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+// 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/math_fwd.hpp>
+#include <boost/math/tools/config.hpp>
+#include <locale> // for the "<<" operator
+
+#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>
+
+#include <type_traits>
+
+namespace boost
+{
+ namespace math
+ {
+
+ namespace detail {
+
+ template <class T>
+ struct is_trivial_arithmetic_type_imp
+ {
+ typedef std::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 {};
+ }
+
+#ifndef BOOST_MATH_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)
+
+ 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
+
+ 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
+ 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()) {}
+
+ 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())) {}
+
+ // templated copy constructor
+
+ template<typename X>
+ 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).
+
+ constexpr T real() const
+ {
+ return(a);
+ }
+
+ constexpr quaternion<T> unreal() const
+ {
+ return(quaternion<T>(static_cast<T>(0), b, c, d));
+ }
+
+ constexpr T R_component_1() const
+ {
+ return(a);
+ }
+
+ constexpr T R_component_2() const
+ {
+ return(b);
+ }
+
+ constexpr T R_component_3() const
+ {
+ return(c);
+ }
+
+ constexpr T R_component_4() const
+ {
+ return(d);
+ }
+
+ constexpr ::std::complex<T> C_component_1() const
+ {
+ return(::std::complex<T>(a, b));
+ }
+
+ constexpr ::std::complex<T> C_component_2() const
+ {
+ return(::std::complex<T>(c, d));
+ }
+
+ BOOST_MATH_CXX14_CONSTEXPR void swap(quaternion& o)
+ {
+#ifndef BOOST_MATH_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_MATH_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_MATH_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);
+ }
+
+ BOOST_MATH_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);
+ }
+
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator = (T const & a_affecter)
+ {
+ a = a_affecter;
+
+ b = c = d = static_cast<T>(0);
+
+ return(*this);
+ }
+
+ BOOST_MATH_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_MATH_CXX14_CONSTEXPR quaternion<T> & do_add(T const & rhs, const std::true_type&)
+ {
+ a += rhs;
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_add(T const & rhs, const std::false_type&)
+ {
+ quaternion<T> result(a + rhs, b, c, d); // exception guard
+ swap(result);
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_add(std::complex<T> const & rhs, const std::true_type&)
+ {
+ a += std::real(rhs);
+ b += std::imag(rhs);
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_add(std::complex<T> const & rhs, const std::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_MATH_CXX14_CONSTEXPR quaternion<T> & do_add(quaternion<X> const & rhs, const std::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_MATH_CXX14_CONSTEXPR quaternion<T> & do_add(quaternion<X> const & rhs, const std::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_MATH_CXX14_CONSTEXPR quaternion<T> & do_subtract(T const & rhs, const std::true_type&)
+ {
+ a -= rhs;
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_subtract(T const & rhs, const std::false_type&)
+ {
+ quaternion<T> result(a - rhs, b, c, d); // exception guard
+ swap(result);
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_subtract(std::complex<T> const & rhs, const std::true_type&)
+ {
+ a -= std::real(rhs);
+ b -= std::imag(rhs);
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_subtract(std::complex<T> const & rhs, const std::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_MATH_CXX14_CONSTEXPR quaternion<T> & do_subtract(quaternion<X> const & rhs, const std::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_MATH_CXX14_CONSTEXPR quaternion<T> & do_subtract(quaternion<X> const & rhs, const std::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_MATH_CXX14_CONSTEXPR quaternion<T> & do_multiply(T const & rhs, const std::true_type&)
+ {
+ a *= rhs;
+ b *= rhs;
+ c *= rhs;
+ d *= rhs;
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_multiply(T const & rhs, const std::false_type&)
+ {
+ quaternion<T> result(a * rhs, b * rhs, c * rhs, d * rhs); // exception guard
+ swap(result);
+ return *this;
+ }
+
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_divide(T const & rhs, const std::true_type&)
+ {
+ a /= rhs;
+ b /= rhs;
+ c /= rhs;
+ d /= rhs;
+ return *this;
+ }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & do_divide(T const & rhs, const std::false_type&)
+ {
+ quaternion<T> result(a / rhs, b / rhs, c / rhs, d / rhs); // exception guard
+ swap(result);
+ return *this;
+ }
+ public:
+
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator += (T const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator += (::std::complex<T> const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
+ template<typename X> BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator += (quaternion<X> const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
+
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator -= (T const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator -= (::std::complex<T> const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
+ template<typename X> BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator -= (quaternion<X> const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
+
+ BOOST_MATH_CXX14_CONSTEXPR quaternion<T> & operator *= (T const & rhs) { return do_multiply(rhs, detail::is_trivial_arithmetic_type<T>()); }
+
+ BOOST_MATH_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_MATH_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_MATH_CXX14_CONSTEXPR quaternion<T> & operator /= (T const & rhs) { return do_divide(rhs, detail::is_trivial_arithmetic_type<T>()); }
+
+ BOOST_MATH_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_MATH_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_MATH_CXX14_CONSTEXPR void swap(quaternion<T>& a, quaternion<T>& b) { a.swap(b); }
+
+// operator+
+template <class T1, class T2>
+inline constexpr typename std::enable_if<std::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 constexpr typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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() + std::real(a), b.R_component_2() + std::imag(a), b.R_component_3(), b.R_component_4());
+}
+template <class T>
+inline 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 constexpr typename std::enable_if<std::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 constexpr typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::is_convertible<T1, T2>::value, quaternion<T2> >::type
+operator - (const std::complex<T1>& a, const quaternion<T2>& b)
+{
+ return quaternion<T1>(std::real(a) - b.R_component_1(), std::imag(a) - b.R_component_2(), -b.R_component_3(), -b.R_component_4());
+}
+template <class T>
+inline 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 constexpr typename std::enable_if<std::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 constexpr typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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_MATH_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 constexpr typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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_MATH_CXX14_CONSTEXPR typename std::enable_if<std::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_MATH_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 constexpr const quaternion<T>& operator + (quaternion<T> const & q)
+ {
+ return q;
+ }
+
+
+ template<typename T>
+ inline 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 constexpr typename std::enable_if<std::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 constexpr typename std::enable_if<std::is_convertible<R, T>::value, bool>::type operator == (quaternion<T> const & lhs, R const & rhs)
+ {
+ return rhs == lhs;
+ }
+
+
+ template<typename T>
+ inline 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 constexpr bool operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs)
+ {
+ return rhs == lhs;
+ }
+
+
+ template<typename T>
+ inline 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 constexpr bool operator != (R const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
+ template<typename T, typename R> inline constexpr bool operator != (quaternion<T> const & lhs, R const & rhs) { return !(lhs == rhs); }
+ template<typename T> inline constexpr bool operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
+ template<typename T> inline constexpr bool operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs) { return !(lhs == rhs); }
+ template<typename T> inline 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)
+ {
+ const ::std::ctype<charT> & ct = ::std::use_facet< ::std::ctype<charT> >(is.getloc());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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;
+
+ cc = ct.narrow(ch, char());
+
+ 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());
+ s.imbue(os.getloc());
+ 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 constexpr T real(quaternion<T> const & q)
+ {
+ return(q.real());
+ }
+
+
+ template<typename T>
+ inline 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_MATH_CXX14_CONSTEXPR T norm(quaternion<T>const & q)
+ {
+ return(real(q*conj(q)));
+ }
+
+
+ template<typename T>
+ inline 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 */
generated by cgit v1.1 at 2025-10-28 23:24:11 +0000