
            using ::std::sin;
            
            T    a = r*cos(angle);
            T    b = r*sin(angle);
            
            return(quaternion<T>(a,b,h1,h2));
        }
        
        
        // transcendentals
        // (please see the documentation)
        
        
        template<typename T>
        inline quaternion<T>                    exp(quaternion<T> const & q)
        {
            using    ::std::exp;
            using    ::std::cos;
            
            using    ::boost::math::sinc_pi;
            
            T    u = exp(real(q));
            
            T    z = abs(unreal(q));
            
            T    w = sinc_pi(z);
            
            return(u*quaternion<T>(cos(z),
                w*q.R_component_2(), w*q.R_component_3(),
                w*q.R_component_4()));
        }
        
        
        template<typename T>
        inline quaternion<T>                    cos(quaternion<T> const & q)
        {
            using    ::std::sin;
            using    ::std::cos;
            using    ::std::cosh;
            
            using    ::boost::math::sinhc_pi;
            
            T    z = abs(unreal(q));
            
            T    w = -sin(q.real())*sinhc_pi(z);
            
            return(quaternion<T>(cos(q.real())*cosh(z),
                w*q.R_component_2(), w*q.R_component_3(),
                w*q.R_component_4()));
        }
        
        
        template<typename T>
        inline quaternion<T>                    sin(quaternion<T> const & q)
        {
            using    ::std::sin;
            using    ::std::cos;
            using    ::std::cosh;
            
            using    ::boost::math::sinhc_pi;
            
            T    z = abs(unreal(q));
            
            T    w = +cos(q.real())*sinhc_pi(z);
            
            return(quaternion<T>(sin(q.real())*cosh(z),
                w*q.R_component_2(), w*q.R_component_3(),
                w*q.R_component_4()));
        }
        
        
        template<typename T>
        inline quaternion<T>                    tan(quaternion<T> const & q)
        {
            return(sin(q)/cos(q));
        }
        
        
        template<typename T>
        inline quaternion<T>                    cosh(quaternion<T> const & q)
        {
            return((exp(+q)+exp(-q))/static_cast<T>(2));
        }
        
        
        template<typename T>
        inline quaternion<T>                    sinh(quaternion<T> const & q)
        {
            return((exp(+q)-exp(-q))/static_cast<T>(2));
        }
        
        
        template<typename T>
        inline quaternion<T>                    tanh(quaternion<T> const & q)
        {
            return(sinh(q)/cosh(q));
        }
        
        
        template<typename T>
        quaternion<T>                            pow(quaternion<T> const & q,
                                                    int n)
        {
            if        (n > 1)
            {
                int    m = n>>1;
                
                quaternion<T>    result = pow(q, m);
                
                result *= result;
                
                if    (n != (m<<1))
                {
                    result *= q; // n odd
                }
                
                return(result);
            }
            else if    (n == 1)
            {
                return(q);
            }
            else if    (n == 0)
            {
                return(quaternion<T>(static_cast<T>(1)));
            }
            else    /* n < 0 */
            {
                return(pow(quaternion<T>(static_cast<T>(1))/q,-n));
            }
        }
    }
}

#endif /* BOOST_QUATERNION_HPP */

cmd: